# Fundamentals of Lie categories

Žan Grad

###### Abstract.

We introduce the basic notions and present examples and results on Lie
categories – categories internal to the category of smooth manifolds.
Demonstrating how the units of a Lie category $\mathcal{C}$ dictate the
behavior of its invertible morphisms $\mathcal{G}(\mathcal{C})$, we develop
sufficient conditions for $\mathcal{G}(\mathcal{C})$ to form a Lie groupoid.
We show that the construction of Lie algebroids from the theory of Lie
groupoids carries through, and ask when the Lie algebroid of
$\mathcal{G}(\mathcal{C})$ is recovered. We reveal that the lack of
invertibility assumption on morphisms leads to a natural generalization of
rank from linear algebra, develop its general properties, and show how the
existence of an extension $\mathcal{C}\hookrightarrow\mathcal{G}$ of a Lie
category to a Lie groupoid affects the ranks of morphisms and the algebroids
of $\mathcal{C}$. Furthermore, certain completeness results for invariant
vector fields on Lie monoids and Lie categories with well-behaved boundaries
are obtained. Interpreting the developed framework in the context of physical
processes, we yield a rigorous approach to the theory of statistical
thermodynamics by observing that entropy change, associated to a physical
process, is a functor.

###### Key words and phrases:

Lie category, Lie groupoid, reversible process, rank, entropy

###### 2020 Mathematics Subject Classification:

22A99, (22A22, 53Z05, 58H05)

This work was supported in part by PhD Grant UI/BD/152069/2021 of FCT,
Portugal.

###### Contents

  1. 1 Introduction
  2. 2 Basic definitions and examples
  3. 3 Reversibility of morphisms
  4. 4 Lie algebroids of a Lie category
  5. 5 Ranks of morphisms
  6. 6 Extensions of Lie categories to Lie groupoids
  7. 7 Completeness of invariant vector fields
  8. 8 An application to physics: Statistical Thermodynamics
  9. A Transversality on manifolds with corners

00footnotetext:

##  1\. Introduction

Since its conception, category theory has proven to provide a unified
framework for the language of mathematics, by making use of the observation
that objects and morphisms thereof arise regardless of the mathematical field
one considers. This paradigm has also been adopted by physicists (see e.g.
[catsquantum]), namely that one can interpret physical states (contained in a
certain phase space) as objects, and physical processes as morphisms between
them, thus forming a category which corresponds to the physical system at
hand. In order to provide such a realization of a given physical system, the
corresponding category should ideally have the capacity to straightforwardly
describe the phenomena which pertain to the given physical system, and also
capture the means for describing and calculating relevant physical quantities.
For a physicist, the latter is generally done using the basic tools of
calculus, in terms of a preferred set of coordinates.

There is a natural way of obtaining such a unified framework for describing
physical processes, by intertwining category theory with the theory of smooth
manifolds. That is, we require the set of objects (states) and the set of
morphisms (processes) of an abstract category to be a pair of smooth
manifolds, and that the composition law $(f,g)\mapsto fg$ is a smooth map
(concatenation of two processes); the mathematical structure obtained is that
of a category internal to the category $\mathbf{Diff}$ of smooth manifolds.
Historically, this kind of a mathematical structure, which we will define in
more precise terms as a Lie category, was first introduced and briefly studied
by Charles Ehresmann in his seminal paper [ehr1959]; in the same paper,
Ehresmann further focused on the notion of a Lie groupoid, which additionally
imposes that all morphisms are invertible. This latter notion has nowadays
been thoroughly researched and continues to have a status of an active field
of research, whereas the same cannot be said for Lie categories; apart from
the work of Ehresmann, this is the first paper dedicated to their systematic
treatment. The main reason behind the fact that Lie categories have gained
negligible attention compared to Lie groupoids lies in the fact that the
assumption on morphisms being invertible implies that all left and right
translations are diffeomorphisms, which accounts for certain preferable
properties of the structure of a Lie groupoid, that will also be highlighted
in this paper.

Our motivation for studying Lie categories, and not merely groupoids, stems
from the physical interpretation that invertible morphisms correspond to
reversible processes, and in physics not all processes in a given system are
reversible; this is well known from the theory of thermodynamics, where
reversible processes are precisely those where the change of entropy when
transiting between two states is zero; we will describe this in precise terms
in Section 8. Another, perhaps more notorious motivation, is given by
irreversibility of wave-function collapse in quantum theory, which was
actually our initial motivation for dropping the invertibility assumption.

As we will see, it will turn out to be desirable to allow the space of
morphisms of a Lie category to possess a boundary – we will encounter both,
mathematical and physical examples where the boundary of the space of
morphisms will play a distinguished role. To temporarily appease and motivate
the reader to this regard, let us briefly note that Lie monoids are examples
of Lie categories with the set of objects a singleton, and that the monoid
$[0,\infty)$ – either for multiplication or addition – provides a first
example of a Lie category with boundary. This simple example already shows
certain intriguing qualities: for instance, considering $[0,\infty)$ for
multiplication, all its invertible elements are contained in its interior, and
considering $[0,\infty)$ for addition, its only invertible element is in its
boundary. This phenomenon, as may be expected, is one of the features of Lie
categories, namely that the units dictate the behavior of invertibles.

Overall, we aim to convince the reader that the interplay of geometrical and
categorical structures alone (without the invertibility assumption) provides
new exciting questions which were so far overlooked within the scope of Lie
theory. Let us summarize our main objectives.

  1. (i)

To demonstrate that Lie categories allow for an abundance of interesting
examples which have so far been missed in the theory of Lie groupoids.

  2. (ii)

To expose some ideas and constructions which carry through to Lie categories
from the theory of Lie groupoids, e.g. the Lie algebroid construction.

  3. (iii)

To inspect the relation between Lie groupoids and Lie categories, and show
that novel notions can be obtained when invertibility is dropped.

  4. (iv)

Last, but not least, to provide an algorithm for constructing Lie categories
that describe physical systems by making use of the notion of entropy, and to
reveal that the mathematical structure implicitly present in the foundations
of statistical thermodynamics is that of a Lie category.

## Notation

All our categories are small, i.e. the classes of objects and morphisms are
sets. We will denote a category by $\mathcal{C}\rightrightarrows\mathcal{X}$,
where $\mathcal{C}$ is the set of morphisms, $\mathcal{X}$ is the set of
objects, and the two arrows indicate the source map
$s\colon\mathcal{C}\rightarrow\mathcal{X}$ and target map
$t\colon\mathcal{C}\rightarrow\mathcal{X}$, which are defined by

$s(x\xrightarrow{g}y)=x,\quad t(x\xrightarrow{g}y)=y,$

for any morphism $g\colon x\rightarrow y$ in $\mathcal{C}$. Alongside these
two maps, a category $\mathcal{C}$ comes equipped with the composition map

$m\colon{\mathcal{C}}^{(2)}\rightarrow\mathcal{C},\quad(g,h)\mapsto gh,$

where
${\mathcal{C}}^{(2)}=\\{(g,h)\in\mathcal{C}\times\mathcal{C}\;|\;s(g)=t(h)\\}$
is the set of all pairs of composable morphisms. Moreover, $\mathcal{C}$ comes
equipped with the unit map

$u\colon\mathcal{X}\rightarrow\mathcal{C},\quad x\mapsto 1_{x}.$

We also define

$\mathcal{C}_{x}=s^{-1}(x),\quad\mathcal{C}^{y}=t^{-1}(y),\quad\mathcal{C}_{x}^{y}=\mathcal{C}_{x}\cap\mathcal{C}^{y},$

and call $\mathcal{C}_{x}$ the source fibre at $x$, and $\mathcal{C}^{y}$ the
target fibre at $y$. Note that any morphism $g\in\mathcal{C}$ determines the
maps

$\displaystyle
L_{g}\colon\mathcal{C}^{s(g)}\rightarrow\mathcal{C}^{t(g)},\quad L_{g}(h)=gh,$
(1.1) $\displaystyle
R_{g}\colon\mathcal{C}_{t(g)}\rightarrow\mathcal{C}_{s(g)},\quad R_{g}(h)=hg,$

called the left translation and right translation by $g$, which are just the
pre-composition and post-composition by $g$, respectively.

##  2\. Basic definitions and examples

######  Definition 2.1.

A Lie category is a small category $\mathcal{C}\rightrightarrows\mathcal{X}$,
where $\mathcal{C}$ is a smooth manifold with or without boundary,
$\mathcal{X}$ is a smooth manifold without boundary, and there holds:

  1. (i)

The source and target maps $s,t\colon\mathcal{C}\rightarrow\mathcal{X}$ are
smooth submersions.

  2. (ii)

The unit map $u\colon\mathcal{X}\rightarrow\mathcal{C}$ and the composition
map $m\colon{\mathcal{C}}^{(2)}\rightarrow\mathcal{C}$ are smooth.

If $\mathcal{C}$ has a boundary, we also assume that
$\mathcal{C}\rightrightarrows\mathcal{X}$ has a regular boundary, that is:

  1. (iii)

The restrictions $\partial s,\partial
t\colon\partial\mathcal{C}\rightarrow\mathcal{X}$ of $s$ and $t$ are smooth
submersions.

######  Remark 2.2.

Given any $x\in\mathcal{X}$, assumptions (i) and (iii) ensure that
$\mathcal{C}_{x}$ and $\mathcal{C}^{x}$ are neat submanifolds of $\mathcal{C}$
(see [difftop, p. 60] or [corners, Proposition 4.2.9]), that is:

$\displaystyle\partial(\mathcal{C}_{x})=\mathcal{C}_{x}\cap\partial\mathcal{C}\quad\text{and}\quad\partial(\mathcal{C}^{x})=\mathcal{C}^{x}\cap\partial\mathcal{C}.$
(2.3)

Moreover, assumptions (i) and (iii) ensure that the set

${\mathcal{C}}^{(2)}=(s\times t)^{-1}(\Delta_{\mathcal{X}})$

of composable morphisms is a neat submanifold of
$\mathcal{C}\times\mathcal{C}$, that is:

$\displaystyle\partial({\mathcal{C}}^{(2)})={\mathcal{C}}^{(2)}\cap\partial(\mathcal{C}\times\mathcal{C})={\mathcal{C}}^{(2)}\cap(\mathcal{C}\times\partial\mathcal{C}\cup\partial\mathcal{C}\times\mathcal{C}),$
(2.4)

by transversality theorem, see Corollary A.7. This corollary also ensures that
if $\mathcal{C}$ has a boundary, the corner points of ${\mathcal{C}}^{(2)}$
are precisely the composable pairs in
$\partial\mathcal{C}\times\partial\mathcal{C}$; moreover, the tangent space of
${\mathcal{C}}^{(2)}$ at a composable pair $(g,h)$ equals

$\displaystyle T_{(g,h)}\mathcal{C}^{(2)}=\\{(v,w)\in T_{g}\mathcal{C}\oplus
T_{h}\mathcal{C}\;|\;\operatorname{d}\\!{s}(v)=\operatorname{d}\\!{t}(w)\\}.$
(2.5)

Smoothness of ${\mathcal{C}}^{(2)}$ implies that the requirement of smoothness
of the composition map $m\colon{\mathcal{C}}^{(2)}\rightarrow\mathcal{C}$
makes sense, and furthermore that left and right translations $L_{g},R_{g}$
are smooth maps between appropriate fibres, as defined by equations (1.1); we
thus obtain a covariant functor $\mathcal{C}\rightarrow\mathbf{Diff}$, given
on objects as $x\mapsto\mathcal{C}^{x}$ and on morphisms as $g\mapsto L_{g}$,
and a contravariant one given by $x\mapsto\mathcal{C}_{x}$, $g\mapsto R_{g}$.

######  Remark 2.6.

The unit map $u\colon\mathcal{X}\rightarrow\mathcal{C}$ of a Lie category
$\mathcal{C}\rightrightarrows\mathcal{X}$ is an embedding, which is a
consequence of the fact that it is a smooth section of the source (and target)
map, hence an injective immersion which is a homeomorphism onto its image,
whose continuous inverse is given by $s|_{u(\mathcal{X})}$.

######  Definition 2.7.

A morphism of Lie categories is a smooth functor
$F\colon\mathcal{C}\rightarrow\mathcal{D}$. A Lie category $\mathcal{C}$ is
said to be a Lie subcategory of $\mathcal{D}$, if there is an injective
immersive morphism $F\colon\mathcal{C}\rightarrow\mathcal{D}$ of Lie
categories.

Remark 2.6 implies that any morphism of Lie categories induces a smooth map
between the respective object manifolds of $\mathcal{C}$ and $\mathcal{D}$. We
now turn to examples of Lie categories.

######  Example 2.8.

In the case when the object space $\mathcal{X}$ is a singleton, a Lie category
$\mathcal{C}\rightrightarrows\\{*\\}$ will be called a Lie monoid. Simply put,
a Lie monoid is a monoid $M$ together with a structure of a smooth manifold
with or without boundary, such that the multiplication $m\colon M\times
M\rightarrow M$ is smooth.

Concrete examples of Lie monoids frequently arise as embedded submonoids of
Lie groups. For instance, we may consider the closed ray
$[0,\infty)\subset\mathbb{R}$ for addition, or more generally, the
$n$-dimensional half-space
$\mathbb{H}^{n}=\mathbb{R}^{n-1}\times[0,\infty)\subset\mathbb{R}^{n}$, where
$n\in\mathbb{N}$, which is a commutative Lie monoid for the usual addition. On
the other hand, the closed ray $[0,\infty)$ for multiplication is not111This
is easily seen since it has a non-cancellative (absorbing) element. a
submonoid of any Lie group. Further examples of Lie monoids that do not arise
as submonoids of Lie groups are:

  1. (i)

The set $\mathbb{R}^{n\times n}$ of square $n$-dimensional matrices is a Lie
monoid for matrix multiplication; more generally, we may consider the Lie
monoid $\operatorname{End}(V)$ of endomorphisms of a finite dimensional vector
space $V$, under composition. Even more generally, any finite-dimensional
unital algebra is a Lie monoid that is enriched over the category
$\mathbf{Vect}$ of vector spaces, since the multiplication map is bilinear. In
particular, this includes the real line, the complex plane, and quaternions
for multiplication.

  2. (ii)

The closed unit disk $\overline{\mathbb{D}}\subset\mathbb{C}$, an abelian Lie
monoid for complex multiplication, and the closed unit 4-ball
$\bar{\mathbb{B}}^{4}$, a non-abelian Lie monoid for quaternionic
multiplication.

The following example generalizes the similar notion of triviality from the
theory of Lie groupoids.

######  Example 2.9.

Let $X$ be a smooth manifold without boundary and $M$ a Lie monoid. A trivial
Lie category is defined by $\mathcal{C}=X\times M\times X$ and
$\mathcal{X}=X$, with $s=\mathrm{pr}_{3},t=\mathrm{pr}_{1}$ and composition as

$(z,g,y)(y,h,x)=(z,gh,x).$

That composition is smooth follows from the smoothness of multiplication in
$M$. In the case when $M=\\{e\\}$ is a trivial monoid, we obtain the well-
known pair groupoid.

The next example reveals the spirit of the notion of a Lie category – that is,
it can be thought of as a smooth family of endomorphisms of an abstract
structure, parametrized by the base manifold $\mathcal{X}$. This is aligned
with the philosophy that a Lie groupoid can be thought of as a smooth
collection of automorphisms (symmetries) of a structure parametrized by
$\mathcal{X}$.

######  Example 2.10.

Let $\pi\colon E\rightarrow X$ be a vector bundle over a smooth manifold $X$
without boundary, whose typical fibre is a fixed vector space $V$. The
endomorphism category of $E\rightarrow X$ is the category
$\operatorname{End}(E)\rightrightarrows X$, where the set of morphisms is
defined as the set

$\operatorname{End}(E)=\\{\xi\colon E_{x}\rightarrow E_{y}\;|\;\xi\text{ is
linear},x,y\in X\\}$

of linear homomorphisms between the fibres of $E\rightarrow X$, and the
structure maps $s,t,m,u$ are defined in the obvious way. To show that
$\operatorname{End}(E)$ admits a structure of a Lie category without boundary,
we must define a smooth structure on $\operatorname{End}(E)$; it is induced by
local trivializations on $E$ in the following way. Denote by

$\\{U_{i}\times V\xrightarrow{\psi_{i}}\pi^{-1}(U_{i})\;|\;{i\in I}\\}$

an atlas of local trivializations of $E$ over an open cover $(U_{i})_{i\in I}$
of $X$, and denote by $\tau_{ij}\colon U_{i}\cap
U_{j}\rightarrow\operatorname{GL}(V)$ the respective transition maps, i.e.
$\psi_{i}^{-1}\psi_{j}(x,v)=(x,\tau_{ij}(x)v)$. For any two indices $i,j\in
I$, we define the map

$\displaystyle\Psi_{i}^{j}\colon U_{j}\times\operatorname{End}(V)\times
U_{i}\rightarrow\operatorname{End}(E)^{U_{j}}_{U_{i}}=s^{-1}(U_{i})\cap
t^{-1}(U_{j})$
$\displaystyle\Psi_{i}^{j}(y,A,x)(e|_{x})=\psi_{j}(y,A\mathrm{pr}_{V}\psi^{-1}_{i}(e|_{x})),$

whose inverse is

$(\Psi_{i}^{j})^{-1}(\xi\colon E_{x}\rightarrow
E_{y})=(y,v\mapsto\mathrm{pr}_{V}\psi_{j}^{-1}\xi\psi_{i}(x,v),x).$

The smoothness of transition maps in this atlas is easily checked by computing

$\displaystyle(\Psi_{k}^{l})^{-1}\Psi_{i}^{j}(y,A,x)=(y,\tau_{jl}(y)^{-1}A\tau_{ik}(x),x),$

where $x\in U_{i}\cap U_{k}$ and $y\in U_{j}\cap U_{l}$. From the local
charts, it is clear that $s$ and $t$ are submersions, and the smoothness of
composition map $m$ follows from smoothness of multiplication in
$\operatorname{End}(V)$; finally, this composition is bilinear when restricted
to
$\operatorname{End}(E)^{z}_{y}\times\operatorname{End}(E)^{y}_{x}\subset\operatorname{End}(E){}^{(2)}$,
so $\operatorname{End}(E)$ is moreover enriched over the category
$\mathbf{Vect}$ of vector spaces.

######  Example 2.11 (Bundles of Lie monoids).

A Lie category with coinciding source and target map $s=t=:p$ is called a
bundle of Lie monoids. In this case, two morphisms are composable if, and only
if, they are in the same fibre of $p$.

A concrete example of such a Lie category is the endomorphism bundle
$E^{*}\otimes E\rightarrow X$ of a vector bundle $E\rightarrow X$, with the
composition given on simple tensors as $(\varphi_{2}\otimes
v_{2})(\varphi_{1}\otimes v_{1})=\varphi_{2}(v_{1})\,\varphi_{1}\otimes
v_{2}$, and extended by bilinearity; this can easily be identified with the
composition of linear maps $E_{x}\rightarrow E_{x}$, so $E^{*}\otimes
E\subset\operatorname{End}(E)$ is a subcategory of $\operatorname{End}(E)$.
Using Lemma 2.15, it is not hard to see that it is actually an embedded Lie
subcategory of $\operatorname{End}(E)$, with respect to the inclusion map.

Another concrete example of a bundle of Lie monoids is the exterior bundle,

$\Lambda(E)=\bigoplus_{i=0}^{\operatorname{rank}E}\Lambda^{k}(E),$

of an $\mathbb{F}$-vector bundle $E\rightarrow X$, i.e. $\Lambda(E)$ consists
of all multivectors in $E$, and composition of $\alpha\in\Lambda^{k}(E_{x})$
and $\beta\in\Lambda^{l}(E_{x})$ is given as $\alpha\wedge\beta$. The units
are given by $1_{x}=1\in\mathbb{F}=\Lambda^{0}(E_{x})$, for any $x\in X$.
Again, the given composition map is smooth, which follows easily from
bilinearity of wedge product; moreover, $\Lambda(E)$ is again enriched over
$\mathbf{Vect}$. In general, bundles of Lie monoids enriched over Vect would
rightfully be called smooth bundles of unital associative algebras.

######  Example 2.12 (Action categories).

An action of a Lie monoid $M$ on a smooth manifold $X$ is a smooth map
$\phi\colon M\times X\rightarrow X$, denoted by $\phi(g,x)=gx$, which
satisfies $ex=x$ and $g(hx)=(gh)x$, for any $x\in X$ and $g,h\in M$. We
observe that contrary to the case of Lie group actions, the map $\phi$ may not
be submersive since the action is no longer by automorphisms of $X$, thus the
target map in the naïve generalization of the action groupoid would not be a
submersion.

To remedy this, we construct the action category of a given Lie monoid action
$\phi\colon M\times X\rightarrow X$ as follows. Denote by

$M\ltimes X=\\{(g,x)\in M\times X\;|\;\phi\text{ is a submersion at }(g,x)\\}$

the set of regular points of the action $\phi$.222Intuitively, the action
category accounts only for the non-critical dynamics pertaining to the given
action. Defining the structure maps as usual, i.e. $s(g,x)=x$, $t(g,x)=gx$,
the units as $u(x)=(e,x)$, and the composition as

$(g,hx)(h,x)=(gh,x),$

we obtain a Lie category $M\ltimes X\rightrightarrows X$. Indeed, since
$M\ltimes X\subset M\times X$ is an open subset, we only need to check that if
$\phi$ is a submersion at the points $(h,x)$ and $(g,hx)$, it is also a
submersion at $(gh,x)$. To that end, we first notice that we may write the
condition $g(hx)=(gh)x$ as an equality of maps $M\times M\times X\rightarrow
X$,

$\phi\circ(\mathrm{pr}_{1},\phi\circ(\mathrm{pr}_{2},\mathrm{pr}_{3}))=\phi\circ(m\circ(\mathrm{pr}_{1},\mathrm{pr}_{2}),\mathrm{pr}_{3}),$

where $m\colon M\times M\rightarrow M$ denotes the multiplication in the Lie
monoid $M$. Differentiating this equality at $(g,h,x)$, we obtain

$\operatorname{d}\\!{\phi}_{(g,hx)}\circ(\mathrm{pr}_{1},\operatorname{d}\\!{\phi}_{(g,x)}\circ(\mathrm{pr}_{2},\mathrm{pr}_{3}))=\operatorname{d}\\!{\phi}_{(gh,x)}\circ(\operatorname{d}\\!{m}_{(g,h)}\circ(\mathrm{pr}_{1},\mathrm{pr}_{2}),\mathrm{pr}_{3}).$

By assumption, the left-hand side is surjective, thus the same holds for the
first map on the right-hand side. Hence $M\ltimes X\rightrightarrows X$ is a
category and thus a Lie category.

######  Example 2.13.

A simple, yet important example of a Lie category is the order category of
$\mathbb{R}$, which is defined as the wide subcategory of the pair groupoid
$\mathcal{G}=\mathbb{R}\times\mathbb{R}\rightrightarrows\mathbb{R}$, given by

$\mathcal{C}=\\{(y,x)\in\mathbb{R}\times\mathbb{R}\;|\;x\leq y\\}.$

The space of morphisms is thus the half-space below the diagonal in
$\mathbb{R}^{2}$, and the source and target maps are the projections to the
vertical and the horizontal axis, respectively, implying that the boundary of
$\mathcal{C}$ is regular. Moreover, since the inversion in the pair groupoid
$\mathcal{G}$ is given by the reflection over the diagonal, the units in
$\mathcal{C}$ are precisely the elements of the diagonal, and these are the
only invertibles.

Although simple, this example has an important property: $\mathcal{C}$ can be
seen as the preimage of the set $[0,\infty)$ under the functor

$f\colon\mathbb{R}\times\mathbb{R}\rightarrow(\mathbb{R},+),\quad f(y,x)=y-x.$

The following example is a generalization of this; we will make use of it when
considering applications to statistical thermodynamics in Section 8.

######  Example 2.14.

Suppose a Lie category $\mathcal{D}\rightrightarrows\mathcal{X}$ without
boundary is given, together with a smooth functor
$f\colon\mathcal{D}\rightarrow(\mathbb{R},+)$ such that:

  1. (i)

$0\in\mathbb{R}$ is a regular value of $f$.

  2. (ii)

$s|_{f^{-1}(0)}$ and $t|_{f^{-1}(0)}$ are submersions.

Then the preimage $\mathcal{C}=f^{-1}\big{(}[0,\infty)\big{)}$ is a Lie
category. Indeed, since $0$ is a regular value of $f$, $\mathcal{C}$ is a
smooth embedded submanifold in $\mathcal{D}$ with boundary
$\partial\mathcal{C}=f^{-1}(0)$, see e.g. [difftop, p. 62]. Moreover, since
$f^{-1}\big{(}(0,\infty)\big{)}=\operatorname{Int}\mathcal{C}$ is open in
$\mathcal{D}$, the restrictions $s|_{\mathcal{C}},t|_{\mathcal{C}}$ of the
source and target maps to $\mathcal{C}$ are submersions. Functoriality of $f$
implies that $\mathcal{C}$ is a wide subcategory of $\mathcal{D}$ with all
invertibles contained within $\partial\mathcal{C}$, and moreover it also
implies that $\partial\mathcal{C}$ is a wide subcategory of $\mathcal{C}$. The
assumption (ii) enables us to use Lemma 2.15 below to conclude that
$\mathcal{C}$ is an embedded Lie subcategory of $\mathcal{D}$.

######  Lemma 2.15.

Let $\mathcal{C}$ be a wide subcategory of a Lie category
$\mathcal{D}\rightrightarrows\mathcal{X}$. Suppose $\mathcal{C}$ is also an
embedded submanifold of $\mathcal{D}$, such that
$s|_{\mathcal{C}},t|_{\mathcal{C}}$ are submersions and either of the
following holds:

  1. (i)

$\mathcal{C}$ has no boundary.

  2. (ii)

$\mathcal{C}$ has a boundary, and
$s|_{\partial\mathcal{C}},t|_{\partial\mathcal{C}}$ are submersions.

Then $\mathcal{C}$ is an embedded Lie subcategory of $\mathcal{D}$.

###### Proof.

The only thing needed to be proven is that the restriction
$m|_{{\mathcal{C}}^{(2)}}\colon{\mathcal{C}}^{(2)}\rightarrow\mathcal{C}$ of
the composition map $m\colon{\mathcal{D}}^{(2)}\rightarrow\mathcal{D}$, is
smooth. To that end, it is enough to check that ${\mathcal{C}}^{(2)}$ is a
submanifold of ${\mathcal{D}}^{(2)}$; notice that
${\mathcal{C}}^{(2)}={\mathcal{D}}^{(2)}\cap(\mathcal{C}\times\mathcal{C})$,
so we will make use of the transversality theorem.

Suppose first that $\mathcal{C}$ and $\mathcal{D}$ have no boundary, so that
by the usual transversality theorem it is enough to check
${\mathcal{D}}^{(2)}$ and $\mathcal{C}\times\mathcal{C}$ are transversal in
$\mathcal{D}\times\mathcal{D}$, i.e.

$T_{(g,h)}{\mathcal{D}}^{(2)}+T_{g}\mathcal{C}\oplus
T_{h}\mathcal{C}=T_{g}\mathcal{D}\oplus T_{h}\mathcal{D},\quad\text{for all
}(g,h)\in{\mathcal{C}}^{(2)}.$

To show this equality, let $(v,w)\in T_{g}\mathcal{D}\oplus T_{h}\mathcal{D}$.
Since $s|_{\mathcal{C}}$ is a submersion, there is a vector $v^{\prime}\in
T_{g}\mathcal{C}$ with
$\operatorname{d}\\!{s}_{g}(v^{\prime})=\operatorname{d}\\!{t}_{h}(w)$. Define
$v^{\prime\prime}=v-v^{\prime}$, and now since $t|_{\mathcal{C}}$ is a
submersion, there is a vector $w^{\prime}\in T_{h}\mathcal{C}$ such that

$\operatorname{d}\\!{t}_{h}(w^{\prime})=\operatorname{d}\\!{t}_{h}(w)-\operatorname{d}\\!{s}_{g}(v^{\prime\prime}).$

Now define $w^{\prime\prime}=w-w^{\prime}$. Clearly,
$(v^{\prime},w^{\prime})\in T_{g}\mathcal{C}\oplus T_{h}\mathcal{C}$, and on
the other hand, the definition of $w^{\prime}$ and $w^{\prime\prime}$ imply

$\operatorname{d}\\!{s}_{g}(v^{\prime\prime})=\operatorname{d}\\!{t}_{h}(w)-\operatorname{d}\\!{t}_{h}(w^{\prime})=\operatorname{d}\\!{t}_{h}(w^{\prime\prime}),$

so that $(v^{\prime\prime},w^{\prime\prime})\in T_{(g,h)}{\mathcal{D}}^{(2)}$,
which concludes our proof for the boundaryless case. If $\mathcal{D}$ has a
boundary, then the above proof together with Proposition A.8 used on the
inclusion
$\mathcal{C}\times\mathcal{C}\xhookrightarrow{\iota}\mathcal{D}\times\mathcal{D}$,
ensures that $\iota\pitchfork{\mathcal{D}}^{(2)}$, and since
${\mathcal{D}}^{(2)}\subset\mathcal{D}\times\mathcal{D}$ is a neat
submanifold,
${\mathcal{C}}^{(2)}\subset\mathcal{C}\times\mathcal{C}\subset\mathcal{D}\times\mathcal{D}$
is a submanifold by Proposition A.4, and hence since
${\mathcal{C}}^{(2)}\subset{\mathcal{D}}^{(2)}$, it is a submanifold of
${\mathcal{D}}^{(2)}$.

If $\mathcal{C}$ has a boundary, then the assumption (ii) ensures that all the
strata

$\operatorname{Int}\mathcal{C}\times\operatorname{Int}\mathcal{C},(\operatorname{Int}\mathcal{C}\times\partial\mathcal{C})\cup(\partial\mathcal{C}\times\operatorname{Int}\mathcal{C}),\partial\mathcal{C}\times\partial\mathcal{C}$

of $\mathcal{C}\times\mathcal{C}$ are transversal to ${\mathcal{D}}^{(2)}$ by
a similar proof as above, so by Proposition A.8 we again conclude
$\iota\pitchfork{\mathcal{D}}^{(2)}$. ∎

######  Remark 2.16.

In particular, an embedded submonoid of a Lie monoid is its embedded Lie
submonoid, however, a direct proof of this is much easier.

##  3\. Reversibility of morphisms

Reversible (synonymously, invertible) morphisms form an important subclass of
morphisms in any category $\mathcal{C}$; recall that $g\in\mathcal{C}$ is said
to be invertible, if there is a unique morphism $g^{-1}\in\mathcal{C}$ such
that $g^{-1}g=1_{s(g)}$ and $gg^{-1}=1_{t(g)}$. We observe that the set

$\mathcal{G}(\mathcal{C})=\\{g\in\mathcal{C}\;|\;g\text{ is invertible}\\}$

is a groupoid over the same base as $\mathcal{C}$. We call
$\mathcal{G}(\mathcal{C})$ the core of $\mathcal{C}$, the study of which
begins with the following simple observation.

######  Proposition 3.1.

For any morphism $g$ in a category $\mathcal{C}$, the following are
equivalent.

  1. (i)

$g$ is invertible.

  2. (ii)

Left and right translations by $g$ are bijections.

  3. (iii)

Left and right translations by $g$ are surjections.

  4. (iv)

$g$ has a left and a right inverse, i.e. there exist
$g^{\prime}\in\mathcal{C}^{s(g)},g^{\prime\prime}\in\mathcal{C}_{t(g)}$ with
$gg^{\prime}=1_{t(g)}$ and $g^{\prime\prime}g=1_{s(g)}$.

###### Proof.

Implications $(i)\Rightarrow(ii)\Rightarrow(iii)$ are clear, and
$(iii)\Rightarrow(iv)$ follows by observing that $(iv)$ means
$1_{t(g)}\in\operatorname{Im}(L_{g})$, $1_{s(g)}\in\operatorname{Im}(R_{g})$.
Lastly, implication $(iv)\Rightarrow(i)$ follows from elementary abstract
algebra:
$g^{\prime\prime}=g^{\prime\prime}1_{t(g)}=g^{\prime\prime}gg^{\prime}=1_{s(g)}g^{\prime}=g^{\prime}$.
∎

######  Remark 3.2.

Note that injectivity of left and right translations corresponds to
cancellative properties. For example, injectivity of $L_{g}$ is equivalent to
stating that for any two $h,k\in\mathcal{C}^{s(g)}$, $gh=gk$ implies $h=k$.

The regular boundary assumption on a Lie category $\mathcal{C}$ has the
important consequence that the units dictate where invertible elements can be:

######  Lemma 3.3.

Let $\mathcal{C}$ be a Lie category. For any invertible morphism
$g\in\mathcal{G}(\mathcal{C})$, the morphisms $g,g^{-1},1_{s(g)},1_{t(g)}$
must either all be contained in the interior $\operatorname{Int}\mathcal{C}$,
or in the boundary $\partial\mathcal{C}$.

###### Proof.

If $\mathcal{C}$ has no boundary then the lemma holds trivially, so suppose
$\partial\mathcal{C}\neq\emptyset$. Invertibility of $g$ means
$L_{g}\colon\mathcal{C}^{s(g)}\rightarrow\mathcal{C}^{t(g)}$ is a
diffeomorphism, which maps $1_{s(g)}\mapsto g$, so we must have either
$1_{s(g)}\in\partial(\mathcal{C}^{s(g)})$ and
$g\in\partial(\mathcal{C}^{t(g)})$, or
$1_{s(g)}\in\operatorname{Int}(\mathcal{C}^{s(g)})$ and
$g\in\operatorname{Int}(\mathcal{C}^{t(g)})$. Regularity of boundary implies
$\partial(\mathcal{C}^{x})=\mathcal{C}^{x}\cap\partial\mathcal{C}$ for any
$x\in\mathcal{X}$, so we must have either $1_{s(g)}\in\partial\mathcal{C}$ and
$g\in\partial\mathcal{C}$, or $1_{s(g)}\in\operatorname{Int}\mathcal{C}$ and
$g\in\operatorname{Int}\mathcal{C}$. A similar result is obtained for
$1_{t(g)}$ and $g$ using the right translation $R_{g}$, and similarly for
$g^{-1}$ and the units using $L_{g^{-1}},R_{g^{-1}}$ since $s(g^{-1})=t(g)$. ∎

The last lemma implies that any Lie groupoid (a Lie category with all
morphisms invertible) must have an empty boundary. We also obtain the
following two immediate corollaries.

######  Corollary 3.4.

For any Lie category $\mathcal{C}\rightrightarrows\mathcal{X}$, there holds:

$\displaystyle u(\mathcal{X})\subset\operatorname{Int}\mathcal{C}$
$\displaystyle\text{ implies
}\mathcal{G}(\mathcal{C})\subset\operatorname{Int}\mathcal{C},$ (3.5)
$\displaystyle u(\mathcal{X})\subset\partial\mathcal{C}$ $\displaystyle\text{
implies }\mathcal{G}(\mathcal{C})\subset\partial\mathcal{C}.$ (3.6)

######  Corollary 3.7.

The invertible elements of any Lie monoid are either contained in its
interior, or in its boundary.

An impending question is whether the core $\mathcal{G}(\mathcal{C})$ of
$\mathcal{C}$ is a Lie groupoid. Without additional assumptions this is false:

######  Example 3.8.

A simple example of a Lie category $\mathcal{C}$ with a regular boundary,
whose core $\mathcal{G}(\mathcal{C})$ is not a manifold, is the disjoint union
of the order category on $\mathbb{R}$ and the pair groupoid on $\mathbb{R}$.
More concretely, we define

$\displaystyle\mathcal{C}$
$\displaystyle=(-\infty,0)^{2}\cup\\{(y,x)\in(0,\infty)^{2}\;|\;x\leq y\\},$
$\displaystyle\mathcal{X}$ $\displaystyle=\mathbb{R}\setminus\\{0\\},$

with the categorical structure induced by the pair groupoid structure on
$\mathbb{R}$.

In this case,
$\mathcal{G}(\mathcal{C})=(-\infty,0)^{2}\cup\\{(x,x)\;|\;x>0\\}$, which has
two components of different dimensions. We have depicted this above, with
invertible elements of $\mathcal{C}$ in red, and non-invertible in blue; the
units in $\operatorname{Int}\mathcal{C}$ are depicted with a dashed line.

The culprit in the example above is that the units were allowed to be both in
the interior and in the boundary.

######  Definition 3.9.

A Lie category $\mathcal{C}\rightrightarrows\mathcal{X}$ is said to have a
normal boundary, if either of the following holds:

  1. (i)

$u(\mathcal{X})\subset\operatorname{Int}\mathcal{C}$.

  2. (ii)

$\partial\mathcal{C}$ is a wide subcategory of $\mathcal{C}$, i.e.
$\partial\mathcal{C}$ is a subcategory of $\mathcal{C}$ and
$u(\mathcal{X})\subset\partial\mathcal{C}$.

######  Remark 3.10.

A Lie category without a boundary trivially has a normal boundary.

The following result was proved by Charles Ehresmann in his pioneering paper
[ehr1959], where he introduced Lie categories. Ehresmann proved it by
implicitly assuming $\partial\mathcal{C}=\emptyset$, and with a somewhat old-
school method of local coordinates; the proof we present is coordinate-free
and uses the modern language of differentiable manifolds, is conceptually
clearer, and holds for categories with a normal boundary.

######  Theorem 3.11.

If $\mathcal{C}$ is a Lie category with a normal boundary, then its core
$\mathcal{G}(\mathcal{C})$ is an embedded Lie subcategory of $\mathcal{C}$.
More precisely, if $u(\mathcal{X})\subset\operatorname{Int}\mathcal{C}$, then
$\mathcal{G}(\mathcal{C})$ is open in $\operatorname{Int}\mathcal{C}$, and if
$u(\mathcal{X})\subset\partial\mathcal{C}$, then $\mathcal{G}(\mathcal{C})$ is
open in $\partial\mathcal{C}$.

###### Proof.

Consider first the case when
$u(\mathcal{X})\subset\operatorname{Int}\mathcal{C}$. We need to show that any
$k\in\mathcal{G}(\mathcal{C})$ has an invertible neighborhood in
$\mathcal{C}$; we will do so by showing that $k$ admits a left-invertible and
a right-invertible neighborhood. With this motive, define the map

$\displaystyle\vartheta\colon{\mathcal{C}}^{(2)}\rightarrow{\mathcal{C}\tensor[_{t}]{\times}{{}_{t}}}\,\mathcal{C},\quad\vartheta(g,h)=(g,gh),$
(3.12)

so $\vartheta=(\mathrm{pr}_{1},m)$. We claim that this is a local
diffeomorphism at the point $(k,k^{-1})$. Note that by equation 2.4 and a
similar result for
$\partial({\mathcal{C}\tensor[_{t}]{\times}{{}_{t}}}\,\mathcal{C})$, we have

$(k,k^{-1})\in\operatorname{Int}({\mathcal{C}}^{(2)})\quad\text{ and
}\quad(k,1_{t(k)})=\vartheta(k,k^{-1})\in\operatorname{Int}({\mathcal{C}\tensor[_{t}]{\times}{{}_{t}}}\,\mathcal{C}),$

so that it is enough to show $\operatorname{d}\\!{\vartheta}_{(k,k^{-1})}$ is
an isomorphism, by virtue of the inverse map theorem (for boundaryless
manifolds). For dimensional reasons, it is enough to check $\vartheta$ is an
immersion at $(k,k^{-1})$.

Suppose $\operatorname{d}\\!{\vartheta}_{(k,k^{-1})}(v,w)=0$ for some
$(v,w)\in T_{(k,k^{-1})}({\mathcal{C}}^{(2)})$. The identity
$\operatorname{d}\\!{\vartheta}(v,w)=(v,\operatorname{d}\\!{m}(v,w))$ implies
$v=0$, so we obtain

$\displaystyle\operatorname{d}\\!{m}_{(k,k^{-1})}(v,w)=\operatorname{d}\\!{(}L_{k})_{k^{-1}}(w).$

Since $L_{k}$ is a diffeomorphism, we conclude $w=0$, hence $\vartheta$ is a
local diffeomorphism at $(k,k^{-1})$, i.e. there is a neighborhood $U$ of
$(k,k^{-1})$ which is mapped diffeomorphically onto a neighborhood $V$ of
$(g,1_{t(g)})$.

Now note that $\mathcal{C}$ embeds into
${\mathcal{C}\tensor[_{t}]{\times}{{}_{t}}}\,\mathcal{C}$, by the map
$\lambda(g)=(g,1_{t(g)}),$ so the set
$\vartheta^{-1}(V\cap\lambda(\mathcal{C}))$ is diffeomorphic to
$V\cap\lambda(\mathcal{C})$, and consists of pairs $(g,g^{\prime})\in U$ such
that $gg^{\prime}=1_{t(g)}$. In other words, $\lambda^{-1}(V)$ is a
neighborhood of $k$, elements of which have right inverses.

Similarly, we can show that the map
$\tilde{\vartheta}\colon{\mathcal{C}}^{(2)}\rightarrow{\mathcal{C}\tensor[_{s}]{\times}{{}_{s}}}\,\mathcal{C},(g,h)\mapsto(gh,h)$
is a local diffeomorphism at $(k^{-1},k)$, and use the embedding
$\tilde{\lambda}\colon\mathcal{C}\rightarrow{\mathcal{C}\tensor[_{s}]{\times}{{}_{s}}}\,\mathcal{C}$,
$\tilde{\lambda}(g)=(1_{s(g)},g)$ to obtain a neighborhood
$\tilde{\lambda}^{-1}(\tilde{V})$ of $k$, elements of which have left
inverses. To conclude, note that Proposition 3.1 guarantees
$\lambda^{-1}(V)\cap\tilde{\lambda}^{-1}(\tilde{V})$ is an invertible
neighborhood of $k$.

Finally, for the case $u(\mathcal{X})\subset\partial\mathcal{C}$, just note
that normality of the boundary can be used with Lemma 2.15 to conclude that
$\partial\mathcal{C}$ is an embedded Lie subcategory of $\mathcal{C}$ without
boundary, so we can apply the previous case to $\partial\mathcal{C}$. ∎

Given a morphism $F\colon\mathcal{C}\rightarrow\mathcal{D}$ between Lie
categories with normal boundaries, we have
$F(\mathcal{G}(\mathcal{C}))\subset\mathcal{G}(\mathcal{D})$ by functoriality,
so we may define $\mathcal{G}(F)=F|_{\mathcal{G}(\mathcal{C})}$. The map
$\mathcal{G}\colon\mathbf{LieCat_{\partial}}\rightarrow\mathbf{LieGrpd}$
defines a functor from the category of Lie categories with normal boundary to
the category of Lie groupoids without boundary, which is easily seen to be
right adjoint to the inclusion functor
$\mathbf{LieGrpd}\hookrightarrow\mathbf{LieCat_{\partial}}$.

Using the last theorem we can also show that the universal property of the
core extends to the differentiable setting. We note that more consequences of
the Theorem 3.11 will be explored in upcoming sections.

######  Corollary 3.13.

Let $\mathcal{C}$ be a Lie category with a normal boundary, and let
$\mathcal{H}\subset\mathcal{C}$ be a groupoid that is also a Lie subcategory
of $\mathcal{C}$. If any smooth morphism
$F\colon\mathcal{G}\rightarrow\mathcal{C}$, defined on a Lie groupoid
$\mathcal{G}$, factors uniquely through $\mathcal{H}$, then
$\mathcal{H}=\mathcal{G}(\mathcal{C})$.

###### Proof.

The inclusion $\mathcal{H}\subset\mathcal{G}(\mathcal{C})$ is an easy
consequence of functoriality of $\mathcal{H}\hookrightarrow\mathcal{C}$, since
functors map isomorphisms to isomorphisms. For the converse inclusion, we pick
the smooth map $F\colon\mathcal{G}(\mathcal{C})\hookrightarrow\mathcal{C}$ and
now the existence of
$\bar{F}\colon\mathcal{G}(\mathcal{C})\rightarrow\mathcal{H}$ such that
$F=\iota\circ\bar{F}$, ensures that if $g\in\mathcal{G}(\mathcal{C})$, then
$g=\bar{F}(g)\in\mathcal{H}$. ∎

##  4\. Lie algebroids of a Lie category

The construction of a Lie algebroid using left-invariant (or right-invariant)
vector fields on a Lie groupoid readily generalizes to Lie categories;
however, we no longer have a canonical isomorphism between the two algebroids,
previously given by inversion map. Below, we present the main idea of the
construction (4.1 through 4.4), mainly to fix notation; we omit some details
which can be found in standard Lie groupoid references.

######  Definition 4.1.

A left-invariant vector field on a Lie category
$\mathcal{C}\rightrightarrows\mathcal{X}$ is a vector field
$X\in\mathfrak{X}(\mathcal{C})$, which is tangent to $t$-fibres, i.e.
$X\in\Gamma^{\infty}(\ker\operatorname{d}\\!{t})$, and left-invariant, i.e.
$\operatorname{d}\\!{(}L_{g})_{h}(X_{h})=X_{gh}$ for all
$(g,h)\in{\mathcal{C}}^{(2)}$. Denote by $\mathfrak{X}^{L}(\mathcal{C})$ the
vector space of left-invariant vector fields on $\mathcal{C}$.

######  Lemma 4.2.

Let $\mathcal{C}$ be a Lie category. The vector space
$\mathfrak{X}^{L}(\mathcal{C})$ is closed under the Lie bracket, and
canonically isomorphic to the vector space
$\Gamma^{\infty}(A^{L}(\mathcal{C}))$ of sections of the vector bundle
$A^{L}(\mathcal{C})=u^{*}(\ker\operatorname{d}\\!{t})$ over $\mathcal{X}$.

###### Proof.

Closedness under the Lie bracket is a consequence of the fact that if $X,Y$
are $L_{g}$-related to themselves when restricted to appropriate fibres, so is
$[X,Y]$. The canonical isomorphism
$\mathrm{ev}\colon\mathfrak{X}^{L}(\mathcal{C})\rightarrow\Gamma^{\infty}(A^{L}(\mathcal{C}))$
is given by restriction to the units, and its inverse $\mathrm{ev}^{-1}$ maps
any section $\alpha\in\Gamma^{\infty}(A^{L}(\mathcal{C}))$ to its left-
invariant extension $\alpha^{L}$, given as
$\alpha^{L}(g)=\operatorname{d}\\!{(}L_{g})_{1_{s(g)}}(\alpha_{s(g)}),$ which
is a smooth section $\mathcal{C}\rightarrow T\mathcal{C}$ since it can be
realized as the composition
$\alpha^{L}=\operatorname{d}\\!{m}\circ\tau^{\alpha}$, where
$\tau^{\alpha}\colon\mathcal{C}\rightarrow T({\mathcal{C}}^{(2)})$ is given by
$g\mapsto(0_{g},\alpha_{1_{s(g)}}).$ ∎

######  Definition 4.3.

The left Lie algebroid of a Lie category
$\mathcal{C}\rightrightarrows\mathcal{X}$ is the vector bundle
$A^{L}(\mathcal{C})\rightarrow\mathcal{X}$, endowed with the Lie bracket
$[\cdot,\cdot]$ on its sections as induced by the isomorphism $\mathrm{ev}$,
together with the anchor map $\rho^{L}\colon A^{L}(\mathcal{C})\rightarrow
T\mathcal{X}$, $\rho^{L}=\operatorname{d}\\!{s}|_{A^{L}(\mathcal{C})}$.

Similarly, we define the right Lie algebroid of a Lie category $\mathcal{C}$
as the vector bundle $A^{R}(\mathcal{C})=u^{*}(\ker\operatorname{d}\\!{s})$,
which as a vector space is isomorphic to the space
$\mathfrak{X}^{R}(\mathcal{C})$ of right-invariant vector fields, and its
anchor map is defined as
$\rho^{R}=\operatorname{d}\\!{t}|_{A^{R}(\mathcal{C})}$.

The proof of the next proposition is again the same as with the Lie groupoid
case, so we omit it. Again, an analogous result holds for the right Lie
algebroid.

######  Proposition 4.4.

The left Lie algebroid $A^{L}(\mathcal{C})$ of a Lie category
$\mathcal{C}\rightrightarrows\mathcal{X}$ satisfies:

  1. (i)

$\rho^{L}$ preserves the brackets, i.e.
$\rho^{L}[\alpha,\beta]=[\rho^{L}\alpha,\rho^{L}\beta]_{T\mathcal{X}}$,

  2. (ii)

Leibniz rule: $[\alpha,f\beta]=f[\alpha,\beta]+\rho^{L}(\alpha)(f)\beta$,

for all $\alpha,\beta\in\Gamma^{\infty}(A^{L}(\mathcal{C}))$ and $f\in
C^{\infty}(\mathcal{X}).$

Any morphism $F\colon\mathcal{C}\rightarrow\mathcal{D}$ over
$\mathrm{id}_{\mathcal{X}}$ of Lie categories over the same object manifold
$\mathcal{X}$, induces morphisms between their left and right Lie algebroids
(respectively), denoted by

$\displaystyle F_{*}^{L}\colon A^{L}(\mathcal{C})\rightarrow
A^{L}(\mathcal{D})\quad\text{and}\quad F_{*}^{R}\colon
A^{R}(\mathcal{C})\rightarrow A^{R}(\mathcal{D}),$

and defined on respective sections of $A^{L}(\mathcal{C})$ and
$A^{R}(\mathcal{C})$ in the obvious way:

$\displaystyle\alpha\mapsto\operatorname{d}\\!{F}\circ\alpha.$

In the case when the object manifolds of $\mathcal{C}$ and $\mathcal{D}$ are
not equal and the morphism $F$ does not restrict to a diffeomorphism between
the units, we encounter the same complications as in the Lie groupoid case;
this is resolved in the same manner as for Lie groupoids, and since we will
not be needing this more general result, we point the reader to [mackenzie,
Chapter 4.3] for details. The upshot is that $A^{L}$ and $A^{R}$ are functors
from the category $\mathbf{LieCat}$ of Lie categories to the category
$\mathbf{LieAlgd}$ of Lie algebroids.

As mentioned, we do not have a canonical isomorphism between the left and
right Lie algebroid of a Lie category, which is given in a Lie groupoid by the
inversion map. However, we have the following result, and we will later
encounter a related one when studying extensions of categories to groupoids
(see section 6).

######  Proposition 4.5.

Let $\mathcal{C}\rightrightarrows\mathcal{X}$ be a Lie category. If the units
of $\mathcal{C}$ are contained in the interior of $\mathcal{C}$, i.e.
$u(\mathcal{X})\subset\operatorname{Int}\mathcal{C}$, then the left and right
Lie algebroids of $\mathcal{C}$ are isomorphic to the Lie algebroid of its
core $\mathcal{G}(\mathcal{C})$.

###### Proof.

By Theorem 3.11, $\mathcal{G}(\mathcal{C})$ is open in $\mathcal{C}$, so we
get the following chain of isomorphisms of Lie algebroids:

$\displaystyle A^{L}(\mathcal{C})\cong A^{L}(\mathcal{G}(\mathcal{C}))\cong
A^{R}(\mathcal{G}(\mathcal{C}))\cong A^{R}(\mathcal{C}),$ (4.6)

where the first and last isomorphism are induced by the inclusion
$\mathcal{G}(\mathcal{C})\hookrightarrow\mathcal{C}$, and the isomorphism in
the middle is induced by inversion in the groupoid $\mathcal{G}(\mathcal{C})$.
∎

######  Remark 4.7.

Note that if $\mathcal{C}$ has a normal boundary and
$u(\mathcal{X})\subset\partial\mathcal{C}$, the Lie algebroid
$A(\mathcal{G}(\mathcal{C}))$ of the core will always fail to be isomorphic to
the two Lie algebroids of $\mathcal{C}$, since the rank of the vector bundle
$A(\mathcal{G}(\mathcal{C}))$ is one less than the rank of
$A^{L}(\mathcal{C})$ and $A^{R}(\mathcal{C})$. This is demonstrated by the
following two examples:

  1. (i)

The two Lie algebras of the Lie monoid $M=\mathbb{H}^{n}$ are isomorphic (as
vector spaces) to $A_{L}(M)\cong\mathbb{R}^{n}\cong A_{R}(M)$, whereas
$A(G(M))\cong\mathbb{R}^{n-1}$ since the core of $M$ is
$G(M)=\mathbb{R}^{n-1}$.

  2. (ii)

Consider the order category $\mathcal{C}=\\{(y,x)\in\mathbb{R}^{2}\;|\;x\leq
y\\}\rightrightarrows\mathbb{R}$ from Example 2.13. Notice that its core
$\mathcal{G}(\mathcal{C})$ is just the base groupoid over $\mathbb{R}$, hence
its Lie algebroid is the zero bundle
$A(\mathcal{G}(\mathcal{C}))=\mathbb{R}\times\\{0\\}$. On the other hand, the
left and right Lie algebroid of $\mathcal{C}$ are both isomorphic to
$T\mathbb{R}$.

##  5\. Ranks of morphisms

What follows can be seen as a natural generalization of the usual notion of
rank from linear algebra (see Example 5.2 (iii)).

######  Definition 5.1.

Let $\mathcal{C}\rightrightarrows\mathcal{X}$ be a Lie category and let
$g\in\mathcal{C}$. The left rank and right rank of $g$ are defined as

$\displaystyle\operatorname{rank}^{L}(g)$
$\displaystyle=\operatorname{rank}\operatorname{d}\\!{(}L_{g})_{1_{s(g)}},$
$\displaystyle\operatorname{rank}^{R}(g)$
$\displaystyle=\operatorname{rank}\operatorname{d}\\!{(}R_{g})_{1_{t(g)}}.$

If the left and right rank of a morphism $g$ are equal, we just write
$\operatorname{rank}(g)=\operatorname{rank}^{L}(g)=\operatorname{rank}^{R}(g)$
and call this integer the rank of $g$. Moreover, we say $g$ has full rank, if
its left and right ranks are full, that is, if

$\operatorname{rank}(g)=\operatorname{codim}_{\mathcal{C}}(\mathcal{X})=\dim\mathcal{C}-\dim\mathcal{X}=:\delta.$

In this case, we will sometimes say that $g$ is a regular morphism. If $g$ is
not regular, we will call it singular.

Finally, we say that $g$ has constant left rank, if
$\operatorname{rank}^{L}(g)=\operatorname{rank}\operatorname{d}\\!{(}L_{g})_{h}$
for all $h\in\mathcal{C}^{s(g)}$, and similarly that $g$ has constant right
rank, if
$\operatorname{rank}^{R}(g)=\operatorname{rank}\operatorname{d}\\!{(}R_{g})_{h}$
for all $h\in\mathcal{C}_{t(g)}$. To avoid ambiguity, we will sometimes write
$\operatorname{rank}_{\mathcal{C}}$ instead of $\mathrm{rank}.$

######  Example 5.2.

  1. (i)

All invertible morphisms in any Lie category have full and constant rank.

  2. (ii)

In a Lie monoid $M$, the ranks of an element $g\in M$ are just the ranks of
$\operatorname{d}\\!{(}L_{g})_{e}$ and $\operatorname{d}\\!{(}R_{g})_{e}$. If
$M$ is an abelian Lie monoid, then clearly any element of $M$ has equal left
and right rank. The example $M=\mathbb{H}^{n}$ shows that regularity does not
imply invertibility.

We will later see that regularity and constancy of ranks of all morphisms in a
Lie category is ensured whenever dealing with a Lie subcategory of a Lie
groupoid, as is with the simple example
$\mathbb{H}^{n}\hookrightarrow\mathbb{R}^{n}$ for addition.

  3. (iii)

If $M=\mathbb{R}^{n\times n}$, let us show how the above notion of rank
relates with the usual one. If $A\in\mathbb{R}^{n\times n}$, then the usual
notion reads $\operatorname{rank}A=\dim\operatorname{Im}A$, when $A$ is seen
as the map $\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$. On the other hand, the
left rank in Definition 5.1 equals:

$\operatorname{rank}^{L}_{M}(A)=\dim\operatorname{Im}\operatorname{d}\\!{(}L_{A})_{I}=\dim\operatorname{Im}L_{A},$

where $L_{A}\colon\mathbb{R}^{n\times n}\rightarrow\mathbb{R}^{n\times n}$ is
the left translation by $A$, and the last equality follows by linearity.
Similarly, $\operatorname{rank}^{R}_{M}(A)=\dim\operatorname{Im}R_{A}$.
Denoting by $E_{ij}$ the matrix with 1 in place $(i,j)$ and zero elsewhere, we
have that $AE_{ij}$ has $i$-th column of $A$ in $j$-th column and is zero
elsewhere, and $E_{ij}A$ has $j$-th row of $A$ in $i$-th row and is zero
elsewhere, so:

$\displaystyle\operatorname{Im}(L_{A})$
$\displaystyle=\operatorname{Lin}(AE_{ij})_{i,j=1}^{n}=\Big{\\{}\begin{bmatrix}v_{1}\dots
v_{n}\end{bmatrix}\;\Big{|}\;v_{i}\in\operatorname{Im}(A)\Big{\\}},$
$\displaystyle\operatorname{Im}(R_{A})$
$\displaystyle=\operatorname{Lin}(E_{ij}A)_{i,j=1}^{n}=\left\\{\begin{bmatrix}v_{1}\dots
v_{n}\end{bmatrix}^{\mathsf{T}}\;\middle|\;v_{i}\in\operatorname{Im}\left(A^{\mathsf{T}}\right)\right\\}.$

Since $\operatorname{rank}A=\operatorname{rank}A^{\mathsf{T}}$, it follows
that

$\operatorname{rank}^{L}_{M}(A)=\operatorname{rank}^{R}_{M}(A)=n\operatorname{rank}(A),$

and we see that $A$ is regular if, and only if, $A$ is invertible.

The above result readily generalizes to arbitrary finite-dimensional vector
spaces: if $V$ is a vector space, then
$\operatorname{rank}_{\operatorname{End}(V)}(A)=\dim(V)\operatorname{rank}(A).$
Moreover, for any vector bundle $E$, the above result clearly also generalizes
to the endomorphism bundle $E^{*}\otimes E$, so that for any $A\colon
E_{x}\rightarrow E_{x}$,

$\operatorname{rank}_{E^{*}\otimes
E}A=\operatorname{rank}(E)\operatorname{rank}(A).$

  4. (iv)

Examples of Lie monoids that do not have coinciding left and right ranks may
be found in the context of finite-dimensional unital algebras. As a concrete
example, take the algebra $A\subset\mathbb{R}^{2\times 2}$ of upper-diagonal
real $2\times 2$ matrices, the canonical basis of which is given by
$a=\bigl{[}\begin{smallmatrix}1&0\\\
0&0\end{smallmatrix}\bigr{]},b=\bigl{[}\begin{smallmatrix}0&0\\\
0&1\end{smallmatrix}\bigr{]},c=\bigl{[}\begin{smallmatrix}0&1\\\
0&0\end{smallmatrix}\bigr{]}$. Identifying $T_{e}A$ with $A$ and noting that
left and right translations are linear maps, we can identify
$\operatorname{d}\\!{(}L_{g})_{e}=L_{g}$ for any $g\in A$, and similarly for
the right translations. It is easy to see
$\operatorname{Im}(R_{a})=\operatorname{Lin}(a)$ and
$\operatorname{Im}(L_{a})=\operatorname{Lin}(a,c)$ by computing all the
products of $a$ with the canonical basis, so we conclude that
$2=\operatorname{rank}^{L}_{A}(a)\neq\operatorname{rank}^{R}_{A}(a)=1$.

######  Proposition 5.3 (Properties of ranks).

In a Lie category $\mathcal{C}\rightrightarrows\mathcal{X}$, there holds:

  1. (i)

The ranks of a composition of composable morphisms $g,h\in\mathcal{C}$ are
bounded from above:

$\displaystyle\operatorname{rank}^{L}(gh)$
$\displaystyle\leq\operatorname{rank}^{L}(h),$ (5.4)
$\displaystyle\operatorname{rank}^{R}(gh)$
$\displaystyle\leq\operatorname{rank}^{R}(g).$ (5.5)

  2. (ii)

The ranks of $g\in\mathcal{C}$ are bounded from below by the ranks of anchors
$\rho^{L}\colon A^{L}(\mathcal{C})\rightarrow T\mathcal{X}$ and
$\rho^{R}\colon A^{R}(\mathcal{C})\rightarrow T\mathcal{X}$ of Lie algebroids
of $\mathcal{C}$:

$\displaystyle\operatorname{rank}^{L}(g)\geq\operatorname{rank}\rho_{s(g)}^{L},$
(5.6)
$\displaystyle\operatorname{rank}^{R}(g)\geq\operatorname{rank}\rho_{t(g)}^{R}.$
(5.7)

###### Proof.

Since $L_{gh}=L_{g}\circ L_{h}$, we have

$\operatorname{rank}^{L}(gh)=\operatorname{rank}(\operatorname{d}\\!{(}L_{g})_{h}\circ\operatorname{d}\\!{(}L_{h})_{1_{s(h)}})\leq\operatorname{rank}^{L}(h),$

and similarly for the right translation by using $R_{gh}=R_{h}\circ R_{g}$, so
$(i)$ follows. For the property $(ii)$ note that the diagram

${\mathcal{C}^{s(g)}}$${\mathcal{C}^{t(g)}}$${\mathcal{X}}$$\scriptstyle{L_{g}}$$\scriptstyle{s|_{\mathcal{C}^{s(g)}}}$$\scriptstyle{s|_{\mathcal{C}^{t(g)}}}$

implies
$\operatorname{rank}^{L}(g)\geq\operatorname{rank}\operatorname{d}\\!{(}s|_{\mathcal{C}^{s(g)}})_{1_{s(g)}}=\operatorname{rank}\rho^{L}_{s(g)}$.
∎

In the context of Lie groupoids, it is well known that the composition map is
a submersion. This is no longer true for Lie categories in general; a
counterexample is provided by the Lie monoid $\mathbb{R}^{n\times n}$ for
matrix multiplication, where $\operatorname{d}\\!{m}_{(0,0)}$ is easily seen
to be the zero map. However, the following results ensure that $m$ is a
submersion in case all morphisms have full and constant rank; as mentioned,
this is the case for Lie categories extendable to Lie groupoids, as we will
see later in Lemma 6.3.

######  Lemma 5.8.

Let $\mathcal{C}\rightrightarrows\mathcal{X}$ be a Lie category and
$g\in\mathcal{C}$. The map $L_{g}$ has full rank at $h\in\mathcal{C}^{s(g)}$
if, and only if, the map
$\vartheta\colon{\mathcal{C}}^{(2)}\rightarrow{\mathcal{C}\tensor[_{t}]{\times}{{}_{t}}}\,\mathcal{C},(g,h)\mapsto(g,gh)$
has full rank at $(g,h)$. In particular, $g$ has full left rank if, and only
if, $\vartheta$ has full rank at $(g,1_{s(g)})$.

Recall that we have already encountered the map $\vartheta$ in the proof of
Theorem 3.11. In a Lie groupoid, this map is a bijection with inverse
$(g,h)\mapsto(g,g^{-1}h)$, so above lemma guarantees it is a diffeomorphism,
which can be used e.g. to show that the inversion map is automatically smooth,
by realizing it as the composition

$g\mapsto(g,1_{t(g)})\xmapsto{\vartheta^{-1}}(g,g^{-1})\mapsto g^{-1}.$

An analogous result to Lemma 5.8 of course holds for ranks of right
translations, using the map
$\tilde{\vartheta}\colon{\mathcal{C}}^{(2)}\rightarrow{\mathcal{C}\tensor[_{s}]{\times}{{}_{s}}}\,\mathcal{C}$,
given by $(g,h)\mapsto(gh,h)$. For instance, the second part of above lemma
would then read: $g$ has full right rank if, and only if, $\tilde{\vartheta}$
has full rank at $(1_{t(g)},g)$.

###### Proof.

For the forward implication, note that
$\operatorname{d}\\!{\vartheta}_{(g,h)}(v,w)=(v,\operatorname{d}\\!{m}_{(g,h)}(v,w))$
holds for all $(v,w)\in T_{(g,h)}{\mathcal{C}}^{(2)}$, hence
$\operatorname{d}\\!{\vartheta}_{(g,h)}(v,w)=(0,0)$ first implies $v=0$, and
since
$\operatorname{d}\\!{m}_{(g,h)}(0,w)=\operatorname{d}\\!{(}L_{g})_{h}(w)$, we
get $w=0$ since $L_{g}$ has full rank at $h$. For the other direction, note
that $\operatorname{d}\\!{(}L_{g})_{h}(w)=0$ implies
$\operatorname{d}\\!{\vartheta}_{(g,h)}(0,w)=(0,0)$, so the assumption that
$\vartheta$ has full rank at $(g,h)$ yields $w=0$. ∎

######  Corollary 5.9.

Let $\mathcal{C}\rightrightarrows\mathcal{X}$ be a Lie category and
$g\in\mathcal{C}$. If the map $L_{g}$ has full rank at
$h\in\mathcal{C}^{s(g)}$, then the composition map
$m\colon{\mathcal{C}}^{(2)}\rightarrow\mathcal{C}$ is a submersion at $(g,h)$.
Hence if all morphisms have full and constant rank, the composition is
submersive.

###### Proof.

Differentiating the identity $m=\mathrm{pr}_{2}\circ\vartheta$ yields
$\operatorname{d}\\!{m}_{(g,h)}=\operatorname{d}\\!{(}\mathrm{pr}_{2})_{(g,gh)}\circ\operatorname{d}\\!{\vartheta}_{(g,h)},$
which is a composition of surjective maps by previous lemma and the fact that
$\mathrm{pr}_{2}\colon{\mathcal{C}\tensor[_{t}]{\times}{{}_{t}}}\,\mathcal{C}\rightarrow\mathcal{C}$
is a submersion. ∎

We now direct our attention to the subsets of $\mathcal{C}$ of morphisms with
full rank. Denote by

$\displaystyle\mathcal{R}_{r}^{L}(\mathcal{C})$
$\displaystyle=\\{g\in\mathcal{C}\;|\;\operatorname{rank}^{L}(g)=r\\},$
$\displaystyle\mathcal{R}_{r}^{R}(\mathcal{C})$
$\displaystyle=\\{g\in\mathcal{C}\;|\;\operatorname{rank}^{R}(g)=r\\}$

the sets of morphisms with left and right rank equal to $r$, respectively, and
furthermore by
$\mathcal{R}_{r}(\mathcal{C})=\mathcal{R}_{r}^{R}(\mathcal{C})\cap\mathcal{R}_{r}^{L}(\mathcal{C})$
morphisms whose both ranks equal $r$.

######  Proposition 5.10.

In any Lie category $\mathcal{C}\rightrightarrows\mathcal{X}$, the subset
$\mathcal{R}_{\delta}(\mathcal{C})\subset\mathcal{C}$ of regular morphisms is
open.

###### Proof.

Let us first show that differentials of left translations define a certain
morphism of vector bundles. We consider the following vector bundle over
$\mathcal{C}$:

$E^{L}=\coprod_{g\in\mathcal{C}}(\ker\operatorname{d}\\!{t}_{1_{s(g)}})^{*}\otimes\ker\operatorname{d}\\!{t}_{g}=(s^{*}u^{*}\ker\operatorname{d}\\!{t})^{*}\otimes\ker\operatorname{d}\\!{t}.$

Denote the projection map by $p_{L}\colon E^{L}\rightarrow\mathcal{C}$, so the
fibre $p_{L}^{-1}(g)=E^{L}_{g}$ of $E^{L}$ consists of all linear maps
$\ker\operatorname{d}\\!{t}_{1_{s(g)}}\rightarrow\ker\operatorname{d}\\!{t}_{g}$.
Note that the map $\phi_{L}\colon\mathcal{C}\rightarrow E^{L}$, given as
$g\mapsto\operatorname{d}\\!{(}L_{g})_{1_{s(g)}}\colon\ker\operatorname{d}\\!{t}_{1_{s(g)}}\rightarrow\ker\operatorname{d}\\!{t}_{g}$
is a section of $E^{L}$, which is smooth since

$\operatorname{d}\\!{(}L_{g})_{1_{s(g)}}=\operatorname{d}\\!{m}_{(g,1_{s(g)})}(0_{g},-).$

In other words, $\phi_{L}$ defines a morphism
$s^{*}u^{*}\ker\operatorname{d}\\!{t}\rightarrow\ker\operatorname{d}\\!{t}$ of
vector bundles.

Now take an atlas of local trivializations $\psi_{i}\colon
p_{L}^{-1}(U_{i})\rightarrow U_{i}\times\mathbb{R}^{\delta\times\delta}$ of
$E^{L}$. The set
$U_{i}\cap\mathcal{R}^{L}_{\delta}(\mathcal{\mathcal{C}})=\phi_{L}^{-1}\circ\psi_{i}^{-1}(U_{i}\times\mathrm{GL}(\delta,\mathbb{R}))$
is open in $U_{i}$, hence

$\mathcal{R}_{\delta}^{L}(\mathcal{C})=\bigcup_{i}\big{(}U_{i}\cap\mathcal{R}_{\delta}^{L}(\mathcal{C})\big{)}$

is open in $\mathcal{C}$. A similar proof works for
$\mathcal{R}^{R}_{\delta}(\mathcal{C})$, so the result for
$\mathcal{R}_{\delta}(\mathcal{C})$ follows. ∎

######  Remark 5.11.

The smooth map $\phi_{L}$ (respectively, $\phi_{R}$) which was used in the
last proposition, can also easily be used to show that
$\operatorname{rank}^{L}$ (respectively, $\operatorname{rank}^{R}$) is a lower
semi-continuous function, i.e. that any morphism $g\in\mathcal{C}$ admits a
neighborhood on which the left (respectively, right) rank is non-decreasing.
Note that this should not be confused with lower semi-continuity of the map
$\mathcal{C}^{s(g)}\rightarrow\mathbb{R}$, given as
$h\mapsto\operatorname{rank}\operatorname{d}\\!{(}L_{g})_{h}$, where
$g\in\mathcal{C}$ is a fixed morphism.

Properties of ranks of morphisms in a Lie category $\mathcal{C}$ reflect on
its categorical structure, as illustrated by the following simple observation.

######  Corollary 5.12.

Let $\mathcal{C}$ be a Lie category. If the regular morphisms have constant
rank, then $\mathcal{R}_{\delta}(\mathcal{C})$ is an open Lie subcategory of
$\mathcal{C}$.

###### Proof.

We only have to check that the composition of two regular morphisms is a
regular morphism. To that end, just notice that
$\operatorname{d}\\!{(}L_{gh})_{1_{s(h)}}=\operatorname{d}\\!{(}L_{g})_{h}\circ\operatorname{d}\\!{(}L_{h})_{1_{s(h)}}$
is a composition of maps with full rank, and similar holds for right
translations. ∎

######  Remark 5.13.

Notice that in this case, the left and right algebroids of
$\mathcal{R}_{\delta}(\mathcal{C})$ are isomorphic to those of $\mathcal{C}$,
respectively. This is in accord with the moral that the left and right
algebroids are, as vector bundles, determined by their fibres at regular
morphisms.

### Action of the core $\mathcal{G}(\mathcal{C})$ on a Lie category
$\mathcal{C}$

The natural left and right actions of the core $\mathcal{G}(\mathcal{C})$ on a
Lie category $\mathcal{C}$ can be used to find properties of Lie
categories.333A reference for actions of Lie groupoids on smooth maps is
[mackenzie, Chapter 1.6]. We will only focus on describing the left action of
$\mathcal{G}(\mathcal{C})$ on $t\colon\mathcal{C}\rightarrow\mathcal{X}$; the
right action of $\mathcal{G}(\mathcal{C})$ on
$s\colon\mathcal{C}\rightarrow\mathcal{X}$ follows a similar construction.

Denote by
$\mathcal{G}(\mathcal{C})\tensor[_{s}]{\times}{{}_{t}}\mathcal{C}=\\{(g,c)\in\mathcal{G}(\mathcal{C})\times\mathcal{C}\;|\;s(g)=t(c)\\}$
the fibred product of the maps $s|_{\mathcal{G}(\mathcal{C})}$ and $t$. This
set has a natural structure of a groupoid over $\mathcal{C}$; indeed, the
source and target maps are given as $\underline{s}(g,c)=c$ and
$\underline{t}(g,c)=gc$, the composition is given as $(g,hc)(h,c)=(gh,c)$, the
unit map is $\underline{u}(c)=(1_{t(c)},c)$ and the inverses are given by
$(g,c)^{-1}=(g^{-1},gc)$. We leave it to the reader to check that this defines
a groupoid over $\mathcal{C}$. As regards to its smooth structure:

######  Proposition 5.14.

Let $\mathcal{C}\rightrightarrows\mathcal{X}$ be a Lie category without
boundary. The left action of the core $\mathcal{G}(\mathcal{C})$ on the target
map $t\colon\mathcal{C}\rightarrow\mathcal{X}$ yields a Lie groupoid

$\mathcal{G}(\mathcal{C})\tensor[_{s}]{\times}{{}_{t}}\mathcal{C}\rightrightarrows\mathcal{C}.$

###### Proof.

The core $\mathcal{G}(\mathcal{C})$ is an open Lie subgroupoid of
$\mathcal{C}$ by Theorem 3.11, and so the usual transversality theorem for
manifolds without boundary implies that

$\underline{\mathcal{C}}=\mathcal{G}(\mathcal{C})\tensor[_{s}]{\times}{{}_{t}}\mathcal{C}=(s|_{\mathcal{G}(\mathcal{C})}\times
t)^{-1}(\Delta_{\mathcal{X}})$

is a smooth submanifold of $\mathcal{G}(\mathcal{C})\times\mathcal{C}$ without
boundary. Furthermore, $\underline{s}=\mathrm{pr}_{2}$ is clearly a
sumbersion, and Corollary 5.9 tells us that the target map

$\underline{t}=m|_{\mathcal{G}(\mathcal{C})\tensor[_{s}]{\times}{{}_{t}}\mathcal{C}}$

is also a submersion, thus
$\underline{\mathcal{C}}{}^{(2)}\subset\underline{\mathcal{C}}\times\underline{\mathcal{C}}$
is also a submanifold by transversality. The smoothness of composition map
$\underline{m}$ and unit map $\underline{u}$ then follows easily from the
smoothness of respective maps in $\mathcal{C}$. ∎

######  Remark 5.15.

Notice that in the case when $\mathcal{C}$ has a normal boundary, the maps
$\underline{s}$ and $\underline{t}$ are still submersions, but their
restrictions to the boundary
$\partial\underline{\mathcal{C}}=\mathcal{G}(\mathcal{C})\times\partial\mathcal{C}$
are not, which shows the need for amending the definition of a Lie category
when the object manifold has a boundary (or corners).

In this case,
$\underline{\mathcal{C}}\subset\mathcal{G}(\mathcal{C})\times\mathcal{C}$ is
still a submanifold (by Proposition A.4) and the maps $\underline{s}$ and
$\underline{t}$ are topological submersions (see Definition A.1) that also
satisfy
$\underline{s}(\partial\underline{\mathcal{C}})\subset\partial\mathcal{C}$ and
$\underline{t}(\partial\underline{\mathcal{C}})\subset\partial\mathcal{C}$, so
[corners, Proposition 4.2.1] implies that $\underline{s}$-fibres and
$\underline{t}$-fibres are submanifolds of $\underline{\mathcal{C}}$. This
suggests a definition of a Lie category for the case when the object manifold
has corners, but we will not pursue this further here.

We can now use the action Lie groupoids above to prove that it does not matter
at which invertible morphism we measure the ranks of a given morphism
$g\in\mathcal{C}$.

######  Corollary 5.16.

Let $\mathcal{C}\rightrightarrows\mathcal{X}$ be a Lie category with a normal
boundary. For any $g\in\mathcal{C}$, there holds:

$\displaystyle\operatorname{rank}^{L}(g)$
$\displaystyle=\operatorname{rank}\operatorname{d}\\!{(}L_{g})_{h}\ \text{ for
any }h\in\mathcal{G}(\mathcal{C})^{s(g)},$
$\displaystyle\operatorname{rank}^{R}(g)$
$\displaystyle=\operatorname{rank}\operatorname{d}\\!{(}R_{g})_{h}\ \text{ for
any }h\in\mathcal{G}(\mathcal{C})_{t(g)}.$

###### Proof.

Consider first the case when $\partial\mathcal{C}=\emptyset$. The restriction
of the target map to any source fibre has constant rank, in any Lie
groupoid.444See proof of Corollary 6.9 for a method of proving this. In our
case, $\underline{s}^{-1}(g)=\mathcal{G}(\mathcal{C})_{t(g)}$ for any
$g\in\mathcal{C}$, and the map

$\underline{t}|_{\underline{s}^{-1}(g)}=R_{g}|_{\mathcal{G}(\mathcal{C})_{t(g)}}$

has constant rank which must thus be equal to $\operatorname{rank}^{R}(g)$. A
similar proof works for left translations, using the right action of
$\mathcal{G}(\mathcal{C})$ on $s\colon\mathcal{C}\rightarrow\mathcal{X}$.

Now suppose $\partial\mathcal{C}\neq\emptyset$. We may consider the left
action groupoids

$\mathcal{G}(\mathcal{C})\tensor[_{s}]{\times}{{}_{t}}\operatorname{Int}\mathcal{C}\rightrightarrows\operatorname{Int}\mathcal{C},\quad\mathcal{G}(\mathcal{C})\tensor[_{s}]{\times}{{}_{t}}\partial\mathcal{C}\rightrightarrows\partial\mathcal{C}$

of $\mathcal{G}(\mathcal{C})$ on $t|_{\operatorname{Int}\mathcal{C}}$ and
$t|_{\partial\mathcal{C}}$, respectively; note that these are in fact actions
of $\mathcal{G}(\mathcal{C})$ since
$L_{g}(\operatorname{Int}\mathcal{C}^{s(g)})\subset\operatorname{Int}\mathcal{C}^{t(g)}$
and $L_{g}(\partial\mathcal{C}^{s(g)})\subset\partial\mathcal{C}^{t(g)}$ for
any $g\in\mathcal{G}(\mathcal{C})$. The same technique as above now shows the
wanted conclusion. ∎

### Singular distributions associated to translations

A Lie category $\mathcal{C}\rightrightarrows\mathcal{X}$ (assume it is
boundaryless for simplicity) comes equipped with singular distributions
determined by differentials of left and right translations. Focusing only on
left translations, define the singular distribution $D\subset T\mathcal{C}$ as

$D_{g}=\operatorname{Im}\operatorname{d}\\!{(}L_{g})_{1_{s(g)}}\leq\ker\operatorname{d}\\!{t}_{g}\leq
T_{g}\mathcal{C}.$

In what follows, we prove that $D$ is integrable, i.e. there is a
decomposition $\mathcal{F}(D)$ of $\mathcal{C}$ into maximally connected
weakly embedded submanifolds,555Recall that an injective immersion
$\varphi\colon M\rightarrow N$ is said to be a weak embedding, if any smooth
map $f\colon P\rightarrow N$ with the property $f(P)\subset\varphi(M)$,
factors through $\varphi$. called the leaves of $\mathcal{F}(D)$, whose
tangent spaces coincide with the fibres of $D$.

Denote the $D$-valued vector fields on $\mathcal{C}$ by

$\Gamma^{\infty}(D):=\\{X\in\mathfrak{X}(\mathcal{C})\;|\;X_{g}\in D_{g}\text{
for all }g\in\mathcal{C}\\}.$

It is not hard to see that $D$ is locally of finite type, i.e. for any
$g\in\mathcal{C}$ there exist finitely many $X_{i}\in\Gamma^{\infty}(D)$ such
that:

  1. (i)

$D_{g}=\mathrm{Span}(X_{i}|_{g})_{i}$,

  2. (ii)

For any $X\in\Gamma^{\infty}(D)$ there exists a neighborhood $U$ of $g$ in
$\mathcal{C}$ and functions $f_{i}{}^{j}\in C^{\infty}(U)$, such that
$[X,X_{i}]_{h}=\sum_{j}f_{i}{}^{j}(h)X_{j}|_{h}$ for all $h\in U$.

Indeed, we may pick sections $(\alpha_{i})_{i}$ of $A^{L}(\mathcal{C})$, such
that they constitute a local frame on some neighborhood $V$ of $s(g)$ in
$\mathcal{X}$, and extend them to left-invariant vector fields $(X_{i})_{i}$.
The point (i) is clearly satisfied; to show (ii), denote $U=s^{-1}(V)$, and
note that that the set of left-invariant vector fields is closed under the Lie
bracket by Lemma 4.2, so on $U$ there holds

$[X_{i},X_{j}]=\textstyle\sum_{k}f_{ij}{}^{k}X_{k},$

for some functions $f_{ij}{}^{k}\in C^{\infty}(U)$, and since any
$X\in\Gamma^{\infty}(D|_{U})$ can be written as a $C^{\infty}(U)$-linear
combination of the tuple $(X_{i}|_{U})_{i}$, the rest follows by using the
Leibniz rule for the Lie bracket.

Since $D$ is locally of finite type, it is integrable by [sussman]. The
following proposition says that the leaves are precisely the connected
components of the orbits

$\left\\{\underline{s}(\underline{t}^{-1}(g))\;\middle|\;g\in\mathcal{C}\right\\}$

of the groupoid
$\mathcal{C}\tensor[_{s}]{\times}{{}_{t}}\mathcal{G}(\mathcal{C})\rightrightarrows\mathcal{C}$
corresponding to the right action of $\mathcal{G}(\mathcal{C})$ on
$s\colon\mathcal{C}\rightarrow\mathcal{X}$.

######  Proposition 5.17.

Let $\mathcal{C}\rightrightarrows\mathcal{X}$ be a Lie category without
boundary. The integral manifold of the singular distribution $D\subset
T\mathcal{C}$ through $g\in\mathcal{C}$ is
$L_{g}(\mathcal{G}(\mathcal{C})^{s(g)})$.

###### Proof.

Corollary 5.16 states $L_{g}|_{\mathcal{G}(\mathcal{C})^{s(g)}}$ has constant
rank, so rank theorem can be applied to deduce
$L_{g}(\mathcal{G}(\mathcal{C})^{s(g)})$ is an immersed submanifold of
$\mathcal{C}^{t(g)}$, the tangent space at $gh$ of which is the image of
differential $\operatorname{d}\\!{(}L_{g})_{h}$, for any
$h\in\mathcal{G}(\mathcal{C})^{s(g)}$. ∎

######  Remark 5.18.

In the case when $\mathcal{C}$ has a boundary, the last proposition is in
general not true, as is easily seen by considering the order category on
$\mathbb{R}$. In general, a natural candidate for the integral manifold of $D$
through $g$ is $L_{g}(U)$ for an appropriate small open neighborhood of
$1_{s(g)}$ in $\mathcal{C}^{s(g)}$, but proving such a general result is
harder since rank theorem is not true for manifolds with boundaries (without
additional assumptions on $L_{g}$).

##  6\. Extensions of Lie categories to Lie groupoids

We have witnessed interesting examples of Lie categories appear by restricting
to an embedded subcategory of a Lie groupoid, in spirit of Lemma 2.15. As we
will see next, in order to ascertain the main properties of such Lie
categories, the assumption of being embedded may be weakened to being
immersed.

######  Definition 6.1.

An extension to a Lie groupoid of a Lie category
$\mathcal{C}\rightrightarrows\mathcal{X}$ is a Lie groupoid
$\mathcal{G}\rightrightarrows\mathcal{X}$, together with an injective,
immersive functor $F\colon\mathcal{C}\rightarrow\mathcal{G}$ over the
identity. In other words, $\mathcal{G}$ is a Lie groupoid such that
$\mathcal{C}$ is its wide Lie subcategory. If such an extension exists, we say
that $\mathcal{C}$ is extendable to a Lie groupoid. Furthermore, we say that
an extension to a Lie groupoid is weakly étale, if
$\dim\mathcal{G}=\dim\mathcal{C}$.

######  Remark 6.2.

Note that if $\mathcal{C}$ and $\mathcal{G}$ do not have a boundary, then an
extension is weakly étale if, and only if, the map $F$ is étale (i.e. a local
diffeomorphism), by virtue of inverse map theorem. If
$\dim\mathcal{G}=\dim\mathcal{C}$, but
$F(\partial\mathcal{C})\not\subset\partial\mathcal{G}$, then $F$ is not a
local diffeomorphism (see e.g. the order category from Example 2.13).

An obvious necessary condition for an arbitrary category $\mathcal{C}$ to
admit an injective functor $F$ into a groupoid $\mathcal{G}$, is
cancellativity of all elements in $\mathcal{C}$, i.e. all left and right
translations in $\mathcal{C}$ must be injective. For example, if $gh=gk$ holds
for some morphisms in $\mathcal{C}$, then $F(g)F(h)=F(g)F(k)$ holds in
$F(\mathcal{C})\subset\mathcal{G}$, implying $F(h)=F(k)$, and hence $h=k$ by
injectivity of $F$, so $L_{g}$ is injective.

In the differentiable setting, extensions to groupoids reflect on ranks and
algebroids; the following lemma yields necessary conditions on a Lie category
$\mathcal{C}$ to be extendable to a Lie groupoid.

######  Lemma 6.3.

If a Lie category $\mathcal{C}$ is extendable to a Lie groupoid $\mathcal{G}$,
all its morphisms have full and constant rank, and all left and right
translations are injective. Moreover, if the extension is weakly étale, then
$A^{L}(\mathcal{C})\cong A(\mathcal{G})\cong A^{R}(\mathcal{C})$.

###### Proof.

Let $F\colon\mathcal{C}\rightarrow\mathcal{G}$ be the groupoid extension, and
let $g,h\in\mathcal{C}$ be composable. To prove that $g$ has full and constant
rank, denote the left translation in $\mathcal{C}$ by $g$ as
$L_{g}^{\mathcal{C}}$, and the left translation in $\mathcal{G}$ by $F(g)$ as
$L^{\mathcal{G}}_{F(g)}$. By functoriality, the diagram

${{\mathcal{C}^{s(g)}}}$${{\mathcal{C}^{t(g)}}}$${{\mathcal{G}^{s(g)}}}$${{\mathcal{G}^{t(g)}}}$$\scriptstyle{L_{g}^{\mathcal{C}}}$$\scriptstyle{L_{F(g)}^{\mathcal{G}}}$$\scriptstyle{F}$$\scriptstyle{F}$

commutes, so if $v\in T_{h}(\mathcal{C}^{s(g)})$ is such that
$\operatorname{d}\\!{(}L_{g}^{\mathcal{C}})_{h}(v)=0$, it implies
$\operatorname{d}\\!{F}_{h}(v)=0$, and so $v=0$ since $F$ is an immersion.
This proves that $\operatorname{d}\\!{(}L_{g}^{\mathcal{C}})_{h}$ has full
rank, and a similar proof shows an analogous result for the right translation.
Hence $g$ has full and constant rank.

To prove the second part, denote the source and target maps in $\mathcal{C}$
and $\mathcal{G}$ as $s^{\mathcal{C}},t^{\mathcal{C}}$ and
$s^{\mathcal{G}},t^{\mathcal{G}}$, respectively. Commutativity of the
following diagram,

${\mathcal{C}}$${\mathcal{G}}$${\mathcal{X}}$$\scriptstyle{F}$$\scriptstyle{t^{\mathcal{C}}}$$\scriptstyle{t^{\mathcal{G}}}$

together with the assumption that our extension is weakly étale, implies that
for any $x\in\mathcal{X}$, $\operatorname{d}\\!{F}_{1_{x}}$ maps
$\ker\operatorname{d}\\!{t}^{\mathcal{C}}_{1_{x}}$ isomorphically onto
$\ker\operatorname{d}\\!{t}^{\mathcal{G}}_{1_{x}}$, so $F$ induces an
isomorphism $F_{*}^{L}\colon A^{L}(\mathcal{C})\rightarrow A^{L}(\mathcal{G})$
of vector bundles, and similarly between $A^{R}(\mathcal{C})$ and
$A^{R}(\mathcal{G})$. Since $F$ is a morphism of Lie categories, these are in
fact isomorphisms of Lie algebroids, so we yield the wanted chain of
isomorphisms. ∎

######  Corollary 6.4.

If a Lie category is extendable to a Lie groupoid, then the composition map
$m\colon{\mathcal{C}}^{(2)}\rightarrow\mathcal{C}$ is a submersion.

###### Proof.

Follows directly from Corollary 5.9 and Lemma 6.3. ∎

######  Remark 6.5.

To conclude $A^{L}(\mathcal{C})\cong A^{R}(\mathcal{C})$, it is enough to
replace the assumption on the extension
$F\colon\mathcal{C}\rightarrow\mathcal{G}$ being weakly étale with the
following weaker condition:

$\displaystyle\operatorname{d}\\!{(}\mathrm{inv})_{1_{x}}(\operatorname{d}\\!{F}_{1_{x}}(\ker\operatorname{d}\\!{t}_{1_{x}}^{\mathcal{C}}))\subset\operatorname{d}\\!{F}_{1_{x}}(\ker\operatorname{d}\\!{s}^{\mathcal{C}}_{1_{x}}),$
(6.6)

but notice that if $\dim\mathcal{G}\neq\dim\mathcal{C}$, then the left and
right algebroids of $\mathcal{C}$ will not be isomorphic to the algebroid
$A(\mathcal{G})$, since their ranks will differ. For simpler notation, assume
that $\mathcal{C}\subset\mathcal{G}$ and
$F\colon\mathcal{C}\hookrightarrow\mathcal{G}$ is an injective immersion.

To show that
$\operatorname{d}\\!{(}\mathrm{inv})_{1_{x}}(\ker\operatorname{d}\\!{t}_{1_{x}}^{\mathcal{C}})\subset\ker\operatorname{d}\\!{s}^{\mathcal{C}}_{1_{x}}$
implies $A^{L}(\mathcal{C})\cong A^{R}(\mathcal{C})$, first observe that the
assumption (6.6) implies equality instead of just the inclusion. Hence the
inversion map induces a vector bundle isomorphism

$\mathrm{inv}_{*}\colon A^{L}(\mathcal{C})\rightarrow A^{R}(\mathcal{C}),$

and now it is not difficult to see that for any
$\alpha\in\Gamma^{\infty}(A^{L}(\mathcal{C}))$, its left-invariant extension
$\alpha^{L}$ to whole $\mathcal{G}$ is $\mathrm{inv}$-related to the right-
invariant extension $(\mathrm{inv}_{*}\alpha)^{R}$. Hence if
$\beta\in\Gamma^{\infty}(A^{L}(\mathcal{C}))$ is another section, we obtain
that for any $g\in\mathcal{G}$,

$\operatorname{d}\\!{(}\mathrm{inv})_{g}([\alpha^{L},\beta^{L}]_{g})=[(\mathrm{inv}_{*}\alpha)^{R},(\mathrm{inv}_{*}\beta)^{R}]_{g^{-1}}.$

Taking $g=1_{x}$ shows that $\mathrm{inv}_{*}$ preserves the brackets of
$A^{L}(\mathcal{C})$ and $A^{R}(\mathcal{C})$, so it is an isomorphism of Lie
algebroids.

######  Remark 6.7.

The importance of Lemma 6.3 is in the fact that it enables us to easily
provide positive answers to the following questions, for the case of Lie
categories extendable to groupoids:

  1. (i)

Are Lie monoids parallelizable?

  2. (ii)

For fixed objects $x,y\in\mathcal{X}$ of a Lie category
$\mathcal{C}\rightrightarrows\mathcal{X}$, is the set $\mathcal{C}_{x}^{y}$ of
morphisms from $x$ to $y$ a submanifold of $\mathcal{C}$? Equivalently, do
Hom-functors map into the category $\mathbf{Diff}$ instead of just
$\mathbf{Set}$?

We remark that these questions remain open for Lie categories which do not
admit an extension to a Lie groupoid.

######  Corollary 6.8.

If a Lie monoid $M$ is extendable to a Lie group, it is paralellizable.

###### Proof.

Extending a basis of $T_{e}M$ to a tuple of left-invariant vector fields
yields a global frame for $TM$ since all elements of $M$ have full rank by
Lemma 6.3. ∎

The proof of the positive answer to (ii) is similar as in the Lie groupoid
case:

######  Corollary 6.9.

If a Lie category $\mathcal{C}\rightrightarrows\mathcal{X}$ is extendable to a
Lie groupoid, then for any $x,y\in\mathcal{X}$ the set $\mathcal{C}_{x}^{y}$
is a closed embedded submanifold of $\mathcal{C}$. In particular,
$\mathcal{C}_{x}^{x}$ is a Lie monoid for any $x\in\mathcal{X}$.

###### Proof.

We realize $\mathcal{C}_{x}^{y}$ as an integral manifold of a certain
distribution on $\mathcal{C}_{x}$, namely
$D=\ker\operatorname{d}\\!{(}t|_{\mathcal{C}_{x}})$, or more instructively,
$D_{g}=\ker\operatorname{d}\\!{t}_{g}\cap\ker\operatorname{d}\\!{s}_{g}$ for
all $g\in\mathcal{C}_{x}$. This is a regular distribution on
$\mathcal{C}_{x}$, since
$D_{g}=\operatorname{d}\\!{(}L_{g})_{1_{x}}(D_{1_{x}})$ holds – the latter is
a consequence of the equality $s|_{\mathcal{C}^{t(g)}}\circ
L_{g}=s|_{\mathcal{C}^{x}}$ and $g$ having full left rank by Lemma 6.3, which
moreover implies that $D$ is a trivial vector subbundle of $T\mathcal{C}_{x}$.

Since $D$ is the kernel of a differential of a smooth map, it is involutive,
so by Frobenius’ theorem integrable. The leaves of the corresponding foliation
are the connected components of subspaces
$\\{\mathcal{C}_{x}^{y}\;|\;y\in\mathcal{X}\\}$ of $\mathcal{C}_{x}$, so they
are its initial submanifolds. Since the subspaces $\mathcal{C}_{x}^{y}$ are
also closed in $\mathcal{C}_{x}$, they are embedded. ∎

##  7\. Completeness of invariant vector fields

It is well-known that on any Lie group $G$, left-invariant vector fields are
complete. In this section, we generalize this result to Lie monoids with
normal boundaries. Furthermore, we generalize the characterization of
completeness of left-invariant vector fields on Lie groupoids to Lie
categories with normal boundaries. At last, we will discuss the exponential
map for Lie monoids.

######  Definition 7.1.

Let $X$ be a vector field on a smooth manifold $M$ with or without boundary,
and denote by $J^{X}_{g}$ the maximal interval on which the integral path
$\gamma^{X}_{g}$ of $X$, starting at $g\in M$, is defined. We say that $X$ is
half-complete if for any $g\in M$ there either holds $[0,\infty)\subset
J^{X}_{g}$ or $(-\infty,0]\subset J^{X}_{g}$.

######  Remark 7.2.

In what follows, we will assume the reader is familiar with the notion of
inward-pointing and outward-pointing tangent vectors on the boundary; we
direct to [lee, p. 118] for a basic reference. A particularly useful
observation is that a vector field $X\in\mathfrak{X}(M)$ which is inward-
pointing (or outward-pointing) at a certain point $g\in\partial M$, must
remain inward-pointing (or outward pointing) on a neighborhood of $g$ in
$\partial M$, from which it follows that $J_{g}^{X}$ cannot contain an open
neighborhood of zero, but may only contain a half-closed interval
$[0,\varepsilon)$, for some $\varepsilon>0$ (or $(-\varepsilon,0]$ in the
outward-pointing case).

Recall from Definition 3.9 that a Lie monoid $M$ has a normal boundary, if
either $e\in\operatorname{Int}M$, or we have both that $e\in\partial M$ and
$\partial M$ is a submonoid of $M$.

######  Theorem 7.3.

Let $M$ be a Lie monoid with a normal boundary and let $X$ be a left-invariant
vector field on $M$. The following holds:

  1. (i)

Suppose either that $e\in\operatorname{Int}M$, or that $e\in\partial M$ and
$X_{e}$ is tangent to $\partial M$. Then $X|_{\partial M}$ is tangent to
$\partial M$, and $X$ is complete.

  2. (ii)

Suppose $e\in\partial M$ and $X_{e}$ is either inward-pointing or outward-
pointing. Then either $J_{e}^{X}=[0,\infty)$ or $J_{e}^{X}=(-\infty,0]$,
respectively, and $X$ is half-complete.

Moreover, the flow of $X$ is given for all $t\in J_{e}^{X}$ by
$\phi_{t}^{X}=R_{\phi_{t}^{X}(e)}$.

###### Proof.

We first inspect the assumptions from (i): notice that if
$e\in\operatorname{Int}M$, then $J_{e}^{X}$ clearly contains an open interval
around zero; on the other hand, if $e\in\partial M$ and $X_{e}$ is tangent to
$\partial M$, then $X|_{\partial M}$ must be everywhere tangent to $\partial
M$ by left-invariance of $X$ and the fact that $\partial M$ is a Lie submonoid
of $M$, so $J_{e}^{X}$ must again contain an open interval around zero.
Furthermore, the assumption from (ii) that $X_{e}$ is either inward-pointing
or outward-pointing implies that $J_{e}^{X}$ contains a half-open interval of
the form $[0,\varepsilon)$ or $(-\varepsilon,0]$, respectively.

We next observe that for any $g\in M$, the composition
$L_{g}\circ\gamma_{e}^{X}$ is an integral path of $X$ starting at $g$, so
maximality of $J_{g}^{X}$ implies

$\displaystyle J_{e}^{X}\subset J_{g}^{X},$ (7.4)

and also $\phi_{t}^{X}(g)=g\gamma_{e}^{X}(t)=R_{\gamma_{e}^{X}(t)}(g)$ for all
$t\in J_{e}^{X}$. Notice that (7.4) now implies that in the case
$e\in\operatorname{Int}M$, $J_{g}^{X}$ contains an open interval around zero
for all $g\in M$, and in particular this holds for any $g\in\partial M$, so we
conclude that $X|_{\partial M}$ must be tangent to $\partial M$, by virtue of
Remark 7.2.

Let $\tau\in J_{e}^{X}$ and consider the affinely translated path

$\zeta^{\tau}\colon(J_{e}^{X}-\tau)\rightarrow
M,\quad\zeta^{\tau}(t)=\gamma_{e}^{X}(t+\tau).$

Since the maximal domain of an affinely translated integral path is just the
affinely translated maximal domain, we get
$J_{e}^{X}-\tau=J_{\zeta^{\tau}(0)}^{X}$.666We do not need the Lie monoid
structure to prove this simple fact. Together with (7.4), this implies

$J_{e}^{X}+\tau\subset J_{e}^{X},\ \text{for all }\tau\in J_{e}^{X}.$

This implies that if $J_{e}^{X}$ contains an open interval around zero, it
must equal $\mathbb{R}$, and if $[0,\varepsilon)\subset J_{e}^{X}$ for some
$\varepsilon>0$, then $[0,\infty)\subset J_{e}^{X}$; similarly for the
outward-pointing case. Together with equation (7.4), this proves our claims
regarding completeness and half-completeness. ∎

######  Corollary 7.5.

If $M$ is a Lie monoid and $e\in\operatorname{Int}M$, then elements in
$\partial M$ do not have full rank. Hence if also $\partial M\neq\emptyset$,
then $M$ is not extendable to a Lie group.

###### Proof.

For a proof by contradiction, assume that there is a $g\in\partial M$ with
full rank, and pick any inward-pointing vector $v\in T_{g}M$. Since $g$ has
full rank, the vector $v$ is extendable to a unique left-invariant vector
field $X$ on $M$, but now $X|_{\partial M}$ must be tangent to $\partial M$ by
Theorem 7.3, contradicting our assumption that $v$ is inward-pointing. The
second part of the corollary follows from Lemma 6.3. ∎

An easy example of the last corollary in play is the Lie monoid $[0,\infty)$
for multiplication. This result may easily be generalized; in any Lie category
$\mathcal{C}\rightrightarrows\mathcal{X}$, a similar inclusion as (7.4) holds:

$\displaystyle J_{1_{s(g)}}^{X}\subset J_{g}^{X},$ (7.6)

for any $g\in\mathcal{C}$ and any left-invariant vector field
$X\in\mathfrak{X}^{L}(\mathcal{C})$. The same argument as before shows that if
$u(\mathcal{X})\subset\operatorname{Int}\mathcal{C}$, then
$X|_{\partial\mathcal{C}}$ is tangent to $\partial\mathcal{C}$, so we
similarly obtain:

######  Corollary 7.7.

If $\mathcal{C}\rightrightarrows\mathcal{X}$ is a Lie category with
$u(\mathcal{X})\subset\operatorname{Int}\mathcal{C}$, then morphisms in
$\partial\mathcal{C}$ do not have full rank. Hence if
$u(\mathcal{X})\subset\operatorname{Int}\mathcal{C}$ and
$\partial\mathcal{C}\neq\emptyset$, then $\mathcal{C}$ is not extendable to a
Lie groupoid.

###### Proof.

As before, suppose there is a morphism $g\in\partial\mathcal{C}$ with full
rank. Regularity of the boundary implies
$\partial(\mathcal{C}^{t(g)})=\partial\mathcal{C}\cap\mathcal{C}^{t(g)}$, so
we may pick an inward-pointing vector $v\in\ker\operatorname{d}\\!{t}_{g}$.
Since $g$ has full rank, we can form the vector
$\operatorname{d}\\!{(}L_{g})_{1_{s(g)}}^{-1}(v)\in\ker\operatorname{d}\\!{t}_{1_{s(g)}}$,
extend it to a section of $u^{*}\ker\operatorname{d}\\!{t}$ using partitions
of unity, and finally extend it to a left-invariant vector field $X$ on
$\mathcal{C}$. Since $X_{g}=v$, we arrive to a contradiction to the fact that
$X|_{\partial\mathcal{C}}$ is tangent to $\partial\mathcal{C}$. ∎

######  Remark 7.8.

In the case when $g$ is an invertible morphism of a Lie category, the
inclusion (7.6) is actually an equality, which follows from the fact that
$L_{g^{-1}}\circ\gamma^{X}_{g}$ is an integral path of $X$ starting at
$1_{s(g)}$.

We now present the promised characterization of completeness of invariant
vector fields on a Lie category. The proof is a small adaptation of the one
from the theory of Lie groupoids.

######  Proposition 7.9.

Let $\mathcal{C}\rightrightarrows\mathcal{X}$ be a Lie category with a normal
boundary, and let $\alpha\in\Gamma^{\infty}(A^{L}(\mathcal{C}))$ be a section
of its left Lie algebroid. Suppose either
$u(\mathcal{X})\subset\operatorname{Int}\mathcal{C}$, or that
$u(\mathcal{X})\subset\partial\mathcal{C}$ and $\alpha^{L}|_{u(\mathcal{X})}$
is tangent to $\partial\mathcal{C}$. Then
$\alpha^{L}\in\mathfrak{X}^{L}(\mathcal{C})$ is complete if, and only if,
$\rho^{L}(\alpha)\in\mathfrak{X}(\mathcal{X})$ is complete.

###### Proof.

First note we have already argued that under given assumptions, the
restriction $\alpha^{L}|_{\partial\mathcal{C}}$ is tangent to
$\partial\mathcal{C}$, implying that the maximal domain of any integral path
of $\alpha^{L}$ is an open interval.

For the forward implication, note that $\alpha^{L}$ and $\rho^{L}(\alpha)$ are
$s$-related, so that if $\phi^{\alpha^{L}}_{t}$ is defined for some
$t\in\mathbb{R}$, then so is $\phi^{\rho^{L}(\alpha)}_{t}$, by the fact that
$s$ is a surjective submersion.

${\mathcal{C}}$${\mathcal{C}}$${\mathcal{X}}$${\mathcal{X}}$$\scriptstyle{s}$$\scriptstyle{\phi^{\alpha^{L}}_{t}}$$\scriptstyle{\phi^{\rho^{L}(\alpha)}_{t}}$$\scriptstyle{s}$

Indeed, take an open cover $(U_{i})_{i}$ of $\mathcal{X}$ by domains of local
sections $\sigma_{i}\colon U_{i}\rightarrow\mathcal{C}$ of $s$, and define
$\vartheta_{i}\colon\mathbb{R}\times U_{i}\rightarrow\mathcal{X}$ as
$\vartheta_{i}(t,x)=(s\circ\phi_{t}^{\alpha^{L}}\circ\sigma)(x)$. It is
straightforward to check that for any $x\in U_{i}$, the map
$t\mapsto\vartheta_{i}(t,x)$ is the integral path of $\rho^{L}(\alpha)$
starting at $x$, so the maps $\vartheta_{i}$ collate to the global flow
$\phi^{\rho^{L}(\alpha)}\colon\mathbb{R}\times\mathcal{X}\rightarrow\mathcal{X}$
of $\rho^{L}(\alpha)$, by uniqueness of integral paths.

For the converse implication, suppose $\rho^{L}(\alpha)$ is complete, and let
$\gamma\colon(a,b)\rightarrow\mathcal{C}$ be an integral path of $\alpha^{L}$;
then $s\circ\gamma$ is an integral path of $\rho^{L}(\alpha)$, thus
$s\circ\gamma$ admits a unique extension to an integral path of
$\rho^{L}(\alpha)$ defined on whole $\mathbb{R}$, which we again denote by
$s\circ\gamma$, so the expression $(s\circ\gamma)(b)$ is defined. There now
exists a path
$\delta\colon(b-\varepsilon,b+\varepsilon)\rightarrow\mathcal{C}$, which is an
integral path of $\alpha^{L}$, such that $\delta(b)=1_{(s\circ\gamma)(b)}$;
importantly, we may assume that $\varepsilon$ is small enough that $\delta$
maps into $\mathcal{G}(\mathcal{C})$, since the latter is either open in
$\operatorname{Int}\mathcal{C}$ or in $\partial\mathcal{C}$ by Theorem 3.11.
We define our wanted extension
$\bar{\gamma}\colon(a,b+\varepsilon)\rightarrow\mathcal{C}$ of $\gamma$ as

$\bar{\gamma}(t)=\begin{cases}\gamma(t)&\text{if }t\in(a,b),\\\
\gamma(b-\frac{\varepsilon}{2})\delta(b-\frac{\varepsilon}{2})^{-1}\delta(t)&\text{if
}t\in(b-\varepsilon,b+\varepsilon).\end{cases}$

Since both $s\circ\gamma$ and $s\circ\delta$ are integral paths of
$\rho^{L}(\alpha)$ valued $(s\circ\gamma)(b)$ at $b$, they coincide on their
common domain, so that
$s(\gamma(b-\frac{\varepsilon}{2}))=s(\delta(b-\frac{\varepsilon}{2}))$, and
moreover $\delta$ must lie in $\mathcal{C}^{(s\circ\gamma)(b)}$ since
$\alpha^{L}$ is tangent to $t$-fibres, which altogether implies that the
multiplication in the definition of $\bar{\gamma}$ is well-defined. Finally,
$\bar{\gamma}$ is in fact an integral path of $\alpha^{L}$ since for any
$t\in(b-\varepsilon,b+\varepsilon)$ there holds:

$\bar{\gamma}^{\prime}(t)=\operatorname{d}\\!{\big{(}L_{\gamma(b-\frac{\varepsilon}{2})\delta(b-\frac{\varepsilon}{2})^{-1}}\big{)}}_{\delta(t)}(\delta^{\prime}(t))=\alpha^{L}_{\gamma(b-\frac{\varepsilon}{2})\delta(b-\frac{\varepsilon}{2})^{-1}\delta(t)}=\alpha^{L}_{\bar{\gamma}(t)}.$

By uniqueness of integral paths, the two partial definitions of $\bar{\gamma}$
coincide on their common domain, from which we conclude that $\bar{\gamma}$
indeed extends $\gamma$. Similarly can be done for the other endpoint, showing
that the integral path $\gamma$ can be extended to the whole $\mathbb{R}$. ∎

Let us now turn back to Lie monoids. As a corollary of Theorem 7.3, we can
generalize the exponential map from the theory of Lie groups to Lie monoids.
First off, the following says that when $e\in\operatorname{Int}M$, the map
$T_{e}M\rightarrow M$, defined as $v\mapsto\phi^{v^{L}}_{1}(e)$ is just the
usual exponential map on $G(M)$.

######  Corollary 7.10.

Let $M$ be a Lie monoid with $e\in\operatorname{Int}M$. The image of the
integral path $\gamma_{e}^{X}$ of any left-invariant vector field $X$ on $M$
is contained in the core $G(M)$.

###### Proof.

By Theorem 7.3, there holds
$\phi_{-t}^{X}(e)\phi_{t}^{X}(e)=\phi_{t}^{X}(\phi_{-t}^{X}(e))=e$ for any
$t\in\mathbb{R}$, so $\phi_{-t}^{X}(e)$ is the inverse of $\phi_{t}^{X}(e)$. ∎

On the other hand, Theorem 7.3 also enables us to define the exponential map
for Lie monoids with boundaries:

######  Definition 7.11.

Let $M$ be a Lie monoid such that $\partial M$ is its submonoid, $e\in\partial
M$. Let $T_{e}^{\scalebox{0.6}{$\bm{+}$}}M\subset T_{e}M$ denote the subset
consisting of inward-pointing vectors in $T_{e}M$ and the vectors in
$T_{e}(\partial M)$. The exponential map on $M$ is then defined as

$\displaystyle\exp\colon T_{e}^{\scalebox{0.6}{$\bm{+}$}}M\rightarrow
M,\quad\exp(v)=\phi^{v^{L}}_{1}(e).$

######  Remark 7.12.

In any boundary chart centered at $e$, $T_{e}^{\scalebox{0.6}{$\bm{+}$}}M$ is
identified with the closed upper half-space $\mathbb{H}^{\dim M}$. We observe
that $T_{e}^{\scalebox{0.6}{$\bm{+}$}}M$ possesses an algebraic structure of a
semimodule over a semiring $[0,\infty)$.777A semiring $R$ satisfies all the
axioms of a ring, except the existence of additive inverses. Because of this,
we must additionally impose $0\cdot a=0=a\cdot 0$ for all $a\in R$. We amend
modules to obtain semimodules in precisely the same way.

As before, Corollary 7.10 ensures that the restriction of $\exp$ to the
vectors tangent to $\partial M$, is precisely the usual exponential map of the
Lie group $G(M)$.

Similar results to those from the theory of Lie groups can be obtained for the
exponential map as defined above, by using an identical approach to the
respective proofs, but working with one-sided derivatives. We leave the
following results for the reader as an exercise:

  1. (i)

Let $v\in T^{\scalebox{0.6}{$\bm{+}$}}_{e}M$ and $t\geq 0$. By rescaling
lemma, $\phi^{v^{L}}_{tr}(e)=\phi_{r}^{tv^{L}}(e)$ for all $r\geq 0$, and
setting $r=1$ we obtain

$\exp(tv)=\phi_{t}^{v^{L}}(e).$

  2. (ii)

The map $\exp$ is a smooth, and there holds
$\operatorname{d}\\!{(\exp)}_{e}=\mathrm{id}_{T_{e}M}$. Since $\exp$ maps
$\exp(T_{e}(\partial M))\subset\partial M$ by Theorem 7.3 (i), one can apply
the inverse map theorem for maps between manifolds with boundaries to conclude
that $\exp$ is a local diffeomorphism at the point $0\in
T_{e}^{\scalebox{0.6}{$\bm{+}$}}M$.

  3. (iii)

Considering the Lie monoid $[0,\infty)$ for addition, any smooth homomorphism
$\alpha\colon[0,\infty)\rightarrow M$ of Lie monoids is called a one-
parametric submonoid of $M$. For any $v\in T_{e}^{\scalebox{0.6}{$\bm{+}$}}M$,
the map $t\mapsto\exp(tv)$ is a one-parametric submonoid of $M$, and any one-
parametric submonoid $\alpha$ can be written as
$\alpha(t)=\exp(t\dot{\alpha}(0))$, where $\dot{\alpha}(0)$ denotes the one-
sided derivative of $\alpha$ at zero.

A consequence of point (iii) above is naturality of $\exp$, i.e. if
$\phi\colon M\rightarrow N$ is a morphism between Lie monoids with normal
boundaries, such that the units of $M$ and $N$ are contained in the respective
boundaries, then for any $v\in T_{e}^{\scalebox{0.6}{$\bm{+}$}}M$, the map
$\alpha(t)=\phi(\exp_{M}(tv))$ defines a one-parametric submonoid of $N$ with
$\dot{\alpha}(0)=\operatorname{d}\\!{\phi}(v)$, so we obtain that the
following diagram commutes.

${M}$${N}$${{T_{e}^{\scalebox{0.6}{$\bm{+}$}}M}}$${{T_{e}^{\scalebox{0.6}{$\bm{+}$}}N}}$$\scriptstyle{\phi}$$\scriptstyle{\operatorname{d}\\!{\phi}}$$\scriptstyle{\exp_{M}}$$\scriptstyle{\exp_{N}}$

##  8\. An application to physics: Statistical Thermodynamics

The interpretation that morphisms correspond to physical processes, and
objects to physical states, can be applied to yield a rigorous approach to
statistical physics, which we will now demonstrate. We will first focus purely
on categorical aspects, and then consider differentiability.

Suppose we are given an isolated physical system consisting of an unknown
number of particles, each of which can be in one of the $n+1$ a-priori given
microstates, which we will index by

$i\in\\{0,\dots,n\\}.$

Since the number of particles in our system is unknown and often large, we
need to work with tuples of probabilities $(p_{0},\dots,p_{n})$, where each
$p_{i}$ is the probability that a particle, chosen at random, is in the $i$-th
microstate. Any tuple of probabilities $(p_{i})_{i=0}^{n}$ is subjected to the
constraint

$\textstyle\sum_{i}p_{i}=1,$

and we will refer to any such tuple $p=(p_{i})_{i}$ as a configuration of the
system, which is just a probability distribution on the finite set of
microstates above. The set of all configurations of our system is thus the
standard $n$-simplex,

$\Delta^{n}=\big{\\{}(p_{0},\dots,p_{n})\in[0,1]^{n+1}\;\big{|}\;\textstyle\sum_{i}p_{i}=1\big{\\}}.$

We associate to any configuration $(p_{i})_{i}$ of our system its expected
surprise,

$S(p_{i})_{i}=-\sum_{i}p_{i}\log p_{i},$

which is called the entropy of the configuration $(p_{i})_{i}$. Letting
$f(x)=x\log x$, we find $\lim_{x\rightarrow
0^{\scalebox{0.6}{$\bm{+}$}}}f(x)=0$, so $f$ may be extended to $[0,\infty)$
by defining $f(0)=0$, implying that $S\colon\Delta^{n}\rightarrow\mathbb{R}$
is defined on whole $\Delta^{n}$ and continuous.

The construction of the space of morphisms between different configurations of
our system is now an application of the second law of thermodynamics:

> A process in an isolated physical system is feasible if, and only if, the
> change of entropy pertaining to the process is non-negative.

In accord with the second law, we define

$\mathcal{D}=\left\\{(p_{i})_{i}\rightarrow(q_{i})_{i}\;\middle|\;S(q_{i})_{i}-S(p_{i})_{i}\geq
0\right\\}.$

In other words, $\mathcal{D}$ consists of pairs
$(q,p)\in\Delta^{n}\times\Delta^{n}$ of configurations, such that the entropy
of the target configuration $q$ is no less than that of the source $p$. That
$\mathcal{D}\rightrightarrows\Delta^{n}$ is a category follows from the fact
that the map

$\delta S\colon\Delta^{n}\times\Delta^{n}\rightarrow\mathbb{R},\quad\delta
S(q,p)=S(q)-S(p),$

is a functor888This is aligned with the moral that entropy should be
inherently perceived as a categorical concept, which was first adopted by Baez
et al.; $\delta S$ is in fact the only map (up to a multiplicative scalar)
which is functorial, convex-linear and continuous, see [baez2011] for details.
from the pair groupoid of $\Delta^{n}$ to the group $\mathbb{R}$ for addition.
Notice that the invertible morphisms in $\mathcal{D}$ are precisely $\delta
S^{-1}(0)$, which are just the processes with zero entropy change.

In the differentiable setting, we need to make certain adjustments to our
category $\mathcal{D}\rightrightarrows\Delta^{n}$, since it is not a Lie
category. First, we note that $\Delta^{n}$ is a manifold with corners, and $S$
is not smooth at its boundary $\partial\Delta^{n}$ since $\lim_{x\rightarrow
0^{\scalebox{0.6}{$\bm{+}$}}}(x\log x)^{\prime}=-\infty$, which is why we
first restrict our attention to the interior $\operatorname{Int}\Delta^{n}.$
Secondly, we notice that the category $\mathcal{D}\rightrightarrows\Delta^{n}$
has a terminal object, as shown by the following.

######  Claim.

The only critical point of entropy $S|_{\operatorname{Int}\Delta^{n}}$ is
given by the so-called microcanonical configuration, i.e.
$p_{i}^{\mu}=\frac{1}{n+1}$ for all $i$. This is a maximum of $S$.

###### Proof.

Constraint $\sum_{i}p_{i}=1$ implies $\sum_{i}\operatorname{d}\\!{p}_{i}=0$
and $\operatorname{d}\\!{p}_{0}=-\sum_{i=1}^{n}\operatorname{d}\\!{p}_{i}$, so
we have

$\operatorname{d}\\!{S}_{(p_{i})_{i}}=-\sum_{i}(1+\log
p_{i})\operatorname{d}\\!{p}_{i}=-\sum_{i}\log
p_{i}\operatorname{d}\\!{p}_{i}=-\sum_{i=1}^{n}(\log p_{i}-\log
p_{0})\operatorname{d}\\!{p}_{i},$

which vanishes if, and only if, $p_{i}=p_{0}$ for all $i$, i.e.
$p_{i}=\frac{1}{n+1}$. That this is a local maximum is left as an exercise,
and it is not hard to see that the value of $S$ at $(p_{i}^{\mu})_{i}$ is
greater than the value of $S$ on $S|_{\partial\Delta^{n}}$. ∎

The configuration $p^{\mu}$ has little physical importance – for example, we
will see below that it can be interpreted as the configuration that is
attained at thermodynamical equilibrium at infinite temperature. We will thus
remove it from the interior of our $n$-simplex of objects, and define our
space of objects to be

$\mathcal{X}=\operatorname{Int}\Delta^{n}-\\{p^{\mu}\\},$

which is a smooth manifold without boundary. Moreover, we define the space of
morphisms over $\mathcal{X}$ as the set

$\mathcal{C}=(\delta
S|_{\mathcal{X}\times\mathcal{X}})^{-1}([0,\infty))=\left\\{(q,p)\in\mathcal{X}\times\mathcal{X}\;\middle|\;S(q)-S(p)\geq
0\right\\}.$

As before, $\mathcal{C}\rightrightarrows\mathcal{X}$ is a subcategory of the
pair groupoid on $\mathcal{X}$. By virtue of Example 2.14, to prove that
$\mathcal{C}\rightrightarrows\mathcal{X}$ is a Lie category, we first need to
check $\delta S|_{\mathcal{X}\times\mathcal{X}}$ has a regular value 0. By
above claim, the only critical point of the map $\delta S$ is the identity
morphism $(p^{\mu},p^{\mu})\in\mathcal{D}$, which is not in $\mathcal{C}$.

Secondly, we have to show that $\mathcal{C}\rightrightarrows\mathcal{X}$ has a
regular boundary, i.e. that
$s|_{\partial\mathcal{C}},t|_{\partial\mathcal{C}}$ are submersions. To this
end, first note that for any $(q,p)\in\partial\mathcal{C}$,

$\displaystyle T_{(q,p)}\partial\mathcal{C}=\ker\operatorname{d}\\!{(}\delta
S)_{(q,p)}$ $\displaystyle=\left\\{(v,w)\in T_{q}\mathcal{X}\oplus
T_{p}\mathcal{X}\;\middle|\;\textstyle\sum_{i=1}^{n}(\log q_{i}-\log
q_{0})v_{i}=\sum_{i=1}^{n}(\log p_{i}-\log p_{0})w_{i}\right\\}.$

Let $v\in T_{q}\mathcal{X}$. Since $p_{i}\neq p_{0}$ for some $i$, we define

$w_{i}=\frac{\textstyle\sum_{i=1}^{n}(\log q_{i}-\log q_{0})v_{i}}{\log
p_{i}-\log p_{0}}$

and now we let $w=(w_{0},0,\dots,0,w_{i},0,\dots,0)$ where $w_{0}=-w_{i}$ is
set to ensure $w\in T_{p}\mathcal{X}$, thus we obtain the wanted pair
$(v,w)\in T_{(q,p)}\partial\mathcal{C}$ with $\operatorname{d}\\!{t}(v,w)=v$;
we similarly show that $s|_{\partial\mathcal{C}}$ is a submersion.
Furthermore, $\mathcal{C}\rightrightarrows\mathcal{X}$ satisfies all the
conclusions from Lemma 6.3, since the pair groupoid
$\mathcal{X}\times\mathcal{X}$ is its weakly étale extension; in particular,
the left and right Lie algebroids of $\mathcal{C}$ are isomorphic to
$T\mathcal{X}\approx\mathcal{X}\times\mathbb{R}^{n}$.

The question interesting for physics is: what is the configuration
$p^{\epsilon}\in\mathcal{X}$ at which the system attains a thermodynamical
equilibrium? It is well-known that the answer to this question is obtained
using the so-called Gibbs algorithm, which we provide here for completeness.
To this end, we need additional a-priori given data, namely each microstate
$i$ of our system has an a-priori assigned quantity $E_{i}$, called the energy
of $i$-th microstate. To derive the wanted configuration $p^{\epsilon}$ we
utilize the so-called principle of maximum entropy:

> A state is at thermodynamical equilibrium if, and only if, it maximizes the
> entropy with respect to the systemic constraints.

That is, we must find the constrained extremum of $S$ with respect to
constraints $\sum_{i}p_{i}=1$ and $\sum_{i}p_{i}E_{i}=E(p_{i})_{i}$. Here
$E\colon\mathcal{X}\rightarrow\mathbb{R}$ is a function on the object space,
determined by the first law of thermodynamics up to an additive constant, as
we will see below – this will impose thermodynamical considerations onto the
statistical description of our system.

To apply the method of Lagrange multipliers, define the function
$\hat{S}\colon\mathcal{X}\rightarrow\mathbb{R}$,

$\hat{S}(p_{i})_{i}=S(p_{i})_{i}-\lambda_{1}\left(E(p)-\textstyle\sum_{i}p_{i}E_{i}\right)-\lambda_{2}(1-\textstyle\sum_{i}p_{i}).$

Requiring $\frac{\partial\hat{S}}{\partial p_{i}}=0$ yields
$p_{i}=\frac{1}{Z}e^{\lambda_{1}E_{i}},$ where we have defined
$Z=e^{-(1+\lambda_{2})}$. The constraint $\textstyle\sum_{i}p_{i}=1$ then
reads

$Z=e^{-(1+\lambda_{2})}=\textstyle\sum_{i}e^{\lambda_{1}E_{i}}.$

On the other hand, the first law of thermodynamics for systems where no work
is exerted reads

$\operatorname{d}\\!{E}=kT\operatorname{d}\\!{S}$

where $T$ is the temperature (a macroscopic external constant) at which the
system is held, and $k$ denotes the Boltzmann constant. Writing out the total
differentials $\operatorname{d}\\!{E}$ and $\operatorname{d}\\!{S}$ gives
$\lambda_{1}=-\frac{1}{kT}$, and so we finally obtain the equilibrium
configuration:

$p_{i}^{\epsilon}=\frac{1}{Z}e^{-\frac{E_{i}}{kT}},\quad
Z=\sum_{i}e^{-\frac{E_{i}}{kT}}.$

We observe that in this categorical framework, the configurations from which
it is possible to attain the equilibrium configuration $p^{\epsilon}$, are now
simply expressible as
$s(t^{-1}({p^{\epsilon}}))=\left\\{p\in\mathcal{X}\;\middle|\;S(p^{\epsilon})\geq
S(p)\right\\}.$

## Further research

Although we hope to have succeeded in portraying the richness of Lie
categories and their potential in physics, we admit that we have not exhausted
all research options regarding them. We state some of them here, and note that
they provide possibilities for future research.

  1. (i)

As stated in Remark 6.7, the question remains whether all Lie monoids are
parallelizable, and whether $\mathcal{C}_{x}^{y}$ is a smooth manifold for
given objects $x,y\in\mathcal{X}$ of a Lie category
$\mathcal{C}\rightrightarrows\mathcal{X}$.

  2. (ii)

Does there exist a class of Lie algebroids that cannot be integrated to a Lie
groupoid, but can be integrated to a Lie category?

  3. (iii)

Remark 5.15 shows the need for considering Lie categories whose object
manifold has a boundary, or more generally, corners. Another example of such a
Lie category should be the flow of a smooth vector field on a manifold with
boundary, generalizing the flow Lie groupoid of a vector field on a
boundaryless manifold.

  4. (iv)

Infinite-dimensional Lie categories. The exterior bundle $\Lambda(E)$ of a
vector bundle $E$ is an example of a bundle of Lie monoids, and it is moreover
a subcategory of the tensor bundle $\oplus_{k=0}^{\infty}\otimes^{k}E$, which
is a bundle of monoids that fails to have finite-dimensional fibres. Moreover,
in statistical mechanics, physicists often work with an infinite number of
microstates, and in quantum mechanics with infinite-dimensional Hilbert
spaces. These examples show the need for introducing infinite-dimensional Lie
categories. Since the theory of infinite-dimensional (Banach) manifolds with
corners is already well-developed in [corners], a theory of infinite-
dimensional Lie categories with corners seems realizable.

  5. (v)

Multiplicative differential forms on Lie categories. On Lie groupoids, such
structures can be used to describe integrated counterparts of Poisson
structures [poisson], and more recently they have been used to provide a
natural generalization of connections on principle bundles in [mec]. We
suspect that interesting geometric structures can be described with
multiplicative differential forms on Lie categories.

  6. (vi)

Haar systems on Lie categories. Just as Haar systems on Lie groupoids provide
a generalization of a Haar measure on a Lie group (and connects the theory of
Lie groupoids to noncommutative geometry), a further generalization to Lie
categories should provide a means of equipping the space $C_{c}(\mathcal{C})$
of compactly supported functions on the space of arrows of a given category
with the convolution product. In this fashion, one expects to generalize the
construction of a groupoid C*-algebra, but due to the lack of existence of
inverses of arrows, there is a-priori no natural way of obtaining the
involution, hence the construction potentially generalizes only to a category
Banach algebra. We strongly suspect that a sensible notion of a Haar system on
a Lie category will rather be defined in terms of a measure $\mu$ on the space
${\mathcal{C}}^{(2)}$ of composible pairs of arrows, satisfying certain
invariance and continuity conditions, instead of being defined in terms of
left-invariant $t$-fibre supported measures on $\mathcal{C}$. Roughly
speaking, the convolution on $C_{c}(\mathcal{C})$ would then conceivably be
defined by
$(f_{1}*f_{2})(g)=\int_{m^{-1}(g)}f_{1}(g_{1})f_{2}(g_{2})\operatorname{d}\\!{\mu}(g_{1},g_{2}),$
where the integration is done over the set
$m^{-1}(g)=\\{(g_{1},g_{2})\in{\mathcal{C}}^{(2)}\;|\;g_{1}g_{2}=g\\}$ of
composible arrows which compose to $g\in\mathcal{C}$.

An interesting question for mathematical physics might be whether the
principle of least action in physics can be used within the theory of Lie
categories, possibly using Haar systems, to describe existing or yield novel
physical theories.

  7. (vii)

Smooth sieves on Lie categories. In the context of Lie groupoids, the notion
of a sieve is vacuous since all morphisms are invertible – any sieve on an
object must equal the whole $t$-fibre over that object. However, a notion of a
smooth sieve seems possible for Lie categories, and their properties should be
researched, together with the properties of Grothendieck sites of such sieves.

  8. (viii)

Dynamical systems. The theory of Lie groupoids has successfully been applied
to describe dynamical systems, see e.g. [dynamics]; a natural question is to
what extent the theory of Lie categories presented in our paper can be used to
generalize these results, to describe non-invertible dynamics.

  9. (ix)

Generalization of Morita equivalence from the context of Lie groupoids to Lie
categories.

## Acknowledgments

I express my gratitude to my supervisor Pedro Resende for his steady direction
and patience; I also thank my second supervisor Ioan Mărcut for his readiness
to discuss Lie categories, for ideas on (counter-)examples, and for keeping a
sharp eye on my proofs. Finally, I thank Jure Kališnik and Rui L. Fernandes
for additional fruitful ideas and talks.

##  Appendix A Transversality on manifolds with corners

We state here some basic definitions and results regarding finite-dimensional
manifolds with corners that we need in the paper, mostly drawing from
[corners]; in what follows, $X$ is assumed to be a second-countable
topological space. Let $V$ be a real $n$-dimensional Banach space999Keep in
mind that picking a basis of $V$ gives a homeomorphism
$V\rightarrow\mathbb{R}^{n}$.; denote by $\Lambda=(\lambda_{i})_{i}$ a
(possibly empty) set of linearly independent covectors $\lambda_{i}\in V^{*}$
and let

$V_{\Lambda}=\\{v\in V\;|\;\lambda(v)\geq 0\text{ for all
}\lambda\in\Lambda\\}.$

Vacuously, there holds $V_{\Lambda}=V$ when $\Lambda=\emptyset$. We will call
$\Lambda$ a corner-defining system on vector space $V$, and also denote

$V_{\Lambda}^{0}=\\{v\in V\;|\;\lambda(v)=0\text{ for all
}\lambda\in\Lambda\\}=\cap_{\lambda\in\Lambda}\ker\lambda.$

Given such a corner-defining system, a chart with corners on $X$ at $p\in X$
is a map $\varphi\colon U\rightarrow V_{\Lambda}$ where $U\subset X$ is an
open neighborhood of $p$, $\varphi(U)\subset V_{\Lambda}$ is an open
neighborhood of $0$, and $\varphi$ is a homeomorphism onto its image with
$\varphi(p)=0$. Such a chart is said to be $n$-dimensional, and the point $p$
is said to have index $|\Lambda|$, for which we will write
$\operatorname{ind}_{X}(p)=|\Lambda|$.

We say that $X$ is an $n$-dimensional manifold with corners if there is an
$n$-dimensional chart with corners around every point, and any two such charts
$\varphi$ and $\varphi^{\prime}$ are compatible, i.e. $\varphi(U\cap
U^{\prime})\subset V_{\Lambda}$ and $\varphi^{\prime}(U\cap U^{\prime})\subset
V_{\Lambda^{\prime}}$ are open subsets, and
$\varphi^{\prime}\circ\varphi^{-1}\colon\varphi(U\cap
U^{\prime})\rightarrow\varphi^{\prime}(U\cap U^{\prime})$ is a
diffeomorphism.101010Denote
$\mathbb{R}^{n}_{k}=\mathbb{R}^{n-k}\times[0,\infty)^{k}$. A map $U\rightarrow
V$ between open subsets of $\mathbb{R}^{n}_{k}$ and $\mathbb{R}^{m}_{l}$ is
smooth at $p\in U$, if it admits a smooth extension to an open neighborhood of
$p$ in $\mathbb{R}^{n}$. Such a collection of charts with corners is called an
atlas with corners. On a manifold with corners, the map
$\operatorname{ind}_{X}\colon X\rightarrow\mathbb{N}_{0}$ is well-defined by
boundary invariance theorem found in [corners, Theorem 1.2.12].

We will use the following notation regarding the elementary terms for
manifolds with corners: the $k$-boundary of $X$ is denoted by
$\partial^{k}X=\\{p\in X\;|\;\operatorname{ind}_{X}(p)\geq k\\}$, and we say
that $X$ is a manifold with boundary if $\partial
X:=\partial^{1}X\neq\emptyset$ and $\partial^{2}X=\emptyset$. The $k$-stratum
of $X$ is its subspace $S^{k}(X)=\\{p\in
X\;|\;\operatorname{ind}_{X}(p)=k\\}$. The set of connected components of the
$k$-stratum is denoted by $\mathcal{S}^{k}(X)$, and the set of $k$-faces of
$X$ is the family

$\mathcal{F}^{k}(X)=\left\\{\bar{S}\;\middle|\;S\in\mathcal{S}^{k}(X)\right\\}$

of topological closures in $X$ of the the components of the $k$-stratum. We
note that the $k$-stratum $S^{k}(X)$ can be given the following canonical
differential structure induced by $X$: if $p\in S^{k}(X)$ and $\varphi\colon
U\rightarrow V_{\Lambda}$ is a corner chart at $p$, then the restriction

$\varphi|_{U\cap S^{k}(X)}\colon U\cap S^{k}(X)\rightarrow V_{\Lambda}^{0}$

is a corner chart at $p$ on $S^{k}(X)$ since there holds $\varphi(U\cap
S^{k}(X))=\varphi(U)\cap V_{\Lambda}^{0}$. With this structure, $S^{k}(X)$
becomes a manifold without boundary of dimension $n-k$.

A subspace $X^{\prime}\subset X$ is said to be a submanifold of $X$, if for
any $p\in X^{\prime}$ there is a chart with corners $\varphi\colon
U\rightarrow V_{\Lambda}$ at $p$ on $X$, and a linear subspace $V^{\prime}$
together with a corner-defining system $\Lambda^{\prime}$ on $V^{\prime}$,
such that $\varphi(U\cap X^{\prime})=\varphi(U)\cap
V^{\prime}_{\Lambda^{\prime}}$ and this is an open subset of
$V^{\prime}_{\Lambda^{\prime}}$. Such a chart is said to be adapted to
$X^{\prime\prime}$ by means of $(V^{\prime},{\Lambda^{\prime}})$, and gives us
a way of defining an intrinsic atlas with corners on $X^{\prime}$. If there
holds $\partial X^{\prime}=\partial X\cap X^{\prime}$, we say that
$X^{\prime}\subset X$ is a neat submanifold.

In the context of manifolds with corners, it is more useful to use non-
infinitesimal notions of submersivity and transversality.

######  Definition A.1.

A smooth map $f\colon X\rightarrow X^{\prime}$ between manifolds with corners
is a topological submersion at $p\in X$, if there is an open neighborhood $U$
of $f(p)$ in $X^{\prime}$ and a smooth map $\sigma\colon U\rightarrow X$ such
that $\sigma(f(p))=p$ and $f\circ\sigma=\mathrm{id}_{U}$. If this holds for
all $p\in X$, we just say that $f\colon X\rightarrow X^{\prime}$ is a
topological submersion.

The notion of a topological submersion is stronger than the usual
infinitesimal one which can be seen by differentiating the equality
$f\circ\sigma=\mathrm{id}_{U}$. Moreover, in the context of boundaryless
manifolds they are equivalent, which is an easy consequence of the usual rank
theorem.

######  Definition A.2.

Suppose that $f\colon X\rightarrow X^{\prime}$ is a smooth map between
manifolds with corners, and $X^{\prime\prime}\subset X^{\prime}$ is a
submanifold. The map $f$ is topologically transversal to $X^{\prime\prime}$ at
$p\in X$, written symbolically as

$f\pitchfork_{p}X^{\prime\prime},$

if either $f(p)\notin X^{\prime\prime}$, or there is a chart
$\varphi^{\prime}\colon U^{\prime}\rightarrow V^{\prime}_{\Lambda^{\prime}}$
on $X^{\prime}$ at $p$ adapted to $X^{\prime\prime}$ by means of
$(V^{\prime\prime},{\Lambda^{\prime\prime}})$ and an open neighborhood $U$ of
$p$ in $X$, such that $f(U)\subset U^{\prime}$ and

${\tau\colon
U}$${{U^{\prime}}}$${{\varphi^{\prime}(U^{\prime})}}$${{(V^{\prime\prime}\oplus
W^{\prime\prime})_{L^{*}\Lambda^{\prime}}}}$${{W^{\prime\prime}}}$$\scriptstyle{f|_{U}}$$\scriptstyle{\varphi^{\prime}}$$\scriptstyle{\approx}$$\scriptstyle{L^{-1}}$$\scriptstyle{\mathrm{pr}_{2}}$

is a topological submersion at $p$, where $W^{\prime\prime}$ is any
complementary subspace to $V^{\prime\prime}$ in $V^{\prime}$, and $L\colon
V^{\prime\prime}\oplus W^{\prime\prime}\rightarrow V^{\prime}$ is the linear
isomorphism given by $L(v,w)=v+w$. If for all $p\in X$ we have
$f\pitchfork_{p}X$, we just write $f\pitchfork X^{\prime\prime}$ and say $f$
is topologically transversal to $X^{\prime\prime}$.

Note that $f\pitchfork_{p}X^{\prime\prime}$ implies the usual infinitesimal
transversality condition

$\displaystyle\operatorname{d}\\!{f}_{p}(T_{p}X)+T_{f(p)}X^{\prime\prime}=T_{f(p)}X^{\prime}.$
(A.3)

Indeed, if $v^{\prime}\in T_{f(p)}X^{\prime}$, there holds

$v^{\prime}=\operatorname{d}\\!{f}_{x}(v)+\operatorname{d}\\!{(}\varphi^{\prime})_{p}^{-1}L\big{(}\mathrm{pr}_{1}L^{-1}\operatorname{d}\\!{(}\varphi^{\prime})_{p}(v^{\prime}-\operatorname{d}\\!{f}_{x}(v)),0\big{)}$

where $v\in T_{p}X$ with
$\operatorname{d}\\!{\tau}_{p}(v)=\mathrm{pr}_{2}L^{-1}\operatorname{d}\\!{(}\varphi^{\prime})_{p}(v^{\prime})$
exists since $\tau$ is a submersion at $p$. We will show in Proposition A.8
that (A.3) implies $f\pitchfork_{p}X^{\prime\prime}$ in case $X$ has no
boundary.

We now state the main transversality theorem for manifolds with corners, which
is a generalization of the same result for boundaryless manifolds.

######  Proposition A.4.

Let $f\colon X\rightarrow X^{\prime}$ be a smooth map between manifolds with
corners and let $X^{\prime\prime}\subset X^{\prime}$ be a neat submanifold. If
$f\pitchfork X^{\prime\prime}$, then $f^{-1}(X^{\prime\prime})\subset X$ is a
submanifold with
$\operatorname{codim}_{X}f^{-1}(X)=\operatorname{codim}_{X^{\prime}}X^{\prime\prime}$,
whose tangent bundle is

$T(f^{-1}(X^{\prime\prime}))=(\operatorname{d}\\!{f})^{-1}(TX^{\prime\prime}).$

Moreover, $f^{-1}(X^{\prime\prime})\subset X$ is totally neat, i.e.
$S^{k}(f^{-1}(X))=f^{-1}(X^{\prime\prime})\cap S^{k}(X)$ for any $k\geq 0$. In
particular, $f^{-1}(X)$ is a neat submanifold of $X$.

###### Proof.

[corners, Proposition 7.1.14]. ∎

In the context of Lie categories with nonempty regular boundaries, this result
enables us to show that the set of composable morphisms has a structure of a
smooth manifold. Let us show how.

######  Lemma A.5.

Let $f\colon X\rightarrow X^{\prime}$ be a smooth map from a manifold $X$ with
boundary to a boundaryless manifold $X^{\prime}$, such that both $f$ and
$\partial f$ are submersions. Then $f$ is a topological submersion.

###### Proof.

We want to show $f$ is a topological submersion at any $p\in X$. If
$p\in\operatorname{Int}X$ this follows from the usual rank theorem used on
$f|_{\operatorname{Int}X}$, and if $p\in\partial X$ it follows from the usual
rank theorem used on $f|_{\partial X}$. ∎

######  Lemma A.6.

If $f\colon X\rightarrow X^{\prime}$ is a smooth topological submersion
between manifolds with corners and $\partial X^{\prime}=\emptyset$, then
$f\pitchfork X^{\prime\prime}$ holds for any submanifold
$X^{\prime\prime}\subset X^{\prime}$.

###### Proof.

Suppose $f(p)\in X^{\prime\prime}$ and let $\varphi^{\prime}\colon
U^{\prime}\rightarrow V^{\prime}$ be a chart on $X^{\prime}$ at $f(p)$ adapted
to $X^{\prime\prime}$ by means of
$(V^{\prime\prime},{\Lambda^{\prime\prime}})$. Let $W^{\prime\prime}$ be any
complementary subspace to $V^{\prime\prime}$ in $V$; continuity of $f$ ensures
there is a neighborhood $U\subset X$ of $p$ such that $f(U)\subset
U^{\prime}$, and now the composition

${U}$${{U^{\prime}}}$${{\varphi^{\prime}(U^{\prime})}}$${{V^{\prime\prime}\oplus
W^{\prime\prime}}}$${{W^{\prime\prime}}}$$\scriptstyle{f|_{U}}$$\scriptstyle{\varphi^{\prime}}$$\scriptstyle{\approx}$$\scriptstyle{L^{-1}}$$\scriptstyle{\mathrm{pr}_{2}}$

is a topological submersion as a composition of topological submersions
$f|_{U}$ and $\mathrm{pr}_{2}L^{-1}\varphi^{\prime}$. ∎

######  Corollary A.7.

Let $X$ and $Y$ be manifolds with boundaries and $Z$ a manifold without
boundary. If $f\colon X\rightarrow Z$ and $g\colon Y\rightarrow Z$ are smooth
maps such that $f,f|_{\partial X}$ and g,$g|_{\partial Y}$ are submersions,
then $X{\tensor*[_{f}]{\times}{{}_{g}}}Y=(f\times g)^{-1}(\Delta_{Z})\subset
X\times Y$ is a submanifold with tangent space at $(p,q)$ equal to

$T_{(p,q)}(X{\tensor*[_{f}]{\times}{{}_{g}}}Y)=\\{(v,w)\in T_{p}X\oplus
T_{q}Y\;|\;\operatorname{d}\\!{f}(v)=\operatorname{d}\\!{g}(w)\\},$

and its boundary is
$\partial(X{\tensor*[_{f}]{\times}{{}_{g}}}Y)=(X{\tensor*[_{f}]{\times}{{}_{g}}}Y)\cap(\partial
X\times Y\cup X\times\partial Y)$.

###### Proof.

Lemma A.5 ensures $f$ and $g$ are topological submersions, and it is easy to
check that $f\times g\colon X\times Y\rightarrow Z\times Z$ is also a smooth
topological submersion. Now Lemma A.6 implies $(f\times
g)\pitchfork\Delta_{Z}$ since $Z$ is boundaryless, and Proposition A.4
finishes the proof since $\Delta_{Z}\subset Z\times Z$ is trivially a neat
submanifold. ∎

The following proposition shows in particular that in the case of boundaryless
manifolds, the usual infinitesimal notion of transversality is equivalent to
the one above.

######  Proposition A.8.

Let $f\colon X\rightarrow X^{\prime}$ be a smooth map between manifolds with
corners and $X^{\prime\prime}\subset X^{\prime}$ a submanifold. For any $p\in
f^{-1}(X^{\prime\prime})$ with $\operatorname{ind}_{X}(p)=k$, the following
statements are equivalent.

  1. (i)

$\operatorname{Im}\operatorname{d}\\!{(}f|_{S^{k}(X)})_{p}+T_{f(p)}X^{\prime\prime}=T_{f(p)}X^{\prime}.$

  2. (ii)

$f|_{S^{k}(X)}\pitchfork_{p}X^{\prime\prime}$.

  3. (iii)

$f\pitchfork_{p}X^{\prime\prime}$.

###### Proof.

To show $(i)\Rightarrow(ii)$, take a chart $\varphi^{\prime}\colon
U^{\prime}\rightarrow V^{\prime}_{\Lambda^{\prime}}$ on $X^{\prime}$ at $f(p)$
adapted to $X^{\prime\prime}$ by means of
$(V^{\prime\prime},{\Lambda^{\prime\prime}})$, take a complementary subspace
$W^{\prime\prime}$ to $V^{\prime\prime}$ of $V^{\prime}$, and consider the map

${\tau^{k}\colon
U}$${{U^{\prime}}}$${{\varphi^{\prime}(U^{\prime})}}$${{(V^{\prime\prime}\oplus
W^{\prime\prime})_{L^{*}\Lambda^{\prime}}}}$${{W^{\prime\prime}}}$$\scriptstyle{f|_{U}}$$\scriptstyle{\varphi^{\prime}}$$\scriptstyle{\approx}$$\scriptstyle{L^{-1}}$$\scriptstyle{\mathrm{pr}_{2}}$

where $U$ is the domain of a chart neighborhood $\varphi\colon U\rightarrow V$
in $S^{k}(X)$ of $p$; by continuity we may assume $f(U)\subset U^{\prime}$.
Since $\partial S^{k}(X)$ is boundaryless, it is enough to show that
$\tau^{k}$ is a submersion. To that end, take any $w\in
T_{0}W^{\prime\prime}\cong W^{\prime\prime}$ and then
$u:=\operatorname{d}\\!{(}\varphi^{\prime})_{p}^{-1}L(0,w)\in
T_{f(p)}X^{\prime}$, so by assumption there exist $v\in T_{p}X$ and
$v^{\prime\prime}\in T_{f(p)}X^{\prime\prime}$ such that
$u=\operatorname{d}\\!{f}_{p}(v)+v^{\prime\prime}$. Since
$L^{-1}\operatorname{d}\\!{(}\varphi^{\prime})_{p}(v^{\prime\prime})\in
V^{\prime\prime}$, we get

$\operatorname{d}\\!{h}_{p}(v)=\mathrm{pr}_{2}L^{-1}\operatorname{d}\\!{(}\varphi^{\prime})(u-v^{\prime\prime})=w.$

Conversely, $(ii)\Rightarrow(i)$ follows from the fact that topological
transversality implies transversality.

For the implication $(ii)\Rightarrow(iii)$, note that if
$Z\xrightarrow{g}Z^{\prime}\xrightarrow{h}Z^{\prime\prime}$ are smooth maps
such that $h\circ g$ is a topological submersion at $p\in Z$, then it is easy
to see $h$ is a topological submersion at $g(p)$. Use this result for the
composition $S^{k}(X)\hookrightarrow X\xrightarrow{\tau}W^{\prime\prime}$
which equals $\tau^{k}$. The converse implication $(iii)\Rightarrow(ii)$ is a
consequence of the following lemma used on the map $\tau$. ∎

######  Lemma A.9.

If $f\colon X\rightarrow X^{\prime}$ is a smooth map between manifolds with
corners, which is a topological submersion at $p$ and there holds
$f(p)\in\operatorname{Int}(X^{\prime})$, then $f|_{S^{k}(X)}$ is a submersion
at $p$, where $k=\operatorname{ind}_{X}(p)$.

###### Proof.

Let $\varphi\colon U\rightarrow V_{\Lambda}$ be a chart on $X$ at $p$ and
$\varphi^{\prime}\colon X^{\prime}\rightarrow V^{\prime}$ on $X^{\prime}$ at
$f(p)$. Note that $\varphi(U\cap S^{k}(X))=\varphi(U)\cap V_{\Lambda}^{0}$.

Let $v^{\prime}\in T_{f(p)}X^{\prime}$ be arbitrary. By assumption, there is a
neighborhood $U^{\prime}$ of $f(p)$ and a smooth map $\sigma\colon
U^{\prime}\rightarrow U\subset X$ such that $\sigma(f(p))=p$ and
$f\circ\sigma=\mathrm{id}_{U^{\prime}}$. Since
$f(p)\in\operatorname{Int}X^{\prime}$, there is a smooth path
$\gamma\colon(-\varepsilon,\varepsilon)\rightarrow U^{\prime}$ with
$\dot{\gamma}(0)=v^{\prime}$, and now consider the path
$\delta=\varphi\circ\sigma\circ\gamma\colon(-\varepsilon,\varepsilon)\rightarrow\varphi(U)\subset
V_{\Lambda}$. Since $\delta(0)=0$ and $\delta$ is defined on an open interval,
we must have that $\delta$ maps into $V_{\Lambda}^{0}$, so $\dot{\delta}(0)\in
T_{0}V_{\Lambda}^{0}\cong V_{\Lambda}^{0}$. Taking
$v=\operatorname{d}\\!{(}\varphi^{-1})_{0}(\dot{\delta}(0))$ hence yields
$v\in T_{p}(S^{k}(X))$ with $\operatorname{d}\\!{f}_{p}(v)=v^{\prime}$. ∎

## References

Ž. Grad, Centre for Mathematical Analysis, Geometry, and Dynamical Systems,
Department of mathematics, Instituto Superior Técnico, Av. Rovisco Pais,
Lisbon, Portugal.

E-mail address<EMAIL_ADDRESS>
Webpage: https://zangrad.github.io.