# Dagger $n$-categories

Giovanni Ferrer, Brett Hungar, Theo Johnson-Freyd, Cameron Krulewski, Lukas
Müller, Nivedita, David Penneys, David Reutter, Claudia Scheimbauer, Luuk
Stehouwer, Chetan Vuppulury

###### Abstract.

We present a coherent definition of _dagger $(\infty,n)$-category_ in terms of
equivariance data trivialized on parts of the category. Our main example is
the bordism higher category $\mathbf{Bord}_{n}^{X}$. This allows us to define
a _reflection-positive topological quantum field theory_ to be a higher dagger
functor from $\mathbf{Bord}_{n}^{X}$ to some target higher dagger category
$\mathcal{C}$. Our definitions have a tunable parameter: a group $G$ acting on
the $(\infty,1)$-category $\mathbf{Cat}_{(\infty,n)}$ of
$(\infty,n)$-categories. Different choices for $G$ accommodate different
flavours of higher dagger structure; the universal choice is
$G=\operatorname{Aut}(\mathbf{Cat}_{(\infty,n)})=(\mathbb{Z}/2\mathbb{Z})^{n}$,
which implements dagger involutions on all levels of morphisms.

The Stratified Cobordism Hypothesis suggests that there should be a map
$\mathrm{PL}(n)\to\operatorname{Aut}(\mathbf{AdjCat}_{(\infty,n)})$, where
$\mathrm{PL}(n)$ is the group of piecewise-linear automorphisms of
$\mathbb{R}^{n}$ and $\mathbf{AdjCat}_{(\infty,n)}$ the $(\infty,1)$-category
of $(\infty,n)$-categories with all adjoints; we conjecture more strongly that
$\operatorname{Aut}(\mathbf{AdjCat}_{(\infty,n)})\cong\mathrm{PL}(n)$. Based
on this conjecture we propose a notion of _dagger $(\infty,n)$-category with
unitary duality_ or _$\mathrm{PL}(n)$ -dagger category_. We outline how to
construct a $\mathrm{PL}(n)$-dagger structure on the fully-extended bordism
$(\infty,n)$-category $\mathbf{Bord}_{n}^{X}$ for any stable tangential
structure $X$; our outline restricts to a rigorous construction of a coherent
dagger structure on the unextended bordism $(\infty,1)$-category
$\mathbf{Bord}_{n,n-1}^{X}$. The article is a report on the results of a
workshop held in Summer 2023, and is intended as a sketch of the big picture
and an invitation for more thorough development.

In June 2023, we gathered for an online workshop on “Dagger Higher
Categories”†††The workshop schedule and participant list can be found at
http://categorified.net/dagger2023.html.. The goal of the workshop was the
following. Many of us had some thoughts and ideas on what a “dagger higher
category” should be, but there was no proposal for a full definition. Through
compiling and comparing ideas, we hoped to emerge with a complete consensus
definition, including nontrivial examples. This goal was met with great
success. In this article, we will elaborate on our findings. Our discussion
will focus on the essential ideas and will be short on technical details and
essentially devoid of proofs; for this reason, we will refer to “statements”
rather than “lemmas” or “theorems,” and strictly speaking all statements
herein (other than those cited to the published literature) carry the
ontological status of conjectures. We also list some true “conjectures,” of
whose veracity we are less confident. Our goal is to sketch the big picture,
and we hope that our article can serve as a blueprint for a more thorough
development of the subject.

##  1\. Dagger 1-categories

Categories with an (anti-)involution have been studied since the early days of
category theory [ML61, Pup62]. A familiar example is the category of Hilbert
spaces and bounded linear maps, with an involution $(-)^{\dagger}$ given by
the usual notion of adjoint map. The notion of a complex $*$-category and
notions of $\rm C^{*}$\- and $\rm W^{*}$-category in the complex linear
setting were introduced in [GLR85] and have been studied extensively in the
$\rm C^{*}$ tensor setting [FK93, DR89, LR97, Wen98, M0̈3, Yam04].

Roughly, an involution $(-)^{\dagger}$ should reverse the direction of
morphisms in a compatible way. In [Sel07], the following notion was introduced
in the non-linear setting. A category $\mathcal{C}$ is a _dagger category_ if
it is equipped with a bijection $(-)^{\dagger}:\hom(x,y)\to\hom(y,x)$ for each
pair $(x,y)$ of objects, such that $(f^{\dagger})^{\dagger}=f$ and $(f\circ
g)^{\dagger}=g^{\dagger}\circ f^{\dagger}$ for all composable morphisms $f,g$.
In other words, $(-)^{\dagger}$ is a functor
$\mathcal{C}\to\mathcal{C}^{\mathrm{op}}$ which is involutive and the identity
on (the set of) objects.

This definition is _evil_ in the following sense [Hen15]: since it explicitly
references the set of objects in $\mathcal{C}$, dagger structures cannot be
readily transported along general equivalences of categories. In detail: given
a dagger category $(\mathcal{C},(-)^{\dagger})$ and an equivalence
$F:\mathcal{C}\simeq\mathcal{D}$ with (weak) inverse $F^{-1}$, the transported
“dagger” functor‡‡‡A different formula for the transported structure supplies
a dagger structure on $\mathcal{D}$ such that $F^{-1}$ is a dagger functor
[Kar18, Lemma 2.1.16], but then $F$ will typically not only fail to be a
dagger equivalence, it will fail to be compatible with the dagger structures
at all. $F\circ(-)^{\dagger}\circ
F^{-1}:\mathcal{D}\to\mathcal{D}^{\mathrm{op}}$ is naturally part of a _weak
involution_ in the sense that its square is coherently-isomorphic to
$\mathrm{id}_{\mathcal{D}}$, but it will almost never be an identity on
objects. We violated the general rule of thumb: from a category, one can
coherently extract its _groupoid_ of objects up to isomorphism, but not its
set of objects. Non-evil notions refer only to coherently-extractable data.

Nevertheless, there is a well-developed “dagger category theory” that
parallels the usual theory of categories, with dagger versions of functor,
natural transformation, and the like [GLR85, LR97, Yam04, HP17, JP17, Kar18,
HK16, HK19, Sri21]. From the perspective of dagger category theory, the
coherently-extractable space of objects is the groupoid of objects and
_unitary_ isomorphism.

With this in mind, [SS23] developed the following ideas. Consider the
$(2,1)$-category $\mathbf{Cat}$ of $1$-categories, functors and natural
isomorphisms. This $(2,1)$-category carries a $\mathbb{Z}/2\mathbb{Z}$-action
which sends $\mathcal{C}\mapsto\mathcal{C}^{\mathrm{op}}$. An _anti-involutive
category_ is a homotopy fixed point for this action: explicitly, it consists
of a category $\mathcal{C}$, a categorical equivalence
$\dagger:\mathcal{C}\overset{\sim}{\to}\mathcal{C}^{\mathrm{op}}$, and a
natural isomorphism
$\eta:\dagger^{\mathrm{op}}\circ\dagger\overset{\sim}{\Rightarrow}\mathrm{id}_{\mathcal{C}}$,
such that for each object $x\in\mathcal{C}$,
$\eta_{x^{\dagger}}^{-1}=(\eta_{x})^{\dagger}$.

If $\mathcal{G}$ is a groupoid, then $\mathcal{G}$ and
$\mathcal{G}^{\mathrm{op}}$ are canonically equivalent via the functor that is
the identity on objects and sends morphisms to their inverses. In other words,
the $\mathbb{Z}/2\mathbb{Z}$-action
$\mathcal{C}\mapsto\mathcal{C}^{\mathrm{op}}$ on $\mathbf{Cat}$ trivializes
when restricted to the full sub-$(2,1)$-category
$\mathbf{Gpd}\subset\mathbf{Cat}$ of groupoids. In particular, given an anti-
involutive category $\mathcal{C}$, the groupoid $\iota_{0}\mathcal{C}$ of
objects of $\mathcal{C}$ inherits a coherent action by
$\mathbb{Z}/2\mathbb{Z}$. Let
$(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}}$ denote the groupoid of
homotopy fixed points for this action.

As a motivating example, consider
$\mathcal{C}=\mathbf{Vec}^{\mathrm{fd}}_{\mathbb{C}}$ the category of finite-
dimensional vector spaces, and $\dagger:V\mapsto\overline{V}^{\vee}$ the
functor that assigns the complex-conjugate dual of a vector space. Then
$(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}}$ is the groupoid of finite-
dimensional _Hermitian_ vector spaces, i.e. vector spaces equipped with a
nondegenerate conjugate-symmetric sesquilinear form which might not be
positive-definite, and “unitary” isomorphisms thereof. In order to encode the
theory of Hilbert spaces, we must specify extra data: we must mark some of the
Hermitian spaces as preferred, and leave out the others. This marking
restricts the set of objects of
$(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}}$, but keeps all unitary
isomorphisms between them; in other words, the marked subspace
$\mathcal{C}_{0}\subset(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}}$ should
be a full subgroupoid. If there were a vector space that did not admit a
Hilbert structure, then we should have already excised it from $\mathcal{C}$;
in other words, the composition
$\mathcal{C}_{0}\subset(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}}\to\mathcal{C}$
should be essentially surjective.

######  Definition 1.1.

A _coherent dagger 1-category_ is an anti-involutive category
$(\mathcal{C},\dagger)$ together with a fully faithful subgroupoid
$\mathcal{C}_{0}\hookrightarrow(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}}$
such that the induced map
$\mathcal{C}_{0}\hookrightarrow(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}}\to\iota_{0}\mathcal{C}$
is essentially surjective. A morphism between coherent dagger 1-categories is
a functor respecting the anti-involutions and the subgroupoids.

Coherent dagger structures are non-evil: if $\mathcal{C}$ is a dagger category
and $\mathcal{C}\cong\mathcal{D}$ an equivalence with an ordinary category,
then $\mathcal{D}$ acquires a natural coherent dagger structure. The main
theorem of [SS23] says that coherent dagger categories are equivalent in the
appropriate sense to dagger categories as traditionally defined. Namely,
coherent dagger categories can be _strictified_ to dagger categories as
traditionally defined:

######  Theorem 1.2 ([SS23]).

Any dagger category $\mathcal{C}$ defines a coherent dagger category by
keeping $\mathcal{C}$ and $\dagger$ as-is, and setting $\mathcal{C}_{0}$ to be
the groupoid of objects in $\mathcal{C}$ and unitary morphisms between them.
The so-defined functor from the $(2,1)$-category of dagger categories, dagger
functors, and unitary natural isomorphisms to the $(2,1)$-category of coherent
dagger categories is an equivalence.

Its inverse assigns to a coherent dagger category
$(\mathcal{C},\dagger,\mathcal{C}_{0}\hookrightarrow(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}})$
the category whose objects are the objects of $\mathcal{C}_{0}$ and whose
morphisms are the morphisms in $\mathcal{C}$ between the images of objects;
this new category is equivalent to $\mathcal{C}$ because of the essential
surjectivity axiom, and its anti-involution is strictly trivial on the set of
objects.

Definition 1.1 immediately generalizes to the $(\infty,1)$-world, where it was
independently proposed by [Hen22]. An _$(\infty,1)$ -category_ is a
homotopically-coherent version of “category enriched in spaces.” As is
standard in higher category theory, we will interchangeably use the word
_$\infty$ -groupoid_ and _space_ for “nice topological space considered up to
homotopy.” Then an $(\infty,1)$-category has not a set of morphisms between
any two objects, but rather a space of morphisms; composition is associative
up to parameterized homotopy; this homotopy itself satisfies higher and higher
homotopies relating different parenthesizations. There are many ways of
axiomatizing the notion of “$(\infty,1)$-category,” and we refer the reader to
[Ber10, AC16] for nice surveys.

Ignoring size issues, there is an $(\infty,1)$-category
$\mathbf{Cat}_{(\infty,1)}$ whose objects are $(\infty,1)$-categories and
whose morphisms are $(\infty,1)$-functors. There is a functor
$\iota_{0}\colon\mathbf{Cat}_{(\infty,1)}\to\mathbf{Space}$, corepresented by
the terminal $(\infty,1)$-category $\\{\mathrm{pt}\\}$, which assigns to each
$(\infty,1)$-category its space of objects. Moreover,
$\mathbf{Cat}_{(\infty,1)}$ carries a canonical
$\mathbb{Z}/2\mathbb{Z}$-action sending
$\mathcal{C}\mapsto\mathcal{C}^{\mathrm{op}}$ [Toë05]. An _anti-involutive
$(\infty,1)$-category_ is a (homotopy) fixed point for this action.

As in the 1-categorical case, the $\mathbb{Z}/2\mathbb{Z}$-action on
$\mathbf{Cat}_{(\infty,1)}$ trivializes on the full subcategory
$\mathbf{Space}\subseteq\mathbf{Cat}_{(\infty,1)}$. Hence, the space of
objects $\iota_{0}\mathcal{C}$ of any anti-involutive $(\infty,1)$-category
$\mathcal{C}$ inherits a coherent $\mathbb{Z}/2\mathbb{Z}$ action, whose
homotopy fixed points we continue to denote by
$(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}}.$

######  Definition 1.3.

A _dagger $(\infty,1)$-category_ is an anti-involutive $(\infty,1)$-category,
together with a full sub-$\infty$-groupoid
$\mathcal{C}_{0}\hookrightarrow(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}}$
such that the induced map
$\mathcal{C}_{0}\hookrightarrow(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}}\to\mathcal{C}$
is essentially surjective.

Anticipating our $n$-dimensional generalization, let us note that part of the
data of a dagger $(\infty,1)$-category is the space $\mathcal{C}_{0}$, the
$(\infty,1)$-category $\mathcal{C}$, and the essential surjection
$\mathcal{C}_{0}\to\iota_{0}\mathcal{C}$. This structure is called a _flagged
$(\infty,1)$-category_. Any specific $1$-category $\mathcal{C}$ in the
traditional sense supplies a flagged $(\infty,1)$-category: one takes
$\mathcal{C}_{0}$ to be the actual set of objects, whereas
$\iota_{0}\mathcal{C}$ is the groupoid of objects and isomorphisms. In other
words, flaggings are a way of remembering “sets” of objects in a way that
nevertheless transports coherently. The requirement that
$\mathcal{C}_{0}\to\iota_{0}\mathcal{C}$ be essentially surjective simply
means that the data of the homomorphisms between elements of $\mathcal{C}_{0}$
suffices to recover all of $\mathcal{C}$ up to categorical equivalence. A
flagging is called _univalent_ §§§This is reminiscent of the completeness
condition of Segal spaces. if it does not in fact remember any further data:
if the map $\mathcal{C}_{0}\to\iota_{0}\mathcal{C}$ is not just essentially
surjective but also fully faithful.

By analogy, in a dagger $(\infty,1)$-category, the requirement that
$\mathcal{C}_{0}\to(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}}$ be fully
faithful will be called the _univalence axiom_ , even though it does not force
any map to be an equivalence. And by analogy, dropping the univalence axiom
leads to a useful weakening of Definition 1.3.

######  Definition 1.4.

A _flagged dagger $(\infty,1)$-category_ is an anti-involutive
$(\infty,1)$-category $\mathcal{C}$ equipped with a map of spaces
$\mathcal{C}_{0}\to(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}}$ such that
the induced map
$\mathcal{C}_{0}\to(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}}\to\mathcal{C}$
is essentially surjective.

Our motivating example of a flagged dagger $(\infty,1)$-category is the higher
category of bordisms, which we discuss in Example 6.7.

Whether univalent or not, the flagging should be thought of as recording the
“identity on objects” condition in the traditional definition of dagger
category. In a flagged dagger category which is not univalent, the groupoid
$\mathcal{C}_{0}$ selects a notion of equivalence between objects that is
finer than unitary equivalence.

Write $\mathbf{Fl{\dagger}Cat}_{(\infty,1)}$ for the $(\infty,1)$-category of
flagged dagger $(\infty,1)$-categories: the homomorphisms are the obvious ones
which preserve the equivariance and the flaggings. Write
${\dagger}\mathbf{Cat}_{(\infty,1)}$ for the full subcategory of
$\mathbf{Fl}{\dagger}\mathbf{Cat}_{(\infty,1)}$ on the univalent ones.

######  Statement 1.5.

The inclusion

${\dagger}\mathbf{Cat}_{(\infty,1)}\hookrightarrow\mathbf{Fl{\dagger}Cat}_{(\infty,1)}$

admits a left adjoint, which replaces $\mathcal{C}_{0}$ with the full
subgroupoid inside $(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}}$ on the
essential image of
$\mathcal{C}_{0}\to(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}}$.

One benefit of Statement 1.5 is that it can be easier to present flagged
dagger categories than their univalentizations. Indeed, most
$(\infty,1)$-categories $\mathcal{C}$ of interest (for example the bordism
categories discussed in Section 6) come with distinguished flaggings
$\mathcal{C}_{0}\to\mathcal{C}$, and sometimes this flagged category is
naturally anti-involutive in the sense that $\mathcal{C}$ is anti-involutive
and $\mathcal{C}_{0}\to\iota_{0}\mathcal{C}$ is
$\mathbb{Z}/2\mathbb{Z}$-equivariant. To promote this to a flagged dagger
structure then merely requires a trivialization of the
$\mathbb{Z}/2\mathbb{Z}$-action on $\mathcal{C}_{0}$. Speaking very
approximately, it can be easier to do this when $\mathcal{C}_{0}$ has very few
morphisms requiring trivialization data.

Instead of dropping the univalence axiom, we could instead drop the essential
surjectivity condition:

######  Definition 1.6.

A _coflagged dagger $(\infty,1)$-category_ is an anti-involutive
$(\infty,1)$-category $\mathcal{C}$ equipped with a fully faithful inclusion
of spaces
$\mathcal{C}_{0}\hookrightarrow(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}}$.

Comparing to Definition 1.1, a coflagged dagger $(\infty,1)$-category can be
thought of as an anti-involutive $(\infty,1)$-category equipped with the data
of Hermitian pairings on some but not necessarily all objects.

######  Statement 1.7.

Writing $\mathbf{coFl{\dagger}Cat}_{(\infty,1)}$ for the $(\infty,1)$-category
of coflagged dagger $(\infty,1)$-categories, the inclusion

${\dagger}\mathbf{Cat}_{(\infty,1)}\hookrightarrow\mathbf{coFl{\dagger}Cat}_{(\infty,1)}$

admits a right adjoint, which replaces $\mathcal{C}$ with its full subcategory
on the image of
$\mathcal{C}_{0}\to(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}}\to\mathcal{C}$.

Note that every anti-involutive $(\infty,1)$-category is naturally a coflagged
dagger $(\infty,1)$-category with
$\mathcal{C}_{0}=(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}}$. Through
this observation, Statement 1.7 specializes to a higher version of the
“Hermitian completion” functor of [SS23].

##  2\. Dagger $n$-categories via flaggings

The notion of $(\infty,n)$-category is designed to formalize categories with
an $n$-fold hierarchy of directions of composition. As with the
$(\infty,1)$-case, there are many models, some of which are surveyed in [BR13,
BR20, BSP21]. Informally, an $(\infty,n)$-category has a space of objects; for
each pair of objects, a space of $1$-morphisms between them and a coherently-
associative and coherently-unital composition law; for each pair of parallel
$1$-morphisms, a space of $2$-morphisms between them, and two (coherently-
associative and coherently-unital) composition laws, one of which lifts the
composition law of $1$-morphisms and the other of which is in the new second
dimension; and so on up to dimension $n$.

In this informal description of $(\infty,n)$-categories in the previous
paragraph, we did not specify which maps between them should be considered
equivalences. There is a most natural guess if one explained the idea without
supplying the name “category”: a homomorphism could be declared an equivalence
when it induces equivalences on all spaces of $k$-morphisms, including on the
space of objects. Comparing with the traditional notion of strict 1-category,
this choice would select the strict isomorphisms and not the categorical
equivalences. In contrast, categorical equivalence does not remember any
specific sets or spaces of $k$-morphisms, but merely the higher groupoids
thereof. As in the $(\infty,1)$-case, the extra data of spaces of morphisms in
a presentation of an $(\infty,n)$-category is called a _flagging_ of that
$(\infty,n)$-category [AF18]. Given a correct theory of
$(\infty,n)$-categories, the notion of flagged $(\infty,n)$-category can be
defined as follows:

######  Definition 2.1.

A _flagged $(\infty,n)$-category_ is a chain

$\mathcal{C}_{0}\to\mathcal{C}_{1}\to\dots\to\mathcal{C}_{n}$

where each $\mathcal{C}_{k}$ is an $(\infty,k)$-category, and the map
$\mathcal{C}_{k}\to\mathcal{C}_{k+1}$ is essentially surjective on $(\leq
k)$-morphisms¶¶¶A functor $F\colon\mathcal{C}\to\mathcal{D}$ of
$(\infty,n)$-categories is _essentially surjective on $(\leq k)$-morphisms_ if
it is essentially surjective on objects and, for every $0\leq j<k$ and every
pair of $j$-morphisms $f$ and $g$ with the same source and target, the functor
$F_{f,g}\colon\hom_{\mathcal{C}}(f,g)\to\hom_{\mathcal{D}}(Ff,Fg)$ of
$(\infty,n-j-1)$-categories is essentially surjective.. The flagging is
_univalent_ if $\mathcal{C}_{k}\to\iota_{k}\mathcal{C}_{k+1}$ is an
equivalence for each $k$, where
$\iota_{k}\colon\mathbf{Cat}_{(\infty,k+1)}\to\mathbf{Cat}_{(\infty,k)}$ is
the right-adjoint to the inclusion
$\mathbf{Cat}_{(\infty,k)}\hookrightarrow\mathbf{Cat}_{(\infty,k+1)}$, which
takes an $(\infty,k+1)$-category and forms an $(\infty,k)$-category by
forgetting the non-invertible $(k+1)$-morphisms.

The definition of dagger $(\infty,1)$-category used as an essential ingredient
the fact that every $(\infty,1)$-category $\mathcal{C}$ has an _opposite_
$\mathcal{C}^{\mathrm{op}}$ that reverses the direction of composition.
Similarly, given an $(\infty,n)$-category $\mathcal{C}$, one can produce new
$(\infty,n)$-categories by reversing any of the $n$ directions of composition.
This supplies an action of $(\mathbb{Z}/2\mathbb{Z})^{n}$ on
$\mathbf{Cat}_{(\infty,n)}$, the $(\infty,1)$-category of
$(\infty,n)$-categories. In fact, this action is completely canonical: one of
the theorems of [BSP21], generalizing [Toë05], says that this map

(1)
$(\mathbb{Z}/2\mathbb{Z})^{n}\to\operatorname{Aut}(\mathbf{Cat}_{(\infty,n)})$

is an isomorphism of (higher) groups. Given $1\leq k\leq n$, we will
occasionally write
$(\mathbb{Z}/2\mathbb{Z})_{k}\subset(\mathbb{Z}/2\mathbb{Z})^{n}$ for the
$k$th coordinate $\mathbb{Z}/2\mathbb{Z}$, which acts on
$(\infty,n)$-categories by reversing the composition of $k$-morphisms; we will
write that action as $\mathcal{C}\mapsto\mathcal{C}^{k\mathrm{op}}$.

######  Definition 2.2.

Given a group homomorphism $G\to(\mathbb{Z}/2\mathbb{Z})^{n}$, an
$(\infty,n)$-category $\mathcal{C}\in\mathbf{Cat}_{(\infty,n)}$ is _$G$
-volutive_∥∥∥From the Latin _volvere_ , “to roll,” and _involvere_ , “to roll
inwards.” when it is equipped with the data making it into a fixed point for
the action of $G$ on $\operatorname{Aut}(\mathbf{Cat}_{(\infty,n)})$ via (1).
In the special case when $G\to(\mathbb{Z}/2\mathbb{Z})^{n}$ is the identity,
we will say $\mathcal{C}$ is _fully-volutive_. When
$G=(\mathbb{Z}/2\mathbb{Z})\to(\mathbb{Z}/2\mathbb{Z})^{n}$ selects the last
involution, we will say $\mathcal{C}$ is _top-volutive_.

As in the 1-category case, volutive structures do not capture the theory of
daggers: they do not include an analogue of the requirement that “dagger is
the identity on objects.” To encode the latter, we use flaggings. To set up
the definition, we note the following. Every $(\infty,k)$-category is in
particular an $(\infty,k+1)$-category, and the inclusion
$\mathbf{Cat}_{(\infty,k)}\hookrightarrow\mathbf{Cat}_{(\infty,k+1)}$ is
stable under the ambient $(\mathbb{Z}/2\mathbb{Z})^{k+1}$-action. Indeed, the
first $k$ involutions
$(\mathbb{Z}/2\mathbb{Z})^{k}\subset(\mathbb{Z}/2\mathbb{Z})^{k+1}$ act on
$\mathbf{Cat}_{(\infty,k)}$ via the canonical action, and the last involution
$(\mathbb{Z}/2\mathbb{Z})_{k+1}$ has a canonical trivialization. Recall that a
fixed point for the trivial action of a group $G$ on a category $\mathcal{X}$
is precisely an action of $G$ on some object $X\in\mathcal{X}$. Thus if
$\mathcal{C}$ is an $(\infty,k)$-category thought of as an
$(\infty,k+1)$-category, then a fully-volutive structure in the
$(\infty,k+1)$-sense is precisely a fully-volutive structure in the
$(\infty,k)$-sense together with a coherently-compatible
$\mathbb{Z}/2\mathbb{Z}$-action. In particular, by picking the trivial
$\mathbb{Z}/2\mathbb{Z}$-action, any fully-volutive $(\infty,k)$-category can
be thought of as a fully-volutive $(\infty,k+1)$-category.

######  Definition 2.3.

A _flagged fully-dagger $(\infty,n)$-category_ is a flagged
$(\infty,n)$-category

$\mathcal{C}_{0}\to\mathcal{C}_{1}\to\dots\to\mathcal{C}_{n}$

such that each $\mathcal{C}_{k}$ is a fully-volutive $(\infty,k)$-category,
and the map $\mathcal{C}_{k}\to\mathcal{C}_{k+1}$ is a map of fully-volutive
$(\infty,k+1)$-categories, where $\mathcal{C}_{k}$ is given the trivial
$(k+1)$th anti-involution. Equivalently, $\mathcal{C}_{k}\to\mathcal{C}_{k+1}$
is factored through a map
$\mathcal{C}_{k}\to(\iota_{k}\mathcal{C}_{k+1})^{(\mathbb{Z}/2\mathbb{Z})_{k+1}}$
of fully-volutive $(\infty,k)$-categories.

We remind the reader that, in the higher-categorical world, requesting that
“this is a map of these things” is requesting for extra structure on the map.
In the case at hand, this structure can be unpacked as follows. A map
$\mathcal{C}_{k}\to\mathcal{C}_{k+1}$ is the same as a map
$\mathcal{C}_{k}\to\iota_{k}\mathcal{C}_{k+1}$. The fully-volutive structure
on $\mathcal{C}_{k+1}$ induces a fully-volutive structure on
$\iota_{k}\mathcal{C}_{k+1}$ in the $(\infty,k)$-sense together with a
typically-nontrivial action of $\mathbb{Z}/2\mathbb{Z}$. The requested
structure unpacks to a map of fully-volutive $(\infty,k)$-categories
$\mathcal{C}_{k}\to(\iota_{k}\mathcal{C}_{k+1})^{\mathbb{Z}/2\mathbb{Z}}$.
Comparing with Definitions 1.1 and 1.3, we exactly recover the “flagged dagger
categories”; in particular, the “essential surjectivity” axiom is enforced by
asking that $\mathcal{C}_{0}\to\mathcal{C}_{1}\to\dots\to\mathcal{C}_{n}$ be a
flagged $(\infty,n)$-category independent of the volutive, but we do not have
any univalence axiom. We add that univalence axiom now.

######  Definition 2.4.

A flagged fully-dagger $(\infty,n)$-category is _univalent_ if the maps
$\mathcal{C}_{k}\to(\iota_{k}\mathcal{C}_{k+1})^{\mathbb{Z}/2\mathbb{Z}}$ are
fully-faithful on $(\geq k+1)$-morphisms******A functor
$F\colon\mathcal{C}\to\mathcal{D}$ is _fully-faithful on $(\geq
k+1)$-morphisms_ if, for every pair of $k$-morphisms $f$ and $g$ with the same
source and target, the functor
$F_{f,g}\colon\mathcal{C}(f,g)\to\mathcal{D}(F(f),F(g))$ is an equivalence.. A
_fully-dagger $(\infty,n)$-category_ is a univalent flagged fully-dagger
$(\infty,n)$-category. We will write ${\dagger}\mathbf{Cat}_{(\infty,n)}$ for
the $(\infty,1)$-category of fully-dagger $(\infty,n)$-categories.

Beware that a fully-dagger $(\infty,n)$-category is _not_ the same as a
flagged fully-dagger $(\infty,n)$-category of which the underlying flagging is
univalent.

Definitions 2.3 and 2.4 refer only to the successive inclusions
$\mathcal{C}_{k}\to(\iota_{k}\mathcal{C}_{k+1})^{\mathbb{Z}/2\mathbb{Z}}\to\mathcal{C}_{k+1}$.
But they imply more general conditions, and these more general conditions will
be needed when studying groups $G$ other than $(\mathbb{Z}/2\mathbb{Z})^{n}$.

######  Statement 2.5.

In a flagged fully-dagger $(\infty,n)$-category $\mathcal{C}$, the map
$\mathcal{C}_{k}\to\mathcal{C}_{l}$ is essentially surjective on $(\leq
k)$-morphisms for every $0\leq k\leq l\leq n$. This map factors through
$(\iota_{k}\mathcal{C}_{l})^{(\mathbb{Z}/2\mathbb{Z})_{k+1}\times\dots\times(\mathbb{Z}/2\mathbb{Z})_{l}}$.
If $\mathcal{C}$ is univalent, then
$\mathcal{C}_{k}\to(\iota_{k}\mathcal{C}_{l})^{(\mathbb{Z}/2\mathbb{Z})_{k+1}\times\dots\times(\mathbb{Z}/2\mathbb{Z})_{l}}$
is fully faithful on $(\geq l)$-morphisms.

######  Definition 2.6.

Let $G\subset(\mathbb{Z}/2\mathbb{Z})^{n}$ be a subgroup. For any subinterval
$\\{k+1,\dots,l\\}\subset\\{1,\dots,n\\}$, define
$G(\\{k+1,\dots,l\\})=G\cap_{(\mathbb{Z}/2\mathbb{Z})^{n}}\prod_{j=k+1}^{l}(\mathbb{Z}/2\mathbb{Z})_{j}$.
A _flagged $G$-dagger $(\infty,n)$-category_ is a flagged
$(\infty,n)$-category $\mathcal{C}$ together with, for each $k$, a
$G(\\{1,\dots,k\\})$-volutive structure on $\mathcal{C}_{k}$ and, for each
$0\leq k<l\leq n$, and a factorization of $\mathcal{C}_{k}\to\mathcal{C}_{l}$
through a map
$\mathcal{C}_{k}\to(\iota_{k}\mathcal{C}_{l})^{G(\\{k+1,\dots,l\\})}$ of
$G(\\{1,\dots,k\\})$-volutive $(\infty,k)$-categories. A flagged $G$-dagger
$(\infty,n)$-category is _univalent_ when the maps
$\mathcal{C}_{k}\to(\iota_{k}\mathcal{C}_{l})^{G(\\{k+1,\dots,l\\})}$ are all
fully faithful on $(\geq l)$-morphisms, in which case the flagged $G$-dagger
$(\infty,n)$-category is a _$G$ -dagger $(\infty,n)$-category_.

We will write $G{\dagger}\mathbf{Cat}_{(\infty,n)}$ for the
$(\infty,1)$-category of $G$-dagger $(\infty,n)$-categories. The most
interesting case for examples is when $G=(\mathbb{Z}/2\mathbb{Z})_{n}$, in
which case we will also refer to $(\mathbb{Z}/2\mathbb{Z})_{n}$-dagger
$(\infty,n)$-categories as _top-dagger_. More generally, the cases that arise
in examples are when $G\subset(\mathbb{Z}/2\mathbb{Z})^{n}$ is a product of
some of the coordinate $(\mathbb{Z}/2\mathbb{Z})_{k}$s. For such a $G$,
Definition 2.6 simplifies: since in this case the maps
$\prod_{j={k+1}}^{l}G(\\{j\\})\to G(\\{k+1,\dots,l\\})$ are all isomorphisms,
one can restrict to just the successive inclusions
$\mathcal{C}_{k}\to\mathcal{C}_{k+1}$ in the definition and invoke a
$G$-version of Statement 2.5. We have suggested a more general definition in
anticipation of our discussion of unitary duality in Section 5.

##  3\. Dagger $n$-categories via enrichment

Let $\mathcal{V}$ be a symmetric monoidal $(\infty,1)$-category, with monoidal
structure
$\otimes_{\mathcal{V}}\colon\mathcal{V}\times\mathcal{V}\to\mathcal{V}$ and
monoidal unit $1_{\mathcal{V}}\in\mathcal{V}$. There is a notion of
_$\mathcal{V}$ -enriched $(\infty,1)$-category_, which we will abbreviate as
_$\mathcal{V}$ -category_. A full definition is provided by [GH15]; we will
recall the main ingredients. A _flagged $\mathcal{V}$-category_ $\mathcal{C}$
consists of a space $\mathcal{C}_{0}$ of objects and, for each
$x,y\in\mathcal{C}_{0}$, an object $\hom(x,y)\in\mathcal{V}$, together with a
coherently-associative and coherently-unital composition law
$\hom(y,z)\otimes_{\mathcal{V}}\hom(x,y)\to\hom(x,z)$. The _global sections_
functor $\Gamma:=\hom(1_{\mathcal{V}},-)\colon\mathcal{V}\to\mathbf{Space}$ is
lax-symmetric-monoidal, and so $\mathcal{C}$ induces a flagged plain
$(\infty,1)$-category $\Gamma\mathcal{C}$ with the same space
$\mathcal{C}_{0}$ of objects, and with spaces of morphisms given by
$\Gamma\hom(-,-)$. The flagged $\mathcal{V}$-category $\mathcal{C}$ is called
_univalent_ if $\Gamma\mathcal{C}$ is univalent. By definition, a
_$\mathcal{V}$ -category_ is a univalent flagged $\mathcal{V}$-category. We
will write $\mathbf{Cat}[\mathcal{V}]$ for the $(\infty,1)$-category of
$\mathcal{V}$-categories. The primordial example:
$\mathbf{Cat}[\mathbf{Cat}_{(\infty,n-1)}]\cong\mathbf{Cat}_{(\infty,n)}$.

Using the symmetry on $\mathcal{V}$, for each $\mathcal{V}$-category
$\mathcal{C}$ it should be possible to define an _opposite_
$\mathcal{V}$-category $\mathcal{C}^{\mathrm{op}}$ with the same objects and
morphisms but the opposite order of composition. This supplies an involution
$\mathbb{Z}/2\mathbb{Z}\to\operatorname{Aut}(\mathbf{Cat}[\mathcal{V}])$. We
can immediately generalize Definition 1.3:

######  Definition 3.1.

Let $\mathcal{C}$ be a $\mathcal{V}$-category, with space of objects
$\iota_{0}\mathcal{C}$. A _flagged dagger structure_ on $\mathcal{C}$ is a
fixed-point structure on $\mathcal{C}$ for the $\mathbb{Z}/2\mathbb{Z}$-action
on $\mathbf{Cat}[\mathcal{V}]$ together with a map of spaces
$\mathcal{C}_{0}\to(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}}$ such that
the induced map
$\mathcal{C}_{0}\to(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}}\to\iota_{0}\mathcal{C}$
is essentially surjective. A flagged dagger structure is _univalent_ when
$\mathcal{C}_{0}\to(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}}$ is fully
faithful. A _dagger $\mathcal{V}$-category_ is a $\mathcal{V}$-category with a
univalent flagged dagger structure.

We will write ${\dagger}\mathbf{Cat}[\mathcal{V}]$ for the
$(\infty,1)$-category of dagger $\mathcal{V}$-categories. We expect that the
equivalence
$\mathbf{Cat}[\mathbf{Cat}_{(\infty,n-1)}]\cong\mathbf{Cat}_{(\infty,n)}$
generalizes to dagger categories:

######  Statement 3.2.

There is an equivalence of $(\infty,1)$-categories

${\dagger}\mathbf{Cat}[{\dagger}\mathbf{Cat}_{(\infty,n-1)}]\cong{\dagger}\mathbf{Cat}_{(\infty,n)}.$

In particular, by iterating Statement 3.2, one arrives at an alternative model
of fully-dagger $(\infty,n)$-category than the one given in Definition 2.4.
Mixing $\mathbf{Cat}[-]$ and ${\dagger}\mathbf{Cat}[-]$ provides alternative
models of the versions with dagger structures on only some levels:

######  Statement 3.3.

There are equivalences of $(\infty,1)$-categories

$\displaystyle{\dagger}\mathbf{Cat}[\mathbf{Cat}_{(\infty,n-1)}]\cong(\mathbb{Z}/2\mathbb{Z})_{1}{\dagger}\mathbf{Cat}_{(\infty,n)}$
$\displaystyle\mathbf{Cat}[G{\dagger}\mathbf{Cat}_{(\infty,n-1)}]\cong(*\times
G){\dagger}\mathbf{Cat}_{(\infty,n)}$

For example, iterating the second of these equivalences supplies an
alternative model for top-dagger $(\infty,n)$-categories.

##  4\. Dagger bicategories

In this section, we will elaborate on how Definitions 2.4 and 2.6 play out in
the case of bicategories. Our goal is to outline a strictification for
coherent dagger bicategories analogous to Statement 1.2 for $1$-categories.
Even though our definition of a fully-dagger bicategory unpacks to something
complicated, it can be strictified to a “more traditionally defined” fully-
dagger bicategory involving less data (but more “evil”):

######  Definition 4.1.

A _bi-involutive bicategory_ is a bicategory $\mathcal{B}$ equipped with two
functors

$\displaystyle\dagger_{1}:\mathcal{B}\to\mathcal{B}^{1\mathrm{op}},\quad\dagger_{2}:\mathcal{B}\to\mathcal{B}^{2\mathrm{op}}$

such that

  1. (a)

$\dagger_{2}$ is the identity on objects and $1$-morphisms and strictly
squares to the identity;

  2. (b)

$\dagger_{1}$ is the identity on objects (but not necessarily on
$1$-morphisms) and weakly squares to the identity in the sense that it comes
equipped with a natural isomorphism
$\phi:\dagger_{1}^{2}\to\mathrm{id}_{\mathcal{B}}$, which is the identity on
objects (but not necessarily on $1$-morphisms).

There is a further condition, left to the reader, comparing the two ways to
trivialize $\dagger_{1}^{3}$. Additionally, the two daggers strongly commute:

$\dagger_{2}^{1\mathrm{op}}\circ\dagger_{1}=\dagger_{1}^{2\mathrm{op}}\circ\dagger_{2}.$

This equality should be compatible with the isomorphism
$\phi:\dagger_{1}^{2}\to\mathrm{id}_{\mathcal{B}}$, i.e. $\phi$ is unitary
with respect to $\dagger_{2}$.

Unpacking Definition 4.1, it is an enriched-type definition in the sense of
Section 3, so that Statement 3.2 for $n=2$ gives a justification for the hope
that bi-involutive bicategories are a model for fully dagger bicategories. In
other words, similarly to how a bicategory is a category weakly enriched in
categories, a bi-involutive bicategory is a $\dagger$-category ($\dagger_{1}$)
weakly enriched in $\dagger$-categories ($\dagger_{2}$).

We decided on the name “bi-involutive bicategory” because it generalizes the
notion of a bi-involutive tensor category of [HP17] to a bicategory with more
than one object.

######  Remark 4.2.

There are at least two other structures one might call $\dagger$-bicategories,
which are both $G$-dagger categories in the sense of 2.6: they correspond to
considering the two canonical $\mathbb{Z}/2\mathbb{Z}$-subgroups of
$\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$.††††††It could be
interesting to consider the diagonal subgroup. The first one leads to a
$\dagger_{2}$ which is the identity on objects and 1-morphisms as above,
giving a category weakly enriched in dagger categories as a special case of
Statement 3.3. The other structure leads to a $\dagger_{1}$ which is the
identity on objects as above, a dagger category weakly enriched in categories.

######  Example 4.3.

A natural example of a bi-involutive bicategory is the bicategory of von
Neumann algebras, Hilbert space bimodules equipped with commuting normal
actions and bounded bimodule homomorphisms [Sau83, BDH88, Con94, BDH14,
CHPJP22]. The daggers are given by

  * ${\dagger}_{1}$

For a Hilbert space bimodule ${}_{M}H_{N}$ for $M,N$ von Neumann algebras,
$H^{{\dagger}_{1}}$ is defined to be ${}_{N}\overline{H}_{M}$ where
$\overline{H}$ is the complex conjugate Hilbert space and the actions are
given by $b\cdot\overline{\xi}\cdot a=\overline{a^{*}\xi b^{*}}$

  * ${\dagger}_{2}$

For 2-morphisms (Hilbert space bimodule homomorphisms), the dagger is simply
given by the adjoint as maps between Hilbert spaces.

We note that there is actually an involution on objects (which we may call
${\dagger}_{0}$) given by taking the opposite algebra (or equivalently, the
complex conjugate algebra). One reason for this additional object-level
involution in monoidal bicategories is due to the extra
$\mathbb{Z}/2\mathbb{Z}$ action given by reversing the monoidal product.

######  Example 4.4.

Consider the case in which $\mathcal{B}$ has one object so that $\mathcal{B}$
is the delooping of a monoidal category $\mathcal{C}$. Having only a top-
dagger $\dagger_{2}$ on $\mathcal{B}$ is the same as having a monoidal dagger
structure on $\mathcal{C}$. On the other hand, having only a $\dagger_{1}$ on
$\mathcal{B}$ is the same as having a weak covariant involution
$\overline{(.)}$ on $\mathcal{C}$ that is op-monoidal in the sense that
$\overline{x\otimes y}\cong\overline{y}\otimes\overline{x}.$ Such structures
have been considered in the context of dagger categories in [Egg11].

In the rest of this section, we provide some ideas for a proof of the
following strictification result, which is a bicategorical analogue of the
main theorem of [SS23].

######  Statement 4.5.

The $(3,1)$-category of bi-involutive bicategories is equivalent to the
$(3,1)$-category of fully-dagger bicategories.

Firstly, a fully-volutive structure on a bicategory $\mathcal{B}$ consists of
a pair of equivalences
$\psi_{1}\colon\mathcal{B}\to\mathcal{B}^{1\mathrm{op}}$ and
$\psi_{2}\colon\mathcal{B}\to\mathcal{B}^{2\mathrm{op}}$, together with
natural isomorphisms
$\Omega_{1}\colon\psi_{1}^{1\mathrm{op}}\circ\psi_{1}\to\mathrm{id}_{\mathcal{B}}$,
$\Omega_{2}\colon\psi_{2}^{2\mathrm{op}}\circ\psi_{2}\to\mathrm{id}_{\mathcal{B}}$,
and
$\Omega_{12}\colon\psi_{2}^{1\mathrm{op}}\circ\psi_{1}\to\psi_{1}^{2\mathrm{op}}\circ\psi_{2}$,
and various modifications which satisfy various coherence conditions. To
provide a fully-dagger structure on this volutive bicategory, we need to
understand the relevant fixed point categories and specify our flaggings.

The bigroupoid
$(\iota_{0}\mathcal{B})^{\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}}$
has objects consisting of an element of $\mathcal{B}$ equipped with fixed
point data for both $\psi_{1}$ and $\psi_{2}$ together with compatibility data
between them, satisfying various coherence conditions. Similarly to the
$1$-categorical case, this in particular consists of isomorphisms
$h_{b}^{1}\colon\psi_{1}(b)\to b$ and $h_{b}^{2}\colon\psi_{2}(b)\to b$. The
space $\mathcal{B}_{0}$ in the definition of a dagger bicategory can be
identified with a subspace of
$(\iota_{0}\mathcal{B})^{\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}}$
such that the map to $\mathcal{B}$ is essentially surjective. This picks out
at least one preferred
$\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$-fixed point structure for
every object $b\in\mathcal{B}$.

The bicategory $\iota_{1}\mathcal{B}^{\mathbb{Z}/2\mathbb{Z}}$ has objects
given by an element of $\mathcal{B}$ equipped with fixed point data for
$\psi_{2}$. The 1-morphisms consist of pairs of a 1-morphism $f\colon b\to
b^{\prime}$ in $\mathcal{B}$ and 2-isomorphisms

(2)
${\psi_{2}(b)}$${b}$${\psi_{2}(b^{\prime})}$${b^{\prime}}$$\scriptstyle{h_{b}^{2}}$$\scriptstyle{\psi_{2}(f)}$$\scriptstyle{f}$$\scriptstyle{h_{b^{\prime}}^{2}}$

satisfying a natural coherence condition. One can think of this $2$-morphism
as the data specifying how $f$ is a unitary $1$-morphism. The bicategory
$\mathcal{B}_{1}$ is equivalent to a subcategory of
$\iota_{1}\mathcal{B}^{\mathbb{Z}/2\mathbb{Z}}$, whose objects can be
identified with those of $\mathcal{B}_{0}$, since
$\mathcal{B}_{0}\to\mathcal{B}_{1}$ is essentially surjective. The 1-morphisms
pick out at least one preferred $\mathbb{Z}/2\mathbb{Z}$-fixed point structure
on every 1-morphism. The 2-morphisms are fixed by the condition that the map
to $\mathcal{B}$ is fully faithful on 2-morphisms. Note how this additionally
fixes the 1-morphisms of $\mathcal{B}_{0}$: they are exactly those whose fixed
point structure restricts to one of the chosen fixed point structures in
$\mathcal{B}_{1}$. In summary, the structure of a dagger bicategory is
equivalent to a fully-volutive bicategory $\mathcal{B}$ together with a choice
of at least one $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$-fixed
point on every object $b\in\mathcal{B}$ and at least one compatible
$\mathbb{Z}/2\mathbb{Z}$-fixed point on every 1-morphism.

From this data we can construct a bi-involutive bicategory (analogous to the
construction in [SS23]) as follows. Consider the bicategory
$\mathcal{B}^{\prime}$ of which the objects are the objects of
$\mathcal{B}_{0}$ and 1-morphisms the 1-morphisms of $\mathcal{B}_{1}$. The
2-morphisms are those of $\mathcal{B}$, requiring no compatibility with the
fixed point data. By construction, there is a forgetful functor
$\mathcal{B}^{\prime}\to\mathcal{B}$, which is an equivalence of bicategories.
The anti-involutions $\psi_{1}$ and $\psi_{2}$ induce compatible anti-
involutions on $\mathcal{B}^{\prime}$.‡‡‡‡‡‡We expect the equivalence
$\mathcal{B}^{\prime}\to\mathcal{B}$ to come equipped with a canonical datum
saying it preserves the fully-volutive structures. Namely, $\dagger_{1}$ is
defined by the same formula as in the $1$-categorical case:

$\displaystyle f\colon b\to b^{\prime}$ $\displaystyle\longmapsto
h_{b}^{1}\circ\psi_{1}(f)\circ(h_{b^{\prime}}^{1})^{-1}.$

We do not spell out the fixed point structure (2) on the $1$-morphism
$f^{\dagger_{1}}$ here. There is a natural isomorphism
$\phi:\dagger_{1}^{2}\to\mathrm{id}_{\mathcal{B}^{\prime}}$ which is the
identity on objects and uses the fixed point data on $b$ and $b^{\prime}$ to
get a $2$-isomorphism $f^{\dagger_{1}\dagger_{1}}\cong f$ for a $1$-morphism
$f:b\to b^{\prime}$. The top-dagger
$\dagger_{2}\colon\mathcal{B}^{\prime}\longrightarrow\mathcal{B}^{\prime
2\mathrm{op}}$ is the identity on objects and 1-morphisms. It sends a
2-morphism $\vartheta\colon f\Rightarrow g$ to a version of
$\psi_{2}(\vartheta)$, where we identify its domain and target with $g$ and
$f$ respectively using their fixed point data. The functor $\dagger_{2}$
strictly squares to the identity. The two daggers commute strictly, and $\phi$
is $\dagger_{2}$-unitary. We have thus constructed a canonical bi-involutive
bicategory $\mathcal{B}^{\prime}$ from the coherent full $\dagger$ bicategory
$\mathcal{B}$.

##  5\. Dagger $n$-categories with unitary duals

An $(\infty,n)$-category $\mathcal{C}$ is said to _have adjoints_ , if for
each $1\leq k\leq n-1$ and each $k$-morphism $f:x\to y$ (between parallel
$(k-1)$-morphisms $x,y$), there exist $k$-morphisms $f^{R},f^{L}:y\to x$ and
$(k+1)$-morphisms $\eta^{R}\colon\mathrm{id}_{x}\to f^{R}\circ f$,
$\epsilon^{R}\colon f\circ f^{R}\to\mathrm{id}_{y}$,
$\eta^{L}\colon\mathrm{id}_{y}\to f\circ f^{L}$, $\epsilon^{L}\colon
f^{L}\circ f\to\mathrm{id}_{x}$ such that the _zig-zag_ compositions, which
after suppressing coherence (e.g. unitor and associator) information become

$\displaystyle
f\overset{f\eta^{R}}{\longrightarrow}ff^{R}f\overset{\epsilon^{R}f}{\longrightarrow}f,$
$\displaystyle
f^{R}\overset{\eta^{R}f^{R}}{\longrightarrow}f^{R}ff^{R}\overset{f^{R}\epsilon^{R}}{\longrightarrow}f^{R},$
$\displaystyle
f\overset{\eta^{L}f}{\longrightarrow}ff^{L}f\overset{f\epsilon^{L}}{\longrightarrow}f,$
$\displaystyle
f^{L}\overset{f^{L}\eta^{L}}{\longrightarrow}f^{L}ff^{L}\overset{\epsilon^{L}f^{L}}{\longrightarrow}f^{L},$

are equivalent to identities. Let
$\mathbf{AdjCat}_{(\infty,n)}\subset\mathbf{Cat}_{(\infty,n)}$ denote the full
sub-$(\infty,1)$-category on the $(\infty,n)$-categories with adjoints.

The theory of $(\infty,n)$-categories with adjoints has been well-studied, but
there are many questions remaining. The most famous work pertains to the
symmetric monoidal case, in which case dualizability is also imposed on
objects. We will call a symmetric monoidal category _rigid_ if it has adjoints
and also admits duals for objects. The _Cobordism Hypothesis_ of [BD95,
Lur09b] asserts that the free rigid symmetric monoidal $(\infty,n)$-category
generated by a single object is the $(\infty,n)$-category
$\mathbf{Bord}_{n}^{\mathrm{fr}}$ of framed $n$-dimensional bordisms — a
_framing_ $\tau$ on a smooth $n$-manifold $M$ is a trivialization of its
tangent bundle $\tau:\mathrm{T}_{M}\cong\mathbb{R}^{n}$. In other words, if
$\mathcal{C}$ is a rigid symmetric-monoidal $(\infty,n)$-category, with space
of objects $\iota_{0}\mathcal{C}$, then there is a canonical equivalence

$\iota_{0}\mathcal{C}\cong\hom_{\mathrm{sym\otimes}}(\mathbf{Bord}_{n}^{\mathrm{fr}},\mathcal{C}).$

The group $\mathrm{O}(n)$ obviously acts on the framings on a given
$n$-manifold, by rotating the trivialization, and so acts on the
$(\infty,n)$-category $\mathbf{Bord}_{n}^{\mathrm{fr}}$. This in turn supplies
a famous action of $\mathrm{O}(n)$ on $\iota_{0}\mathcal{C}$ for any rigid
symmetric monoidal $(\infty,n)$-category $\mathcal{C}$.

In fact, there is a larger group that acts, as explained in [Lur09b, Remark
2.4.30]. It makes sense to talk about a “tangent bundle” of a piecewise linear
(a.k.a. PL) manifold, but it is not a vector bundle: whereas the tangent
bundle of a smooth $n$-manifold $M$ is classified by a map
$\mathrm{T}_{M}:M\to\mathrm{B}\mathrm{O}(n)\simeq\mathrm{B}\mathrm{Diff}(n)$,
a tangent bundle of a PL manifold is classified by a map
$\mathrm{T}_{M}:M\to\mathrm{B}\mathrm{PL}(n)$. In particular, it makes sense
to talk about _framed PL manifolds_ : they are PL manifolds equipped with a
trivialization of $\mathrm{T}_{M}:M\to\mathrm{B}\mathrm{PL}(n)$. By
construction, any smoothing of a PL manifold lifts
$\mathrm{T}_{M}:M\to\mathrm{B}\mathrm{PL}(n)$ through
$\mathrm{B}\mathrm{Diff}(n)\simeq\mathrm{B}\mathrm{O}(n)$. We now quote a
nontrivial fact of differential topology, called the _Main Theorem of
Smoothing Theory_ :

######  Theorem 5.1 ([KS77, Essay IV]).

Let $M$ be a piecewise-linear manifold. Then the space of lifts of
$\mathrm{T}_{M}:M\to\mathrm{B}\mathrm{PL}(n)$ through
$\mathrm{B}\mathrm{O}(n)$ is homotopy equivalent to the space of smoothings of
$M$.

An immediate corollary is that a framed PL manifold has a unique (up to a
contractible space) smoothing that is compatible with the framing. In
particular, the framed bordism categories built from smooth or from PL
manifolds are equivalent. But $\mathrm{PL}(n)$ acts by rotating the framings
of $n$-dimensional PL-manifolds, and so acts on the PL version of
$\mathbf{Bord}^{\mathrm{fr}}_{n}$, and so acts on the space
$\iota_{0}\mathcal{C}$. This is the largest group that acts universally on the
objects of rigid $(\infty,n)$-categories:

######  Statement 5.2 ([Lur09b, Remark 2.4.30]).

Assuming the Cobordism Hypothesis, the rotate-the-framing map
$\mathrm{PL}(n)\to\operatorname{Aut}_{\mathrm{sym\otimes}}(\mathbf{Bord}_{n}^{\mathrm{fr}})$
is an equivalence when $n\neq 4$. The case $n=4$ is equivalent to the
4-dimensional piecewise-linear Schoenflies conjecture (which remains open).

The proof of Statement 5.2 has circulated among experts, but is not available
in print.

The story in the absence of symmetric monoidal structures is less well-
studied. The _Tangle Hypothesis_ and _Cobordism Hypothesis with Singularities_
amount to the existence of a graphical calculus for $(\infty,n)$-categories
with adjoints, generalizing the graphical calculi described for example in
[Sel11]. The precise details of this graphical calculus have not been worked
out in the literature. Roughly, the main ingredients are the following:

  1. (a)

The diagrams in the graphical calculus are drawn on networks of submanifolds
of the standard $\mathbb{R}^{n}$.

  2. (b)

Strata of codimension $k$ are labeled by $k$-morphisms.

  3. (c)

Strata are normally framed: if $X\subset\mathbb{R}^{n}$ is a codimension-$k$
submanifold, then it comes with a trivialization
$\nu:\mathrm{N}_{X}\cong\mathbb{R}^{k}$ of its normal bundle. More generally,
substrata of strata are relatively normally framed. This normal framing
encodes source and target information (and so must be consistent with the
labelings).

  4. (d)

Strata are tangentially framed: if $X\subset\mathbb{R}^{n}$ is a
codimension-$k$ submanifold, then it comes with a trivialization
$\tau:\mathrm{T}_{X}\cong\mathbb{R}^{n-k}$ of its tangent bundle. This
tangential framing encodes the direction of composition internal to the
morphism.

  5. (e)

The various framing data are compatible. For example, if
$X\subset\mathbb{R}^{n}$ is a non-sub stratum, it must come equipped with a
nullhomotopy of the composite isomorphism

$\mathbb{R}^{n}=\mathrm{T}_{\mathbb{R}^{n}}=\mathrm{N}_{X}\oplus\mathrm{T}_{X}\xrightarrow{\nu\oplus\tau}\mathbb{R}^{k}\oplus\mathbb{R}^{n-k}=\mathbb{R}^{n},$

where the left-hand equality simply uses that $X$ is a submanifold of the
standard $\mathbb{R}^{n}$. For substrata $Y\subset\dots\subset
X\subset\mathbb{R}^{n}$, there are similar but more complicated compatibility
conditions.

When $n=2$, one finds the well known calculus of “string diagrams.” The
framing-compatibility is probably the least familiar component of this
calculus. Along a 1-dimensional stratum, it is a nullhomotopy of an element of
$\mathrm{O}(2)$. That element is nullhomotopic only when it lives in
$\mathrm{SO}(2)$. This forces the normal and tangential framings to determine
each other. But there is more to a nullhomotopy than just its existence: an
element of $\mathrm{SO}(2)$ has a $\mathbb{Z}$-torsor of nullhomotopies.
Explicitly, the framing compatibility consists of a “winding number” carried
by each wire. This axiom prevents closed circles; it is what allows the string
diagram calculus to apply to non-pivotal 2-categories.

To say that $(\infty,n)$-categories with adjoints admit some graphical
calculus is to say that $(\infty,n)$-categories with adjoints are precisely
“interpreters” for such a graphical calculus. Suppose we are given such an
interpreter. Here is another interpreter: precompose your interpreter with
some element of $\mathrm{O}(n)$ acting on all input diagrams. Thus, assuming
that there is indeed such a graphical calculus, one finds an action of
$\mathrm{O}(n)$ on $\mathbf{AdjCat}_{(\infty,n)}$. We could act by any element
of $\mathrm{Diff}(n)$; since $\mathrm{O}(n)\hookrightarrow\mathrm{Diff}(n)$ is
a homotopy equivalence, this supplies “the same” action. But we were not
precise about what regularity of manifolds are allowed. Statement 5.1 and its
corollary about unique smoothings for framed PL manifolds says that one might
as well work with a PL diagrams, and these are in any case more natural to
draw. As such, we find an action of $\mathrm{PL}(n)$ on
$\mathbf{AdjCat}_{(\infty,n)}$. Given Statement 5.2, we expect:

######  Conjecture 5.3.

  1. (a)

There is a natural map

$\mathrm{PL}(n)\to\operatorname{Aut}(\mathbf{AdjCat}_{(\infty,n)}).$

  2. (b)

This map is an equivalence.

For the remainder of this article, we will assume part (a) of Conjecture 5.3,
and we will be motivated by our belief in part (b). We emphasize that, not
only is the current literature far from a proof of part (b), it does not even
supply a rigorous construction of this map in part (a) except when $n$ is very
low:

######  Example 5.4.

When $n=1$, having adjoints is vacuous, and
$\mathrm{PL}(1)=\mathbb{Z}/2\mathbb{Z}$ acts by
$\mathcal{C}\mapsto\mathcal{C}^{\mathrm{op}}$ as already described.

######  Example 5.5.

When $n=2$,
$\mathrm{PL}(2)=\mathrm{O}(2)=\mathrm{SO}(2)\rtimes\mathbb{Z}/2\mathbb{Z}=\mathrm{B}\mathbb{Z}\rtimes\mathbb{Z}/2\mathbb{Z}$.
The $\mathbb{Z}/2\mathbb{Z}$ subgroup acts by taking
$\mathcal{C}\in\mathbf{AdjCat}_{(\infty,2)}$ to $\mathcal{C}^{2\mathrm{op}}$,
with the opposite composition of $2$-morphisms. Note that there is an
equivalence $\mathcal{C}^{2\mathrm{op}}\simeq\mathcal{C}^{1\mathrm{op}}$ which
is the identity on objects and which acts on 1-morphisms by $f\mapsto f^{R}$;
as such, the assignment $\mathcal{C}\mapsto\mathcal{C}^{1\mathrm{op}}$ would
produce an equivalent $\mathbb{Z}/2\mathbb{Z}$-action on
$\mathbf{AdjCat}_{(\infty,n)}$. The generator of $\mathrm{B}\mathbb{Z}$ acts
on $\mathbf{AdjCat}_{(\infty,2)}$ by a natural automorphism of
$\mathrm{id}_{\mathbf{AdjCat}_{(\infty,n)}}$. Such a natural automorphism has
components: its component at $\mathcal{C}\in\mathbf{AdjCat}_{(\infty,2)}$ is
the autofunctor $\mathcal{C}\to\mathcal{C}$ that is the identity on objects
and takes a 1-morphism $f$ to its double-right-dual $f^{RR}$. To fully present
an action of $\mathrm{B}\mathbb{Z}\rtimes\mathbb{Z}/2\mathbb{Z}$ is to present
a map out of the space
$\mathrm{B}(\mathrm{B}\mathbb{Z}\rtimes\mathbb{Z}/2\mathbb{Z})$, which is the
total space of a nontrivial $\mathbb{C}P^{\infty}$-bundle over
$\mathbb{R}P^{\infty}$. The fact that $O(2)\ncong O(1)\times SO(2)$ is
expressed by the natural isomorphism
$(()^{RR})^{2\mathrm{op}}\cong(()^{2\mathrm{op}})()^{RR-1}$, using opposites
take left to right adjoints. The higher cells in (a cell model for) this space
map to compatibility data between the actions of lower cells.

In analogy with Definition 2.2, we declare:

######  Definition 5.6.

A $(\infty,n)$-category with adjoints
$\mathcal{C}\in\mathbf{AdjCat}_{(\infty,n)}$ is _$\mathrm{PL}(n)$ -volutive_
if it is (equipped with the structure of) a fixed point for the
$\mathrm{PL}(n)$-action on $\mathbf{AdjCat}_{(\infty,n)}$.

An $(\infty,n-1)$-category, thought of as an $(\infty,n)$-category, never has
all adjoints unless it is a groupoid: there is no organic inclusion
$\mathbf{AdjCat}_{(\infty,n-1)}\to\mathbf{AdjCat}_{(\infty,n)}$. The other
direction, however, does work: if $\mathcal{C}$ is a $(\infty,n)$-category
with adjoints, then its underlying $(\infty,n-1)$-category
$\iota_{n-1}\mathcal{C}$ has adjoints. We will say that a flagged
$(\infty,n)$-category
$\mathcal{C}_{0}\to\mathcal{C}_{1}\to\dots\to\mathcal{C}_{n}$ _has adjoints_
if every $\mathcal{C}_{k}$ does.

The functor

$\iota_{n-1}:\mathbf{AdjCat}_{(\infty,n)}\to\mathbf{AdjCat}_{(\infty,n-1)}$

cannot be $\mathrm{PL}(n)$-equivariant simply because $\mathrm{PL}(n)$ does
not act on the codomain. But it is equivariant for
$\mathrm{PL}(n-1)\subset\mathrm{PL}(n)$. Even better: $\iota_{n-1}$ is
$(\mathrm{PL}(n-1)\times\mathrm{PL}(1))$-equivariant, where on the domain the
action is via the inclusion into $\mathrm{PL}(n)$, and on the codomain the
$\mathrm{PL}(1)$-factor acts trivially. The upshot: if $\mathcal{C}$ is
$\mathrm{PL}(n)$-volutive, then $\iota_{n-1}\mathcal{C}$ is
$\mathrm{PL}(n-1)$-volutive and also carries a
$\mathbb{Z}/2\mathbb{Z}$-action. More generally, the functor
$\iota_{k}\colon\mathbf{AdjCat}_{(\infty,n)}\to\mathbf{AdjCat}_{(\infty,k)}$
is $\mathrm{PL}(k)\times\mathrm{PL}(n-k)$ equivariant, where
$\mathrm{PL}(n-k)$ acts on the codomain trivially. We are led naturally to the
following version of Definitions 2.3 and 2.4 for categories with adjoints:

######  Definition 5.7.

A _flagged $\mathrm{PL}$-dagger $(\infty,n)$-category_, also called a _flagged
(fully-)dagger $(\infty,n)$-category with unitary duality_, is a flagged
$(\infty,n)$-category with adjoints

$\mathcal{C}_{0}\to\mathcal{C}_{1}\to\dots\to\mathcal{C}_{n}$

such that the $(\infty,k)$-category $\mathcal{C}_{k}$ is
$\mathrm{PL}(k)$-volutive, and each functor
$\mathcal{C}_{k}\to\mathcal{C}_{k+j}$ is
$(\mathrm{PL}(k)\times\mathrm{PL}(j))$-volutive, with trivial***More
precisely, the $\mathrm{PL}(k)\times\mathrm{PL}(j)$-volutionon
$\mathcal{C}_{k}$ is induced by pulling back its
$\mathrm{PL}(k)$-volutionalong the projection
$\mathrm{PL}(k)\times\mathrm{PL}(j)\to\mathrm{PL}(k)$, and on
$\mathcal{C}_{k+j}$ it is induced by pulling back its
$\mathrm{PL}(k+j)$-volutionalong the inclusion
$\mathrm{PL}(k)\times\mathrm{PL}(j)\to\mathrm{PL}(k+j)$.
$\mathrm{PL}(j)$-volutionon $\mathcal{C}_{k}$.

A flagged $\mathrm{PL}$-dagger $(\infty,n)$-category is _univalent_ if the
$\mathrm{PL}(k)$-volutive functor
$\mathcal{C}_{k}\to(\iota_{k}\mathcal{C}_{k+1})^{\mathrm{PL}(1)}$ is fully
faithful on $(>k)$-morphisms. A _$\mathrm{PL}$ -dagger $(\infty,n)$-category_,
also called a _dagger $(\infty,n)$-category with unitary duality_, is a
univalent flagged one.

######  Remark 5.8.

For good families of groups $G(n)$ related to $\mathrm{PL}(n)$, we expect that
there is a definition of $G$-dagger categories with adjoints generalizing
Definition 2.6. Examples should include, in particular, $\mathrm{O}(n)$ and
$\mathrm{SO}(n)$. We will not try to work out the precise conditions on $G(n)$
or all details of the definition here.

Let us turn now to justifying the name “with unitary duality.” We will do so
by unpacking the notion in the case of bicategories in Examples 5.9 and 5.10.
Before that, note that the forgetful functor
$\mathbf{AdjCat}_{(\infty,n)}\to\mathbf{Cat}_{(\infty,n)}$ is
$(\mathbb{Z}/2\mathbb{Z})^{n}$ equivariant, where
$(\mathbb{Z}/2\mathbb{Z})^{n}$ acts through the map
$(\mathbb{Z}/2\mathbb{Z})^{n}=\mathrm{PL}(1)^{n}\to\mathrm{PL}(n)$. Thus to
every dagger $(\infty,n)$-category with unitary duality we can assign an
underlying (fully-)dagger $(\infty,n)$-category in which the unitary duality
is forgotten. So the question is to understand what this extra “unitary
duality” data looks like.

######  Example 5.9.

We specialize from $(\infty,n)$-categories to bicategories and look at the
$\mathrm{SO}(2)$ subgroup of $\mathrm{PL}(2)=\mathrm{O}(2)$. What is the data
of an $\mathrm{SO}(2)$-volution on a bicategory $\mathcal{B}$? Fixed-point
data for an $\mathrm{SO}(2)$($=\mathrm{B}\mathbb{Z}$)-action consists of data
assigned to each cell in $\mathrm{B}\mathrm{SO}(2)=\mathbb{C}P^{\infty}$;
there is one cell in each even dimension. The $0$-cell selects the bicategory
$\mathcal{B}$. The $2$-cell selects a trivialization $\theta$ of $(-)^{RR}$:
such a trivialization unpacks to a family of 1-isomorphisms $\theta_{b}\colon
b\to b$ for all $b\in\mathcal{B}$ and 2-isomorphisms
$f^{RR}\circ\theta_{b_{1}}\to\theta_{b_{2}}\circ f$ for $1$-morphisms
$f:b_{1}\to b_{2}$ satisfying various coherence conditions. The $4$-cell
selects a quadratic equation that $\theta$ must solve; the reader is
encouraged to work out this equation as an exercise. In the special case of
bicategories, this is all the necessary data: the higher cells in
$\mathbb{C}P^{\infty}$ admit unique assignments, because the space of
bicategories with adjoints is a homotopy 3-type.

The group $\mathrm{SO}(1)$ is trivial, and so to enhance an
$\mathrm{SO}(2)$-volutive bicategory $\mathcal{B}$ to an
$\mathrm{SO}(2)$-dagger bicategory, one needs only to supply the data of an
$\mathrm{SO}(2)$-volutive essentially surjective functor
$\mathcal{B}_{0}\to\mathcal{B}$ such that
$\mathcal{B}_{0}\to(\iota_{0}\mathcal{B})^{\mathrm{SO}(2)}$ is 1-fully
faithful. The objects $(\iota_{0}\mathcal{B})^{\mathrm{SO}(2)}$ are given by
pairs consisting of of an object $b\in\mathcal{B}$ and a 2-isomorphism
$\omega_{b}\colon\theta_{b}\to\mathrm{id}_{b}$. Similar to Section 4, form a
new bicategory $\mathcal{B}^{\prime}$ whose objects are those of
$\mathcal{B}_{0}$ and morphisms are those in $\mathcal{B}$. This bicategory
$\mathcal{B}^{\prime}$ comes with a trivialization of the double dual functor
$(-)^{RR}$ which is the identity on objects (and which solves a quadratic
equation). In other words, $\mathcal{B}^{\prime}$ is a _pivotal bicategory_ as
defined for example in [FSY23, Definition 2.1], also called an _even-handed
bicategory_ [Bar09]. Like traditionally-defined dagger $1$-categories, pivotal
bicategories are “evil” in the sense that pivotal structures do not transport
across bicategorical equivalences. Remark 5.8 suggests a coherent version of
“pivotal bicategory”: they are the $\mathrm{SO}(2)$-dagger bicategories.

To finish the discussion of the name “unitary duality,” we now restore the
reflection.

######  Example 5.10.

As mentioned in Example 5.5, to discuss actions by, and fixed points for,
$\mathrm{O}(2)=\mathrm{B}\mathbb{Z}\rtimes\mathbb{Z}/2\mathbb{Z}$, one should
assign data to the cells in a cell model for $\mathrm{B}\mathrm{O}(2)$, which
is a $\mathbb{C}P^{\infty}$-bundle over $\mathbb{R}P^{\infty}$. In particular,
$\mathrm{B}\mathrm{O}(2)$ has a cell model with one cell of dimension $2s+r$
for each pair $s,r\in\mathbb{N}$. The cells indexed $(s,0)$ supply the
restriction of the data along $\mathrm{SO}(2)\to\mathrm{O}(2)$, and the cells
indexed $(0,r)$ supply the restriction of the data along
$(\mathbb{Z}/2\mathbb{Z})_{2}\subset\mathrm{O}(2)$. We see that an
$\mathrm{O}(2)$-volution consists of a $(\mathbb{Z}/2\mathbb{Z})_{2}$-volution
$\psi_{2}:\mathcal{B}\to\mathcal{B}^{2\mathrm{op}}$, a trivialization
$f^{RR}\to\theta_{b_{2}}\circ f\circ\theta_{b_{1}}^{-1}$ as in Example 5.9,
together with a natural modification equating the two ways to identify
$\psi_{2}(f^{RR})\cong\psi_{2}(f)^{LL}$ with $\psi_{2}(f)$, satisfying some
conditions. This in particular includes a $(\mathbb{Z}/2\mathbb{Z})_{2}$-fixed
point datum $\psi_{2}(\theta_{b}))\cong\theta_{\psi_{2}(b_{1})}^{-1}$. We see
that an $\mathrm{O}(2)$-dagger structure consists of

  1. (a)

an $\mathrm{O}(2)$-equivariant essentially surjective functor
$\mathcal{B}_{0}\to\mathcal{B}$ from a $2$-groupoid such that
$\mathcal{B}_{0}\to(\iota_{0}\mathcal{B})^{\mathrm{O}(2)}$ is fully faithful.
The $\mathrm{O}(2)$-fixed points combine the trivializations $\omega$ of
Example 5.9 from $\mathrm{SO}(2)$ with the $\mathrm{O}(1)$-fixed point data
$h_{b}^{2}$ such that $\omega$ is compatible with
$\psi_{2}(\theta_{b}))\cong\theta_{\psi_{2}(b_{1})}^{-1}$ and the
$2$-isomorphism $f^{RR}\to\theta_{b_{2}}\circ f\circ\theta_{b_{1}}^{-1}$
preserves $(\mathbb{Z}/2\mathbb{Z})_{2}$-fixed point data.

  2. (b)

As in Section 4, the $(2,1)$-category $\mathcal{B}_{1}$ and the functors
$\mathcal{B}_{1}\to\mathcal{B}$ and $\mathcal{B}_{0}\to\mathcal{B}_{1}$ are
fixed after specifying $(\mathbb{Z}/2\mathbb{Z})_{2}$-fixed point data on
$1$-morphisms of $\mathcal{B}$.

  3. (c)

The functors $\mathcal{B}_{0}\to\mathcal{B}_{1}$ and
$\mathcal{B}_{1}\to\mathcal{B}$ are still required to be
$(\mathbb{Z}/2\mathbb{Z})^{2}$-equivariant. As explained in Section 4, the
$(\mathbb{Z}/2\mathbb{Z})_{2}$ simply specifies the agreement of fixed point
data on $2$-morphisms. For the $(\mathbb{Z}/2\mathbb{Z})_{1}$-equivariance,
first note that the underlying $(\mathbb{Z}/2\mathbb{Z})_{1}$-volution of the
$\mathrm{O}(2)$-volution can be described by $\psi_{1}=(.)^{R}\circ\psi_{2}$,
where we made an arbitrary choice of the right adjoint to trivialize the
$180^{\circ}$ rotation in $\mathrm{SO}(2)$. Using the pivotal structure and
the fact that $\psi_{2}(f^{R})\cong\psi_{2}(f)^{L}$, we obtain data specifying
that $\psi_{2}$ and $(.)^{R}$ commute. Note that the
$(\mathbb{Z}/2\mathbb{Z})_{1}$-fixed point data induced by the
$(\mathbb{Z}/2\mathbb{Z})_{2}$-fixed point data $h^{2}_{b}$ is
$(h^{2}_{b})^{R}$. The $(\mathbb{Z}/2\mathbb{Z})_{1}$-equivariance data of
$\mathcal{B}_{1}\to(\iota_{1}\mathcal{B})^{(\mathbb{Z}/2\mathbb{Z})_{2}}$ will
ensure that if $h_{f}$ is a $(\mathbb{Z}/2\mathbb{Z})_{2}$-fixed point data on
a $1$-morphism, then there is a canonical fixed point data $h_{f^{R}}$ for
$f^{R}$, which makes the right adjoint into a top dagger functor.

When the bicategory $\mathcal{B}$ is the one-object delooping of a rigid
monoidal 1-category $\mathcal{C}$, and $\mathcal{B}_{0}$ is declared to
consist exactly of this object, this should recover the notion of a dagger
category with a unitary dual functor [Sel11, Definition on page 53 part (2)],
see also [Pen20].

As in Definition 1.6, one can also define _coflagged_ $\mathrm{PL}$-dagger
$(\infty,n)$-categories by keeping the univalence axiom Definition 5.7 but
dropping the essential surjectivity (i.e. not requiring that
$\mathcal{C}_{0}\to\dots\to\mathcal{C}_{n}$ be a flagging). Statement 1.7
generalizes: every coflagged $\mathrm{PL}$-dagger category can be completed to
a $\mathrm{PL}$-dagger category by replacing each $\mathcal{C}_{k}$ with the
full image of $\mathcal{C}_{k-1}$; this is the right adjoint to the forgetful
functor from $\mathrm{PL}$-dagger to coflagged $\mathrm{PL}$-dagger
categories. The further forgetful functor from coflagged $\mathrm{PL}$-dagger
to $\mathrm{PL}$-anti-involutive categories that remembers only the top level
also has a right adjoint which assigns to a $\mathrm{PL}$-anti-involutive
$(\infty,n)$-category $\mathcal{C}$ the diagram

(3)
$(\iota_{0}\mathcal{C})^{\mathrm{PL}(n)}\to(\iota_{1}\mathcal{C})^{\mathrm{PL}(n-1)}\to\dots\to\mathcal{C}.$

##  6\. Bordism categories and reflection-positive topological quantum field
theories

For some of us, our interest in higher dagger categories stems from the role
that we expect them to play in the study of (topological) quantum field
theories. For any space $X\to\mathrm{B}\mathrm{O}(n)$ over
$\mathrm{B}\mathrm{O}(n)$ there is a rigid symmetric monoidal
$(\infty,n)$-category $\mathbf{Bord}_{n}^{X}$ of bordisms $M$ with a lift of
their tangent bundle $\mathrm{T}_{M}\colon M\to\mathrm{B}\mathrm{O}(n)$
through $X$. (The construction of $\mathbf{Bord}_{n}^{X}$ is outlined in
[Lur09b] and fully implemented by one of us in [CS19].) For example, a framing
is a trivialization of $\mathrm{T}_{M}$, i.e. a lift through
$\ast\to\mathrm{B}\mathrm{O}(n)$, and
$\mathbf{Bord}_{n}^{\ast}=\mathbf{Bord}_{n}^{\mathrm{fr}}$. Let $\mathcal{C}$
be a symmetric monoidal $(\infty,n)$-category. An _$X$ -structured
$n$-dimensional fully extended topological quantum field theory with values in
$\mathcal{C}$_ is a symmetric monoidal functor
$\mathcal{Z}\colon\mathbf{Bord}_{n}^{X}\to\mathcal{C}$. Since the bordism
category is rigid, every topological quantum field theory will factor through
the maximal rigid sub-$(\infty,n)$-category of $\mathcal{C}$. Hence we will
from now on assume without loss of generality that $\mathcal{C}$ is rigid.
Recall that the Cobordism Hypothesis allows us to identify fully extended
framed topological field theories with $\iota_{0}\mathcal{C}$, which hence
carries a $\mathrm{PL}(n)$-action. This was the starting point for the
discussion in the previous section. The Cobordism Hypothesis also provides a
description of the space of $X$-structured fully extended topological quantum
field theories as the space of $X$-fixed points $(\iota_{0}\mathcal{C})^{X}$.

As an example, let us focus on unoriented smooth $n$-dimensional topological
quantum field theories valued in $\mathcal{C}$. The space of such theories is
$(\iota_{0}\mathcal{C})^{\mathrm{O}(n)}$. The $1$-morphisms in $\mathcal{C}$
are the objects of a rigid symmetric monoidal $(\infty,n-1)$-category
$\iota_{n-1}\mathcal{C}^{\to}$ [JFS17, Section 7], and thus carry an action by
$\mathrm{O}(n-1)$; the maps that assign to a 1-morphism its source and target
and that compose 1-morphisms are $\mathrm{O}(n-1)$-equivariant, and compile
into an action of $\mathrm{O}(n-1)$ on the whole $(\infty,1)$-category
$\iota_{1}\mathcal{C}$. This action is meaningful from the perspective of
topological quantum field theory: there is an $\mathrm{O}(n-1)$-action on the
space of 1-morphisms between the objects underlying two different unoriented
topological field theories, and the Stratified Cobordism Hypothesis says that
fixed points for this $\mathrm{O}(n-1)$-action classify unoriented
codimension-$1$ defects between the two field theories. Similarly,
$2$-morphisms carry an induced $\mathrm{O}(n-2)$-action whose fixed points
classified unoriented codimension-$2$ defects, and so on. In summary,
unoriented smooth defects assemble into the following diagram:

(4)
$(\iota_{0}\mathcal{C})^{\mathrm{O}(n)}\to(\iota_{1}\mathcal{C})^{\mathrm{O}(n-1)}\to\dots\to\mathcal{C}.$

There is nothing special in (4) about unoriented smooth theories and the
groups $\mathrm{O}(k)$: any good sequence of groups $G(k)$ would work (compare
Remark 5.8). For example, replacing the $\mathrm{O}(k)$’s in (4) with
$\mathrm{SO}(k)$’s would organize the defects between oriented smooth field
theories as analyzed for example in [DKR11, CMS20]; replacing the
$\mathrm{O}(k)$’s with $\mathrm{PL}(k)$’s would organize the defects between
unoriented piecewise linear field theories.

The diagram (4) is suspiciously close to (3) but they are a priori different:
the action of $\mathrm{O}(k)$ on $\iota_{n-k}\mathcal{C}$ in (4) comes from
the Cobordism Hypothesis and uses the symmetric monoidal structure on
$\mathcal{C}$, whereas in (3) we envisioned selecting a
$\mathrm{PL}(n)$-volution on $\mathcal{C}$, i.e. fixed-point data for the
action of $\mathrm{PL}(n)$ on the graphical calculus for $\mathcal{C}$. We
expect the relation to be the following. Any symmetric monoidal
$(\infty,n)$-category $\mathcal{C}$ determines a symmetric monoidal
$(\infty,n+1)$-category $\mathrm{B}\mathcal{C}$ with one object $\bullet$ and
$\operatorname{End}_{\mathrm{B}\mathcal{C}}(\bullet)=\mathcal{C}$; iterating
this supplies $(\infty,n+m)$-categories $\mathrm{B}^{m}\mathcal{C}$ for every
$m\in\mathbb{N}$. If $\mathcal{C}$ is rigid, then $\mathrm{B}^{m}\mathcal{C}$
will have all adjoints. The graphical calculi for $\mathcal{C}$ and for
$\mathrm{B}^{m}\mathcal{C}$ are compatible via the embedding of
$\mathbb{R}^{n}$ into $\mathbb{R}^{n+m}$ as the last $n$ coordinates. Hence we
can think of the graphical calculus for a rigid symmetric monoidal
$(\infty,n)$-category as taking place in $\mathbb{R}^{\infty}$. The space of
embeddings (of any finite-dimensional object) into $\mathbb{R}^{\infty}$ is
contractible. This contractibility selects a canonical trivialization of the
$\mathrm{PL}(n)$-action predicted in Conjecture 5.3 for any rigid symmetric
monoidal $(\infty,n)$-category. In other words, if we let
$\mathbf{SymRigidCat}_{(\infty,n)}$ denote the $(\infty,1)$-category of rigid
symmetric monoidal $(\infty,n)$-categories:

######  Statement 6.1.

The forgetful functor

$\mathbf{SymRigidCat}_{(\infty,n)}\to\mathbf{AdjCat}_{(\infty,n)}$

factors through the $\mathrm{PL}(n)$-fixed points
$\mathbf{AdjCat}_{(\infty,n)}^{\mathrm{PL}(n)}\to\mathbf{AdjCat}_{(\infty,n)}$.

In particular, a rigid symmetric monoidal structure on a $(\infty,n)$-category
selects a canonical $\mathrm{PL}(n)$-volutive structure on its underlying
$(\infty,n)$-category with adjoints. Moreover, any symmetric monoidal functor
between rigid symmetric monoidal $(\infty,n)$-categories will automatically
intertwine these canonical $\mathrm{PL}(n)$-volutive structures.

Given a rigid symmetric monoidal $(\infty,n)$-category $\mathcal{C}$, we
expect that the induced $\mathrm{PL}(n)$-action on $\iota_{0}\mathcal{C}$
agrees with the one coming from the Cobordism Hypothesis.

Other symmetric monoidal $\mathrm{PL}(n)$-volutive structures on $\mathcal{C}$
are given by twisting the canonical one by a symmetric monoidal
$\mathrm{PL}(n)$-action on $\mathcal{C}$: symmetric monoidal
$\mathrm{PL}(n)$-volutive structures form a trivialized torsor over
$\mathrm{PL}(n)$-actions.

There is a straightforward way to define symmetric monoidal
$\mathrm{PL}(n)$-dagger categories by simply requiring the volution, the
flagging by $\mathcal{C}_{i}$’s, and the trivialization data, to all be
symmetric monoidal. Interestingly, if $\mathcal{C}$ is in addition rigid the
sequence of deloopings
$\mathcal{C},\mathrm{B}\mathcal{C},\mathrm{B}^{2}\mathcal{C},\dots$ allows
more. Suppose that

(5) $\mathcal{C}_{0}\to\mathcal{C}_{1}\to\dots\to\mathcal{C}_{n}=\mathcal{C}$

is a symmetric monoidal flagged $(\infty,n)$-category. Then the deloopings
assemble into a flagged $(\infty,n+1)$-category by selecting the unit object
at the bottom:

(6)
$\\{\bullet\\}\to\mathrm{B}\mathcal{C}_{0}\to\mathrm{B}\mathcal{C}_{1}\to\dots\mathrm{B}\mathcal{C}_{n}=\mathrm{B}\mathcal{C}.$

If $\mathcal{C}_{k}$ is rigid as a symmetric monoidal $(\infty,k)$-category,
then $\mathrm{B}\mathcal{C}_{k}$ is also rigid, now as a symmetric monoidal
$(\infty,k+1)$-category.

Given such a structure, just like it is natural to ask for a symmetric
monoidal flagged $\mathrm{PL}(n)$-dagger structure on a chosen flagging (5),
it is natural to ask to give $\mathrm{B}\mathcal{C}$ a flagged
$\mathrm{PL}(n+1)$-dagger structure with chosen flagging (6). Iterating, it is
natural to ask to give $\mathrm{B}^{m}\mathcal{C}$ a compatible symmetric
monoidal $\mathrm{PL}(n+m)$-dagger structure. We call the structure just
sketched a _$\mathrm{PL}$ -dagger tower with underlying $(\infty,n)$-category
$\mathcal{C}$_.

After untwisting by the canonical volutive structures, a symmetric monoidal
$\mathrm{PL}(n+m)$-dagger structure on $\mathrm{B}^{m}\mathcal{C}$ unpacks to
a symmetric monoidal action of $\mathrm{PL}(k+m)$ on $\mathcal{C}_{k}$ for all
$k\leq n$, and compatible $(\mathrm{PL}(k+m)\times\mathrm{PL}(j))$-equivariant
maps $\mathcal{C}_{k}\to\mathcal{C}_{k+j}$ for all $j+k\leq n$. Univalence is
just about the $\mathrm{PL}(j)$-fixed points: it is independent of $m$. A more
careful treatment and definition is beyond the scope of this short article.

######  Example 6.2.

To illustrate the difference between the two definitions above, we look at
dagger 1-categories. A (strict) symmetric monoidal dagger 1-category is a
symmetric monoidal category equipped with a symmetric monoidal anti-involution
which is the identity on objects. From Example 5.10 we learn that a
$\mathrm{PL}$-dagger tower with underlying 1-category is in addition equipped
with a unitary dual functor. There are no additional structures corresponding
to higher deloopings, since they would correspond to higher morphisms.

We will now construct a dagger structure on certain bordism categories
$\mathbf{Bord}_{n}^{X}$. Let us recall some more details about these
$(\infty,n)$-categories. They depend on a choice of tangential structure
$X\to\mathrm{B}\mathrm{PL}(n)$ — for example, by smoothing theory (Theorem
5.1), the smooth unoriented bordism category
$\mathbf{Bord}_{n}^{\mathrm{O}(n)}$ corresponds to the tangential structure
$\mathrm{B}\mathrm{O}(n)\to\mathrm{B}\mathrm{PL}(n)$. Given $k\leq n$, set
$X(k)\to\mathrm{PL}(k)$ to be the pullback of $X\to\mathrm{PL}(n)$ along the
standard inclusion $\mathrm{B}\mathrm{PL}(k)\to\mathrm{B}\mathrm{PL}(n)$. A
$k$-morphism in $\mathbf{Bord}_{n}^{X}$ is by definition a $k$-dimensional
cobordism $M^{k}$ equipped with a lift of its tangent bundle
$\mathrm{T}_{M}\colon M\to\mathrm{B}\mathrm{PL}(k)$ through $X(k)$; we will
refer to such a lift simply as an _$X$ -structure_.

Just as the tangential structure $X(k)$ on $k$-morphisms is pulled back from
an $n$-dimensional tangential structure, it often happens that the input
$n$-dimensional tangential structure is itself pulled back from a higher-
dimensional tangential structure:

######  Definition 6.3.

A _stabilization_ of an $n$-dimensional tangential structure
$X(n)\to\mathrm{B}\mathrm{PL}(n)$ is a map
$X\to\mathrm{B}\mathrm{PL}=\injlim_{n\to\infty}\mathrm{B}\mathrm{PL}(n)$ and
an equivalence $X(n)\cong
X\times_{\mathrm{B}\mathrm{PL}}\mathrm{B}\mathrm{PL}(n)$. A tangential
structure is _stable_ when it is equipped with a stabilization.

We emphasize that stability is structure, not just a property.

######  Example 6.4.

Define a _stable smooth structure_ on a PL-manifold $M$ to be a lift of its
stabilized tangent bundle through
$\mathrm{B}\mathrm{O}\to\mathrm{B}\mathrm{PL}$. In other words, for an
$n$-manifold, stable smoothness corresponds to the tangential structure
$\mathrm{B}\mathrm{O}\times_{\mathrm{B}\mathrm{PL}}\mathrm{B}\mathrm{PL}(n)$.
Stable smoothness is by construction a stable tangential structure.

Stable smoothness is weaker than smoothness: the map
$\mathrm{B}\mathrm{O}(n)\to\mathrm{B}\mathrm{O}\times_{\mathrm{B}\mathrm{PL}}\mathrm{B}\mathrm{PL}(n)$
is not a homotopy equivalence. Indeed, actual-smoothness is not a stable
tangential structure. Write $\mathbf{Bord}_{n}^{\mathrm{O}(n)}$ for the
(unoriented) actually-smooth bordism $(\infty,n)$-category and
$\mathbf{Bord}_{n}^{\mathrm{O}}$ for the stably-smooth version. The map
$\mathbf{Bord}_{n}^{\mathrm{O}(n)}\to\mathbf{Bord}_{n}^{\mathrm{O}}$ is not an
equivalence of $(\infty,n)$-categories.

That said, a careful analysis of smoothing theory shows that the map
$\mathrm{B}\mathrm{O}(n)\to\mathrm{B}\mathrm{O}\times_{\mathrm{B}\mathrm{PL}}\mathrm{B}\mathrm{PL}(n)$
is $n$-connected; in particular, this follows from [Lur09a, Lecture 21,
Theorem 1]. In terms of bordism categories, this implies that
$\mathbf{Bord}_{n}^{\mathrm{O}(n)}\to\mathbf{Bord}_{n}^{\mathrm{O}}$ becomes
an equivalence after quotienting to weak $n$-, a.k.a. $(n,n)$-, categories.
The upshot is that for TQFTs valued in a weak $n$-category, the notions of
smooth and stably-smooth TQFT agree.

The description of $k$-morphisms in $\mathbf{Bord}_{n}^{X}$ as $X$-structured
$k$-dimensional cobordisms does not precisely present the actual space of
$k$-morphisms in $\mathbf{Bord}_{n}^{X}$. Indeed, the natural space of
$k$-dimensional $X$-structured bordisms has as its equivalences the
$X$-structured diffeomorphisms. But the categorical equivalences in
$\mathbf{Bord}_{n}^{X}$ between $(<n)$-dimensional morphisms also include
$X$-structured h-cobordisms, and in high dimensions not every h-cobordism
comes from a diffeomorphism. Rather, this description presents a flagged
$(\infty,n)$-category of bordisms as

(7)
$\mathbf{Bord}_{0}^{X}\to\mathbf{Bord}_{1}^{X}\to\dots\to\mathbf{Bord}_{n}^{X}.$

The main result of this section is:

######  Statement 6.5.

Suppose that $X\to\mathrm{B}\mathrm{PL}$ is a stable tangential structure.
Then the flagged $X$-structured bordism category (7) is naturally a flagged
$\mathrm{PL}(n)$-dagger $(\infty,n)$-category.

###### Construction.

To build a dagger on $\mathbf{Bord}_{n}^{X}$, we must first provide a
$\mathrm{PL}(n)$-volution. We will do this by twisting the canonical
$\mathrm{PL}(n)$-volution by a carefully-selected action of $\mathrm{PL}(n)$
on $\mathbf{Bord}_{n}^{X}$. To build such an action, consider the functor

$\mathbf{Space}_{/\mathrm{B}\mathrm{PL}}\to\mathbf{Space}_{/\mathrm{B}\mathrm{PL}(n)}\to\mathbf{SymCat}_{(\infty,n)}\hskip
28.45274ptX\mapsto\mathbf{Bord}_{n}^{X}$

which first pulls back a stable tangential structure
$\\{X\to\mathrm{B}\mathrm{PL}\\}\in\mathbf{Space}_{/\mathrm{B}\mathrm{PL}}$ to
its unstable variant
$X(n):=X\times_{\mathrm{B}\mathrm{PL}}\mathrm{B}\mathrm{PL}(n)$ and then
assigns the corresponding bordism category. To build an action of
$\mathrm{PL}(n)$ on the resulting bordism category, it therefore suffices to
specify a functor

$\mathrm{B}\mathrm{PL}(n)\to\mathbf{Space}_{/\mathrm{B}\mathrm{PL}},$

i.e. an action of $\mathrm{PL}(n)$ on the bundle $X\to\mathrm{B}\mathrm{PL}$.
Equivalently, we want to choose a bundle
$X^{\prime}\to\mathrm{B}\mathrm{PL}\times\mathrm{B}\mathrm{PL}(n)$ whose
pullback along
$\mathrm{B}\mathrm{PL}=\mathrm{B}\mathrm{PL}\times\\{\mathrm{pt}\\}\to\mathrm{B}\mathrm{PL}\times\mathrm{B}\mathrm{PL}(n)$
is $X$.

Recall that $\mathrm{B}\mathrm{PL}$ is a coherently-associative monoid under
$\oplus$, and that in homotopy theory a monoid is a group as soon as its
$\pi_{0}$ is a group. Thus $\mathrm{B}\mathrm{PL}$ comes with an operation
$\text{``}{\ominus}\text{''}\colon\mathrm{B}\mathrm{PL}\to\mathrm{B}\mathrm{PL}$,
canonical up a contractible space of choices, that inverts with respect to the
group operation $\oplus$. With this operation in hand, we choose to set
$X^{\prime}$ to be the pullback of $X\to\mathrm{B}\mathrm{PL}$ along the
composition

$\mathrm{B}\mathrm{PL}\times\mathrm{B}\mathrm{PL}(n)\to\mathrm{B}\mathrm{PL}\times\mathrm{B}\mathrm{PL}\xrightarrow{\mathrm{id}\times\ominus}\mathrm{B}\mathrm{PL}\times\mathrm{B}\mathrm{PL}\xrightarrow{\oplus}\mathrm{B}\mathrm{PL}.$

These actions are compatible: the inclusion
$\mathbf{Bord}_{k}^{X}\to\mathbf{Bord}_{n}^{X}$ intertwines the induced
$\mathrm{PL}(k)$-volutions. What remains is to trivialize a certain induced
action of $\mathrm{PL}(n-k)$ on $\mathbf{Bord}_{k}^{X}$. Our
$\mathrm{PL}(n)$-volution on $\mathbf{Bord}_{n}^{X}$ was built out of two
pieces: the canonical volution together with our choice $X^{\prime}$. By the
same token, the action of $\mathrm{PL}(n-k)$ on $\mathbf{Bord}_{k}^{X}$ that
we care about has two pieces. One piece is the restriction along
$\mathrm{PL}(n-k)\to\mathrm{PL}(n)$ of the action on
$X\to\mathrm{B}\mathrm{PL}$. The other piece comes from restricting the
canonical volution along $\mathrm{PL}(n-k)\to\mathrm{PL}(n)$. Combining these
pieces and writing the problem in terms of bundles, one finds that what needs
trivializing is the restriction of $X$ along the composition

$\mathrm{B}\mathrm{PL}(n-k)\xrightarrow{\operatorname{diag}}\mathrm{B}\mathrm{PL}(n-k)\times\mathrm{B}\mathrm{PL}(n-k)\to\mathrm{B}\mathrm{PL}\times\mathrm{B}\mathrm{PL}\xrightarrow{\mathrm{id}\times\ominus}\mathrm{B}\mathrm{PL}\times\mathrm{B}\mathrm{PL}\xrightarrow{\oplus}\mathrm{B}\mathrm{PL}.$

But this composition factors through

$\mathrm{B}\mathrm{PL}\xrightarrow{\operatorname{diag}}\mathrm{B}\mathrm{PL}\times\mathrm{B}\mathrm{PL}\xrightarrow{\mathrm{id}\times\ominus}\mathrm{B}\mathrm{PL}\times\mathrm{B}\mathrm{PL}\xrightarrow{\oplus}\mathrm{B}\mathrm{PL},$

which is trivial by the definition of $\ominus$. ∎

######  Remark 6.6.

Both the notion of a dagger category with unitary duals and Statement 6.5 are
based on $\mathrm{PL}$ geometry. There is an analogous story replacing
$\mathrm{PL}(n)$ with $\mathrm{O}(n)$ and $\mathrm{PL}$-geometry with smooth
geometry. The smooth bordism category with smoothly stable tangential
structure has a natural $\mathrm{O}$-dagger structure.

######  Example 6.7.

Let us restrict to unextended bordism categories. Whereas the very definition
of $\mathrm{PL}(n)$-dagger $(\infty,n)$-category, and hence the content of
Statement 6.5, required Conjecture 5.3, the definition in the unextended case
and the construction described below are fully rigorous.

Because smooth bordism categories are more familiar than their piecewise-
linear counterparts, we will focus on that case. For each
$X\to\mathrm{B}\mathrm{O}(n)$ there is a smooth $X$-structured unextended
bordism $(\infty,1)$-category $\mathbf{Bord}_{n,n-1}^{X}$ whose objects are
closed $(n-1)$-dimensional $X$-structured manifolds and whose morphisms are
$X$-structured $n$-dimensional bordisms. This category is naturally flagged:
the naïve equivalences between objects in $\mathbf{Bord}_{n,n-1}^{X}$ are the
$X$-structured diffeomorphisms, but the categorical equivalences also include
$X$-structured h-cobordisms; writing $\operatorname{Man}_{n-1}^{X}$ for the
space of $X$-structured closed $(n-1)$-manifolds and diffeomorphisms between
them, we find a flagged $(\infty,1)$-category

$\operatorname{Man}_{n-1}^{X}\to\mathbf{Bord}_{n,n-1}^{X}$

which is often not univalent.

Since $\mathbf{Bord}_{n,n-1}^{X}$ is rigid symmetric monoidal, it has a
canonical anti-involution (which is to say a volution for the group
$\mathrm{PL}(1)=\mathbb{Z}/2\mathbb{Z}$). It is precisely the operation
$(-)^{\vee}$ that takes duals. This canonical anti-involution typically does
not extend to a dagger structure: for example, when $X=\mathrm{SO}$, the dual
of an object is its orientation-reversal, and most oriented manifolds simply
are not orientation-diffeomorphic to their orientation reversals. This is why
we needed to find an appropriate twist by a symmetric monoidal action.

Suppose that our tangential structure is stable, i.e. (in the smooth case)
pulled back from $\mathrm{B}\mathrm{O}$. We will twist the canonical anti-
involution on $\mathbf{Bord}_{n,n-1}^{X}$ by the involution on
$X\to\mathrm{B}\mathrm{O}$ built by restricting along the composition

(8)
$\mathrm{B}\mathrm{O}\times\mathrm{B}(\mathbb{Z}/2\mathbb{Z})=\mathrm{B}\mathrm{O}\times\mathrm{B}\mathrm{O}(1)\to\mathrm{B}\mathrm{O}\times\mathrm{B}\mathrm{O}\xrightarrow{\mathrm{id}\times\ominus}\mathrm{B}\mathrm{O}\times\mathrm{B}\mathrm{O}\xrightarrow{\oplus}\mathrm{B}\mathrm{O}.$

To finish the construction of a dagger structure, we take the corresponding
twisted anti-involution, restrict it to the (flagged) groupoid
$\operatorname{Man}_{n-1}^{X}$ of objects where it becomes an involution, and
trivialize it. This involution combines duality with the chosen involution on
$X\to\mathrm{B}\mathrm{O}$.

In general, given an $n$-dimensional tangential structure
$X\to\mathrm{B}\mathrm{O}(n)$, the duality involution on
$\operatorname{Man}_{n-1}^{X}$ is the one built from restricting $X$ along
$\mathrm{B}\mathrm{O}(n-1)\times\mathrm{B}(\mathbb{Z}/2\mathbb{Z})=\mathrm{B}\mathrm{O}(n-1)\times\mathrm{B}\mathrm{O}(1)\to\mathrm{B}\mathrm{O}(n)$.
Together with the twisting, we find that we win if the total composition

$\mathrm{B}\mathrm{O}(n-1)\times\mathrm{B}(\mathbb{Z}/2\mathbb{Z})\xrightarrow{\mathrm{id}\times\operatorname{diag}}\mathrm{B}\mathrm{O}(n-1)\times\mathrm{B}(\mathbb{Z}/2\mathbb{Z})\times\mathrm{B}(\mathbb{Z}/2\mathbb{Z})\to\mathrm{B}\mathrm{O}(n)\times\mathrm{B}(\mathbb{Z}/2\mathbb{Z})\xrightarrow{\eqref{eqn:smoothunextended}}\mathrm{B}\mathrm{O}$

agrees with the canonical map
$\mathrm{B}\mathrm{O}(n-1)\to\mathrm{B}\mathrm{O}$. And it does by virtue of
$\ominus$.

######  Statement 6.8.

The twist defined in (8) is the same twist used to form $\hat{H}^{(1)}_{n}$ in
[FH21, Appendix E], and hence our dagger structure agrees with the dagger
structure implicit in [FH21].

Symmetric monoidal dagger structures on bordism categories are relevant for
the definition of reflection positivity [Ati88, Bae06, TV17, FH21, Ste24]. As
a motivating example, consider the anti-involution on the symmetric monoidal
category of finite dimensional super vector spaces $\mathbf{sVect}$ given by
twisting the canonical one by the $\mathbb{Z}/2\mathbb{Z}$-action
corresponding to complex conjugation. The category of fixed points
$(\iota_{0}\mathbf{sVect})^{\mathbb{Z}/2}$ describes Hermitian super vector
spaces and unitary maps between them. We denote by
$\mathbf{sHilb}^{u}\subset(\iota_{0}\mathbf{sVect})^{\mathbb{Z}/2}$ its full
subcategory on the super Hilbert spaces and by $\mathbf{sHilb}$ the coherent
dagger 1-category

$\mathbf{sHilb}^{u}\to\mathbf{sVect}.$

An $X$-structured unextended topological quantum field theory
$Z\colon\mathbf{Bord}^{X}_{n,n-1}\to\mathbf{sVect}$ has a _reflection
structure_ when it is a functor of anti-involutive, or equivalently
$\mathbb{Z}/2\mathbb{Z}$-equivariant, categories. A field theory with
reflection structure is called _reflection positive_ if it induces a dagger
functor $\mathbf{Bord}^{X}_{n,n-1}\to\mathbf{sHilb}$: if the Hermitian
structures induced by Example 6.7 on the images of $Z$ are all positive
definite. Reflection positivity encodes physical unitarity. Thus we propose
the following tentative definition:

######  Proposal 6.9.

Let $\mathcal{C}$ be a rigid symmetric monoidal $\mathrm{PL}(n)$-dagger
$(\infty,n)$-category. A _unitary $X$-structured extended topological quantum
field theory_ valued in $\mathcal{C}$ is a functor of symmetric monoidal
$\mathrm{PL}(n)$-dagger categories $\mathbf{Bord}_{n}^{X}\to\mathcal{C}$.

######  Remark 6.10.

Proposal 6.9 is to be interpreted loosely as the type of definition we expect
to be correct. However, there are many details to be pinned down. For
instance, we expect the dagger structure on $\mathbf{Bord}_{n}^{X}$ from
Statement 6.5 to be part of a $\mathrm{PL}$-dagger tower and a unitary
extended topological field theory might be required to respect this structure.
In addition, our proposal prefers the $\mathrm{PL}$-bordism category, but the
smooth version from Remark 6.6 might be more appropriate in certain
applications.

Fully defining unitary extended topological quantum field theory furthermore
requires settling on good target categories $\mathcal{C}$. Without unitarity,
good target categories should feel like higher categories of “higher vector
spaces” or “higher super vector spaces”; with unitarity, one should instead
expect $\mathcal{C}$ to organize the “higher (super) Hilbert spaces.” In
upcoming work [CFH+], some of us construct a symmetric monoidal
$(\infty,3)$-category $\mathbf{Hilb}_{3}$ of “finite-dimensional 3-Hilbert
spaces.” It is rigid by construction, and moreover it is expected to carry an
organic $\mathrm{PL}(3)$-dagger structure. Together with Definition 6.9 and
Statement 6.5, one arrives at a definition of $3$-dimensional bosonic fully-
extended unitary topological quantum field theories for any stable tangential
structure. The construction of $\mathbf{Hilb}_{3}$ in [CFH+] starts with the
usual dagger $1$-category $\mathbf{Hilb}$ of finite-dimensional Hilbert spaces
and repeatedly applies a certain manifestly-dagger delooping procedure. This
delooping procedure is expected to extend to even higher categories as well.

Another candidate for $\mathbf{Hilb}_{3}$ weakening the finite dimensionality
conditions would be an appropriate Morita $3$-category of Bicommutant
Categories. Bicommutant categories as first introduced in [Hen17]; their
Morita theory is under development. Similar to the 3-category
$\mathbf{TensCat}$, the category of Bicommutant categories, bicommutant
category bimodules (W*-categories), equivariant functors and natural
transformations, called $\mathbf{BicomCat}$ could serve as target for unitary
quantum field theories, more general than just TQFTs, for instance chiral
conformal field theories. It is expected that this category is a strictly
fully-dagger, with involutions given by the categorical
$\mathrm{}^{\mathrm{op}}$ operation on objects and 1-morphisms and using
adjoint functors and natural transformations on the top two levels.

Further analysis of Definition 6.9 — construction of examples, a Unitary
Cobordism Hypothesis, etc. — will be the subject of future work.

### Acknowledgements

The authors would like to thank the other participants of the June 2023 Zoom
workshop on Dagger Higher Categories: Bruce Bartlett, André Henriques, Chris
Heunen, Peter Selinger, and Dominic Verdon, along with Jan Steinebrunner. The
authors acknowledge the following grant support:

GF, BH, DP | NSF DMS 2154389
---|---
CK | NSF GRFP 2141064
TJF | NSERC RGPIN-2021-02424
TJF, LM, CS, LS | Simons Collaboration on Global Categorical Symmetries (Simons Foundation grants 888996 and 1013836)
LS | Atlantic Association for Research in the Mathematical Sciences
DR | Deutsche Forschungsgemeinschaft (DFG) – 493608176
CS | Deutsche Forschungsgemeinschaft (DFG) – SFB 1085 Higher Invariants
N | James Buckee Scholarship, Merton College, Oxford

Research at Perimeter Institute is supported in part by the Government of
Canada through the Department of Innovation, Science and Economic Development
and by the Province of Ontario through the Ministry of Colleges and
Universities.

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