of 0-, 1-, and 2-cells in $\mathscr{B}$ satisfying the triangle identities,
the Euler characteristic $\chi(F)$ of the $1$-cell $F$ is defined to be the
composite morphism

$\langle\langle U_{C}\rangle\rangle\xrightarrow{\langle\langle
u\rangle\rangle}\langle\langle GF\rangle\rangle\cong\langle\langle
FG\rangle\rangle\xrightarrow{\langle\langle c\rangle\rangle}\langle\langle
U_{D}\rangle\rangle$

in $\mathscr{D}$.

Morita invariance of shadows is formulated as follows in terms of the Euler
characteristic.

######  Proposition 4.2 ([CP19, Proposition 4.6]).

If the adjunction in 4.1 is actually an equivalence, i.e., if the 2-cells $c$
and $u$ are invertible, then $\chi(F)$ is an isomorphism in $\mathscr{D}$ with
inverse given by $\chi(G)$.

This result can be generalized to arbitrary endomorphisms. For a given
endomorphism $Q:C\to C$, there is a chain of morphisms

$\langle\langle Q\rangle\rangle\cong\langle\langle
Q\mathrm{U}_{C}\rangle\rangle\xrightarrow{\ \langle\langle
u\mathrm{id}_{Q}\rangle\rangle\ }\langle\langle
GFQ\rangle\rangle\xrightarrow{\langle\langle\mathrm{id}_{GFQ}u\rangle\rangle}\langle\langle
GFQGF\rangle\rangle\cong\langle\langle
FQGFG\rangle\rangle\xrightarrow{\langle\langle c\rangle\rangle}\langle\langle
FQG\rangle\rangle,$

which is known as the trace of $Q$ (which is different from the trace defined
in 2.17) and denoted $\mathrm{tr}(u_{Q})$. Now, we have the following result
due to Campbell and Ponto.

######  Proposition 4.3 ([CP19, Proposition 4.8]).

If the adjunction in 4.1 is actually an equivalence, i.e., if the 2-cells $c$
and $u$ are invertible, then $\mathrm{tr}(u_{Q})$ is an isomorphism in
$\mathscr{D}$ with inverse given by $\mathrm{tr}(c_{Q})$.

We provide an alternative proof of these results below, exploiting the
equivalence between shadows and functors out of Hochschild homology, which
leads to a formal, functorial construction of $\chi$ and $\mathrm{tr}$. Since
by 3.19, a shadow out of $\mathscr{B}$ is equivalent to a functor out of the
category $\mathrm{biHH}(\mathscr{B})$, we can focus the proof of Morita
invariance on the representing object.

######  Definition 4.4.

Let $\mathscr{A}\mathrm{dj}$ be the free strict bicategory generated by the
two objects, two $1$-morphisms and two $2$-morphisms given in 4.1 and relation
given by the triangle equalities.

For more details regarding the $2$-category $\mathscr{A}\mathrm{dj}$ see the
original description in [SS86].

######  Definition 4.5.

Let $\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{q}$ be the strict bicategory
with the same objects, $1$-cells and $2$-cells as $\mathscr{A}\mathrm{dj}$,
but where all $2$-morphisms are invertible.

As in 4.1, a bifunctor out of $\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{q}$
can be depicted as a diagram

${0}$${1}$$\scriptstyle{f}$$\scriptstyle{g\circ
f}$$\scriptstyle{g}$$\scriptstyle{f\circ
g}$$\scriptstyle{u}$$\scriptstyle{\cong}$$\scriptstyle{c}$$\scriptstyle{\cong}$

######  Definition 4.6.

For any bicategory $\mathscr{B}$, let

$\mathscr{A}\mathrm{dj}(\mathscr{B})=\mathrm{Fun}(\mathscr{A}\mathrm{dj},\mathscr{B})$

and

$\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{q}(\mathscr{B})=\mathrm{Fun}(\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{q},\mathscr{B}),$

the categories of adjunctions in $\mathscr{B}$ and of adjoint equivalences in
$\mathscr{B}$, respectively.

There is an evident localization functor
$\mathscr{A}\mathrm{dj}\to\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{q}$ that
induces a restriction functor

(4.7)
$\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{q}(\mathscr{B})\to\mathscr{A}\mathrm{dj}(\mathscr{B}).$

To complete our set-up, we introduce the following notation.

######  Notation 4.8.

For any category $\mathscr{D}$, let

$\mathscr{A}\mathrm{rr}(\mathscr{D})=\mathrm{Fun}([1],\mathscr{D}),$

and let $\mathscr{I}\mathrm{so}(\mathscr{D})$ denote the full subcategory of
$\mathscr{A}\mathrm{rr}(\mathscr{D})$ consisting of isomorphisms.

######  Theorem 4.9.

For any bicategory $\mathscr{B}$, there is a functor

$\widehat{\chi}:\mathscr{A}\mathrm{dj}(\mathscr{B})\to\mathscr{A}\mathrm{rr}\big{(}\mathrm{biHH}(\mathscr{B})\big{)}$

that takes an adjunction $\sigma=(C,D,F,G,u,c)$ to the morphism

$\widehat{\chi}(\sigma):\mathrm{U}_{C}\xrightarrow{u}GF\xrightarrow{\cong}FG\xrightarrow{c}\mathrm{U}_{D}$

in $\mathrm{THH}(\mathscr{B})$.

Note that it is essential here that the codomain of this functor is the
category of morphisms in $\mathrm{biHH}(\mathscr{B})$, which ensures the
existence of the isomorphism $GF\xrightarrow{\cong}FG$.

###### Proof.

The functoriality of $\mathrm{biHH}$ for enrichments in
$\mathscr{V}=(\mathscr{C}\mathrm{at},\times)$ (3.4) implies that there is a
functor

$\mathrm{biHH}:\mathscr{A}\mathrm{dj}(\mathscr{B})\to\mathrm{Fun}\big{(}\mathrm{biHH}(\mathscr{A}\mathrm{dj}),\mathrm{biHH}(\mathscr{B})\big{)}$

The explicit colimit description of $\mathrm{biHH}(\mathscr{A}\mathrm{dj})$ as
a colimit in $\mathscr{C}\mathrm{at}$ (3.4) implies that
$\mathrm{biHH}(\mathscr{A}\mathrm{dj})$ in particular has the objects the two
endomorphisms $\mathrm{U}_{0},\mathrm{U}_{1}$, has the $2$ $1$-morphisms
$u:\mathrm{U}_{0}\to gf$ and $c:fg\to\mathrm{U}_{1}$ and an isomorphism
$fg\cong gf$.

Hence, there is a morphism
$\alpha:[1]\to\mathrm{biHH}(\mathscr{A}\mathrm{dj})$, that picks out the
morphism $\mathrm{U}_{0}\to gf\cong fg\to\mathrm{U}_{1}$. The existence of the
desired functor now follows from precomposing with $\alpha$. ∎

######  Remark 4.10.

Notice 4.9 will generalize to an arbitrary $\mathscr{V}$-enriched
$\infty$-category, as soon as the functoriality of $\mathrm{HH}_{\mathscr{V}}$
is established in this case.

The corollary below is an immediate consequence of the fact that every functor
preserves isomorphisms.

######  Corollary 4.11.

Precomposing the functor
$\widehat{\chi}:\mathscr{A}\mathrm{dj}(\mathscr{B})\to\mathscr{A}\mathrm{rr}\big{(}\mathrm{biHH}(\mathscr{B})\big{)}$
with the restriction functor
$\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{q}(\mathscr{B})\to\mathscr{A}\mathrm{dj}(\mathscr{B})$
given in 4.7 gives rise to a functor

$\widehat{\chi}:\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{q}(\mathscr{B})\to\mathscr{I}\mathrm{so}\big{(}\mathrm{biHH}(\mathscr{B})\big{)}.$

From 4.11 we can recover the Morita invariance of shadows, proven originally
in [CP19, Proposition 4.6].

######  Proposition 4.12.

Shadows satisfy Morita invariance.

###### Proof.

Let $\langle\langle-\rangle\rangle$ be a shadow functor on $\mathscr{B}$.
Consider its unstrictification (3.19), which is a functor
$\mathscr{U}\mathrm{n}\big{(}\langle\langle-\rangle\rangle\big{)}:\mathrm{THH}(\mathscr{B})\to\mathscr{D}$.
Precomposing this with the functor $\widehat{\chi}$ of 4.9 gives rise to a
functor

$\mathscr{A}\mathrm{rr}\Big{(}\mathscr{U}\mathrm{n}\big{(}\langle\langle-\rangle\rangle\big{)}\Big{)}\circ\widehat{\chi}:\mathscr{A}\mathrm{dj}(\mathscr{B})\to\mathscr{A}\mathrm{rr}(\mathscr{D})$

that takes an adjunction in $\mathscr{B}$ to the morphism

$\langle\langle U_{C}\rangle\rangle\to\langle\langle
GF\rangle\rangle\cong\langle\langle FG\rangle\rangle\to\langle\langle
U_{D}\rangle\rangle$

in $\mathscr{D}$. The desired conclusion now follows from 4.11. ∎

We now want to move on to the more general case and study traces of
endomorphisms $\mathrm{tr}(u_{Q})$. This requires us to have a better
understanding of the $2$-category with one adjunction and one endomorphism.

######  Definition 4.13.

Let $\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd}$ be the free category with
the same objects, $1$-morphisms and $2$-morphisms of $\mathscr{A}\mathrm{dj}$
along with one additional free endomorphism $[0]\to[0]$. Define
$\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{q}\mathscr{E}\mathrm{nd}$ similarly
along with its localization map
$\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd}\to\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{q}\mathscr{E}\mathrm{nd}$.

######  Remark 4.14.

If we denote the free endomorphism by $q:0\to 0$ an arbitrary endomorphism
$0\to 0$ in $\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd}$ is a word on the
two letter $q$ and $gf$, whereas a free endomorphism on $1$ is a word on $fg$
and $fqg$.

######  Definition 4.15.

For a bicategory $\mathscr{B}$, let
$\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd}(\mathscr{B})=\mathrm{Fun}(\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd},\mathscr{B})$
and
$\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{q}\mathscr{E}\mathrm{nd}(\mathscr{B})=\mathrm{Fun}(\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{q}\mathscr{E}\mathrm{nd},\mathscr{B})$
and notice we again have an inclusion functor
$\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{q}\mathscr{E}\mathrm{nd}(\mathscr{B})\to\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd}(\mathscr{B})$.

Notice, as we have a pushout square

${{[0]}}$${\mathscr{A}\mathrm{dj}}$${B\mathbb{N}}$${\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd}}$

and a similar one for
$\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{q}\mathscr{E}\mathrm{nd}$ and so

$\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd}(\mathscr{B})\simeq\mathscr{A}\mathrm{dj}(\mathscr{B})\times_{\mathscr{B}}\mathscr{E}\mathrm{nd}(\mathscr{B}),$
$\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{q}\mathscr{E}\mathrm{nd}(\mathscr{B})\simeq\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{q}(\mathscr{B})\times_{\mathscr{B}}\mathscr{E}\mathrm{nd}(\mathscr{B}).$

Using this description we can now define the desired functor.

######  Theorem 4.16.

For any bicategory $\mathscr{B}$, there is a functor

$\widehat{\mathrm{tr}}:\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd}(\mathscr{B})\to\mathscr{A}\mathrm{rr}\big{(}\mathrm{biHH}(\mathscr{B})\big{)}$

that takes an adjunction $\sigma=(C,D,F,G,u,c)$ and endomorphism $Q:C\to C$ to
the morphism

$\widehat{\mathrm{tr}}(\sigma,Q):Q\xrightarrow{uQu}GFQFG\xrightarrow{\cong}FQGFG\xrightarrow{FQGc}FQG$

in $\mathrm{biHH}(\mathscr{B})$.

###### Proof.

Again functoriality of $\mathrm{biHH}$ (3.4) implies that there is a functor

$\displaystyle\mathrm{biHH}:\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd}(\mathscr{B})$
$\displaystyle\to\mathrm{Fun}\big{(}\mathrm{biHH}(\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd}),\mathrm{THH}(\mathscr{B})\big{)}.$

Now, again the explicit description of
$\mathrm{biHH}(\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd})$ (3.4) and the
description of endomorphisms in $\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd}$
(4.14) implies that
$\mathrm{biHH}(\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd})$ in particular
has $1$-morphisms $uqu:q\to gfqfg$ and $fqgc:fqgfg\to fqg$, and there exists
an isomorphism $gfqgf\cong fqgfg$. Let
$\beta:[1]\to\mathrm{biHH}(\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd})$ be
the functor that picks out the composition of $uqu$ and $fqgc$ (with the
isomorphism in between). Then we obtain the desired functor by precomposing
with $\beta$. ∎

We again have the following result analogous to 4.11.

######  Corollary 4.17.

Precomposing the functor
$\widehat{\mathrm{tr}}:\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd}(\mathscr{B})\to\mathscr{A}\mathrm{rr}\big{(}\mathrm{biHH}(\mathscr{B})\big{)}$
with the restriction functor
$\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{q}\mathscr{E}\mathrm{nd}(\mathscr{B})\to\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd}(\mathscr{B})$
given in 4.15 gives rise to a functor

$\widehat{\chi}:\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{q}\mathscr{E}\mathrm{nd}(\mathscr{B})\to\mathscr{I}\mathrm{so}\big{(}\mathrm{biHH}(\mathscr{B})\big{)}.$

Finally, we can now recover [CP19, Proposition 4.8] using the same argument we
used in 4.12 relying on 3.19.

######  Proposition 4.18.

Let $\langle\langle-\rangle\rangle$ be a shadow on a bicategory $\mathscr{B}$
valued in a $\mathscr{D}$. Then for a given Morita equivalence
$\sigma=(C,D,F,G,u,c)$ and endomorphism $Q:C\to C$, the map
$\mathrm{tr}(u_{Q}):\langle\langle Q\rangle\rangle\to\langle\langle
FQG\rangle\rangle$ is an isomorphism in $\mathscr{D}$ with inverse
$\mathrm{tr}(c_{Q})$.

##  5\. Hochschild Homology of $2$-Categories and Homotopy Coherence

In the previous section we exploited the structure two bicategories, namely
$\mathscr{A}\mathrm{dj}$ and $\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd}$,
to deduce Morita invariance or invariance of the Euler characteristic and
trace, presenting alternative proofs to the ones in [CP19]. However, this
alternative approach naturally results in several interesting questions:

  1. (1)

The construction of the Euler characteristic and trace involves a choice of
isomorphism. Is the choice of isomorphism canonical or depend on choices?

  2. (2)

We construct the Euler characteristic and trace by restricting to a certain
morphism in the categories $\mathrm{biHH}(\mathscr{A}\mathrm{dj})$ and
$\mathrm{biHH}(\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd})$. Would it be
possible to deduce other invariance properties of shadows by restricting to
other morphisms in those categories?

Answering these type of questions necessitates having a clear understanding of
$\mathrm{biHH}(\mathscr{A}\mathrm{dj})$ and
$\mathrm{biHH}(\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd})$ far beyond the
ones used in Section 4. Hence, in this section we completely compute those two
categories (5.23/5.26), which we then use to address the first question
(5.25/5.28), leaving the second one for future work. The key towards all these
computations is a very explicit description of Hochschild homology of strict
bicategories, which is the content of 5.10 and requires a technical analysis.

Recall that for any small category $\mathscr{C}$ and cocomplete category
$\mathscr{D}$, there is a _tensor (or coend) functor_

$-\otimes-:\mathrm{Fun}(\mathscr{C},\mathscr{D})\times\mathrm{Fun}(\mathscr{C}^{op},\mathscr{D})\to\mathscr{D}$

that takes a a pair $(F,G)$ to the coequalizer of

${\displaystyle\coprod_{f:c\to c^{\prime}}F(c)\times
G(c^{\prime})}$${\displaystyle\coprod_{c}F(c)\times G(c)}$

.

######  Remark 5.1.

The tensor functor defined above satisfies the following properties.

  * •

[Rie14, Example 4.1.3]: If $\mathscr{D}$ has a terminal object and
$F:\mathscr{C}\to\mathscr{D}$ is the terminal functor, then $F\otimes
G=\mathrm{colim}_{\mathscr{C}^{op}}G$.

  * •

[Rie14, Example 4.1.5]: If $F=\mathrm{Hom}_{\mathscr{C}}(c,-)$, then $F\otimes
G\cong G(c)$.

  * •

The tensor functor preserves colimits.

For more details regarding the tensor functor, see [Rie14, Section 4.1].

Below we provide a more explicit description of pseudo-colimits in terms of
the tensor functor, for which the following technical result proves useful.

######  Lemma 5.2.

The tensor functor

$-\otimes-:\mathscr{C}\mathrm{at}^{{\mathbbe{\Delta}}}\times\mathscr{C}\mathrm{at}^{{\mathbbe{\Delta}}^{op}}\to\mathscr{C}\mathrm{at}$

is left Quillen, with respect to the canonical model structure on the codomain
and the Reedy model structure on each factor of the domain.

######  Remark 5.3.

The canonical model structure on $\mathscr{C}\mathrm{at}$ is Cartesian, and

  * •

the cofibrations are functors that are injective on objects, and

  * •

weak equivalences are categorical equivalences.

See also [Rie14, Example 11.3.9] for more details.

###### Proof.

Since the canonical model structure on $\mathscr{C}\mathrm{at}$ is Cartesian,
we can conclude by an argument analogous to that in [Hir03, 18.4.11]. See also
the discussion in [Rie14, Section 14.3]. ∎

To use this lemma to describe pseudo-colimits, we need the following
definition.

######  Definition 5.4.

Let $I[n]$ denote the category with $n$ objects and a unique morphism between
any two objects (which in particular implies that all morphisms are
isomorphisms). The collection of all $I[n]$ underlies a cosimplicial category

$I[\bullet]:{\mathbbe{\Delta}}\to\mathscr{C}\mathrm{at}.$

######  Proposition 5.5.

If $F:{\mathbbe{\Delta}}^{op}\to\mathscr{C}\mathrm{at}$ is a strict functor,
then the pseudo-colimit of $F$ is equivalent to the tensor product
$I[\bullet]\otimes F$.

###### Proof.

By [Gam08] the pseudo-colimit of $F$ coincides with the homotopy colimit of
$F$ in the canonical model structure on $\mathscr{C}\mathrm{at}$. By 5.1 the
colimit of $F$ is given by $*\otimes F$ and so, by Lemma 5.2, the homotopy
colimit is given by left deriving this functor.

Observe that $F:{\mathbbe{\Delta}}^{op}\to\mathscr{C}\mathrm{at}$ is already
Reedy cofibrant. Indeed, the latching object $L_{n}F$ is a full subcategory of
$F_{n}$ and so by construction $L_{n}F\to F_{n}$ is a cofibration (5.3). We
need thus only to find a Reedy cofibrant replacement of the terminal diagram.

We claim that $I[\bullet]:{\mathbbe{\Delta}}\to\mathscr{C}\mathrm{at}$ (5.4)
is such a cofibrant replacement. The map to $I[\bullet]\to*$ is obviously a
level-wise weak equivalence (5.3), so it suffices to show that $I[\bullet]$ is
Reedy cofibrant. This follows from direct computation, as the latching object
$L_{n}I$ has $n+1$ objects, and the map $L_{n}I\to I[n]$ is the identity on
objects and hence a cofibration (5.3). ∎

We next provide a more explicit description of $I[\bullet]\otimes F$. Recall
there is an adjunction

(5.6)
${\text{s}\mathscr{S}\text{et}}$${\mathscr{C}\mathrm{at}}$$\scriptstyle{\tau_{1}}$$\scriptstyle{\bot}$$\scriptstyle{N}$

where the right adjoint is the nerve functor, and the left adjoint is known as
the fundamental category.

######  Remark 5.7.

The fundamental category can be described explicitly as follows.

  * •

$\mathrm{Obj}_{\tau_{1}X}=X_{0}$.

  * •

The set $X_{1}$ generates the morphisms of $\tau_{1}X$.

  * •

The morphisms of $\tau_{1}X$ satisfy the relations $d_{1}\sigma\sim
d_{0}\sigma\circ d_{2}\sigma$ for all $\sigma\in X_{2}$ and
$s_{0}x\sim\mathrm{id}_{x}$ for all $x\in X_{0}$.

See [Rez17, Section 11] for more details.

The following simple property of the fundamental category proves useful to us
below.

######  Lemma 5.8.

For all functors $F:J\to\mathscr{C}\mathrm{at}$,

$\mathrm{colim}_{J}F\cong\tau_{1}\mathrm{colim}_{J}N(F).$

In particular, for every strict functor
$F:{\mathbbe{\Delta}}^{op}\to\mathscr{C}\mathrm{at}$,

$I[\bullet]\otimes F\cong\tau_{1}(NI[\bullet]\otimes NF).$

###### Proof.

For any category $\mathscr{D}$, there is a chain of isomorphisms

$\displaystyle\mathrm{Hom}_{\mathscr{C}\mathrm{at}}\big{(}\tau_{1}\mathrm{colim}_{J}N(F),\mathscr{D}\big{)}$
$\displaystyle\cong\mathrm{Hom}_{\text{s}\mathscr{S}\text{et}}\big{(}\mathrm{colim}_{J}N(F),N\mathscr{D}\big{)}$
$\displaystyle\text{adjunction}(\tau_{1},N)$
$\displaystyle\cong\lim_{J}\mathrm{Hom}_{\text{s}\mathscr{S}\text{et}}\big{(}N(F)(j),N\mathscr{D}\big{)}$
definition of colimit
$\displaystyle\cong\lim_{J}\mathrm{Hom}_{\mathscr{C}\mathrm{at}}\big{(}F(j),\mathscr{D}\big{)}$
$N$ fully faithful,

which implies that $\tau_{1}\mathrm{colim}_{J}N(F)$ satisfies the universal
property of a colimit, and we can conclude. ∎

By definition $\mathrm{biHH}(\mathscr{B})$ is a pseudo-colimit and we now use
our newly gained understanding of pseudo-colimits of categories to give a more
precise characterization of this construction via generators and relations.

For the next proof, recall that if $\mathscr{B}$ is a $2$-category, then for
every object $X$, there is a unit object $\mathrm{U}_{X}$ in
$\mathscr{B}(X,X)$. Additionally, for every $1$-morphism $F$ in
$\mathscr{B}(X,Y)$, there is an identity $2$-morphism $\mathrm{id}_{F}$.
Moreover, the composition of two $1$-morphisms or $2$-morphisms in
$\mathscr{B}(X,Y)$ and $\mathscr{B}(Y,Z)$ is denoted via $\ast$, to
distinguish it from the composition internal to the categories
$\mathscr{B}(X,Y)$.

######  Remark 5.9.

For the next theorem, we need a detailed understanding of the bisimplicial set
$(N\mathscr{B}_{*}\mathscr{O}_{bihh})_{\bullet\bullet}$ and so we present here
an explicit diagram for the benefit of the reader.

For a category $\mathscr{C}$, let $\mathrm{Comp}_{\mathscr{C}}$ denote the set
of composable morphisms $(f,g)$ such that $\mathrm{Dom}_{g}=\mathrm{Cod}_{f}$.
Now, using 3.1, we can depict the bisimplicial set
$(N\mathscr{B}_{*}\mathscr{O}_{bihh})_{\bullet\bullet}$ as follows with
certain morphisms described explicitly.

${\displaystyle\coprod_{X\in\mathrm{Obj}_{\mathscr{B}}}\mathrm{Obj}_{\mathscr{B}(X,X)}}$${\displaystyle\coprod_{X,Y\in\mathrm{Obj}_{\mathscr{B}}}\mathrm{Obj}_{\mathscr{B}(X,Y)\times\mathscr{B}(Y,X)}}$${\displaystyle\coprod_{X,Y,Z\in\mathrm{Obj}_{\mathscr{B}}}\mathrm{Obj}_{\mathscr{B}(X,Y)\times\mathscr{B}(Y,Z)\times\mathscr{B}(Z,X)}}$${\cdots}$${\displaystyle\coprod_{X\in\mathrm{Obj}_{\mathscr{B}}}\mathrm{Mor}_{\mathscr{B}(X,X)}}$${\displaystyle\coprod_{X,Y\in\mathrm{Obj}_{\mathscr{B}}}\mathrm{Mor}_{\mathscr{B}(X,Y)\times\mathscr{B}(Y,X)}}$${\displaystyle\coprod_{X,Y,Z\in\mathrm{Obj}_{\mathscr{B}}}\mathrm{Mor}_{\mathscr{B}(X,Y)\times\mathscr{B}(Y,Z)\times\mathscr{B}(Z,X)}}$${\cdots}$${\displaystyle\coprod_{X\in\mathrm{Obj}_{\mathscr{B}}}\mathrm{Comp}_{\mathscr{B}(X,X)}}$${\displaystyle\coprod_{X,Y\in\mathrm{Obj}_{\mathscr{B}}}\mathrm{Comp}_{\mathscr{B}(X,Y)\times\mathscr{B}(Y,X)}}$${\displaystyle\coprod_{X,Y,Z\in\mathrm{Obj}_{\mathscr{B}}}\mathrm{Comp}_{\mathscr{B}(X,Y)\times\mathscr{B}(Y,Z)\times\mathscr{B}(Z,X)}}$${\cdots}$${\
\vdots\ }$${\ \vdots\ }$${\ \vdots\
}$$\scriptstyle{\mathrm{id}_{F}}$$\scriptstyle{(F,\mathrm{U})}$$\scriptstyle{\mathrm{id}_{(F,G)}}$$\scriptstyle{G\ast
F}$$\scriptstyle{F\ast G}$$\scriptstyle{(F,G\ast H)}$$\scriptstyle{(F\ast
G,H)}$$\scriptstyle{(H\ast
F,G)}$$\scriptstyle{\mathrm{Dom}_{\alpha}}$$\scriptstyle{\mathrm{Cod}_{\alpha}}$$\scriptstyle{(\alpha,\mathrm{id}_{\mathrm{U}})}$$\scriptstyle{(\mathrm{Dom}_{\alpha},\mathrm{Dom}_{\beta})}$$\scriptstyle{(\mathrm{Cod}_{\alpha},\mathrm{Cod}_{\beta})}$$\scriptstyle{\beta\ast\alpha}$$\scriptstyle{\alpha\ast\beta}$$\scriptstyle{(\alpha,\beta\ast\gamma)}$$\scriptstyle{(\alpha\ast\beta,\gamma)}$$\scriptstyle{(\gamma\ast\alpha,\beta)}$$\scriptstyle{\alpha_{2}\circ\alpha_{1}}$$\scriptstyle{\alpha_{1}}$$\scriptstyle{\alpha_{2}}$$\scriptstyle{(\alpha_{2}\circ\alpha_{1},}$$\scriptstyle{\beta_{2}\circ\beta_{1})}$$\scriptstyle{(\alpha_{1},\beta_{1})}$$\scriptstyle{(\alpha_{2},\beta_{2})}$$\scriptstyle{(\beta_{1}\ast\alpha_{1},\beta_{2}\ast\alpha_{2})}$$\scriptstyle{(\alpha_{1}\ast\beta_{1},\alpha_{2}\ast\beta_{2})}$$\scriptstyle{(\alpha_{2}\circ\alpha_{1},}$$\scriptstyle{\beta_{2}\circ\beta_{1},}$$\scriptstyle{\gamma_{2}\circ\gamma_{1})}$$\scriptstyle{(\alpha_{1},\beta_{1},\gamma_{1})}$$\scriptstyle{(\alpha_{2},\beta_{2},\gamma_{2})}$

######  Theorem 5.10.

If $\mathscr{B}$ is any $2$-category, then $\mathrm{biHH}(\mathscr{B})$ admits
the following presentation.

  * •

$\mathrm{Obj}_{\mathrm{biHH}(\mathscr{B})}=\coprod_{X\in\mathrm{Obj}_{\mathscr{B}}}\mathrm{Obj}_{\mathscr{B}(X,X)}$

  * •

Generating morphisms

    1. (1)

Symbols $\alpha:\mathrm{Dom}_{\alpha}\to\mathrm{Cod}_{\alpha}$ for all
$\alpha\in\coprod_{X\in\mathrm{Obj}_{\mathscr{B}}}\mathrm{Mor}_{\mathscr{B}(X,X)}$.

    2. (2)

Symbols $(F,G):F*G\to G*F$ for all $F\in\mathrm{Obj}_{\mathscr{B}(X,Y)}$,
$G\in\mathrm{Obj}_{\mathscr{B}(Y,X)}$ and all
$X,Y\in\mathrm{Obj}_{\mathscr{B}}$.

  * •

Relations

    1. (1)

$(F,\mathrm{U}_{X}):F\to F$ is the identity morphism of the object
$F\in\mathrm{Obj}_{\mathscr{B}(X,X)}$.

    2. (2)

The composite of a pair of symbols
$\alpha,\beta\in\mathrm{Mor}_{\mathscr{B}(X,X)}$ such that
$\mathrm{Dom}_{\beta}=\mathrm{Cod}_{\alpha}$ is equal to their composite
$\beta\circ\alpha$ in the category $\mathscr{B}(X,X)$.

    3. (3)

All symbols $(F,G)$ are isomorphisms.

    4. (4)

For all symbols $\alpha:F\to G$ and $\beta:H\to L$ in
$\mathrm{Mor}_{\mathscr{B}(X,X)}$,

$(G,L)\circ(\alpha\ast\beta)=(\beta\ast\alpha)\circ(F,H).$

    5. (5)

For any three $1$-morphisms $F\in\mathrm{Obj}_{\mathscr{B}(X,Y)}$,
$G\in\mathrm{Obj}_{\mathscr{B}(Y,Z)}$, $H\in\mathrm{Obj}_{\mathscr{B}(Z,X)}$,

$(F,G\ast H)=(H\ast F,G)\circ(F\ast G,H).$

###### Proof.

According to 2.10, $\mathrm{biHH}(\mathscr{B})$ is the colimit of the
simplicial diagram $\mathscr{B}_{*}(\mathscr{O}_{bihh})_{\bullet}$. Because
$\mathscr{B}$ is a $2$-category,
$\mathscr{B}_{*}(\mathscr{O}_{thh})_{\bullet}$ is a strict diagram rather than
just a pseudo-diagram, whence by 5.5, $\mathrm{biHH}(\mathscr{B})$ is
equivalent to the tensor
$I[\bullet]\otimes\mathscr{B}_{*}(\mathscr{O}_{bihh})_{\bullet}$, which is
isomorphic to the fundamental category of
$NI[\bullet]\otimes(N\mathscr{B}_{*}\mathscr{O}_{bihh})_{\bullet}$ by Lemma
5.8.

In order to evaluate the fundamental category we need to better understand
levels $0,1,$ and $2$ of the simplicial set
$NI[\bullet]\otimes(N\mathscr{B}_{*}\mathscr{O}_{bihh})_{\bullet}$. For a
fixed $k\geq 0$, the evaluation map
$(-)_{k}:\text{s}\mathscr{S}\text{et}\to\mathscr{S}\text{et}$ preserves
colimits, so there is a bijection of sets

$(N(I[\bullet])\otimes N\mathscr{B}_{*}\mathscr{O}_{bihh})_{k}\cong
NI[\bullet]_{k}\otimes(N\mathscr{B}_{*}\mathscr{O}_{bihh})_{k}.$

The following general observation is useful to us below. Let
$\Delta[\bullet]:{\mathbbe{\Delta}}\to\text{s}\mathscr{S}\text{et}$ be the
Yoneda embedding. For every bisimplicial set $X$,

(5.11) $\Delta[\bullet]_{n}\otimes X_{k}\cong\Delta[n]\otimes X_{k}\cong
X_{nk},$

where the last step follows from 5.1, as $\Delta[n]$ is a representable
functor. In particular,

$(\Delta[\bullet]\otimes X)_{k}\cong X_{kk},$

i.e., $\Delta[\bullet]\otimes X$ is the diagonal of the bisimplicial set $X$.

If $k=0$, then $I[\bullet]_{0}=\Delta[\bullet]_{0}$, the representable
functor, and so by the argument above

$\mathrm{Obj}_{\mathrm{THH}(\mathscr{B})}=(NI[\bullet]\otimes
N\mathscr{B}_{*}\mathscr{O}_{bihh})_{0}=(N\Delta[\bullet]\otimes
N\mathscr{B}_{*}\mathscr{O}_{bihh})_{0}=(N\mathscr{B}_{*}\mathscr{O}_{bihh})_{00}=\coprod_{X}\mathrm{Obj}_{\mathscr{B}(X,X)},$

as desired.

If $k=1$, then
$NI[\bullet]_{1}=\mathrm{Mor}_{I[\bullet]}=\Delta[\bullet]_{1}\coprod_{\Delta[\bullet]_{0}}\Delta[\bullet]_{1}$,
where the pushout is that of the degeneracy map
$s_{0}:\Delta[\bullet]_{0}\to\Delta[\bullet]_{1}$ with itself, since any two
objects of $I[n]$ are connected by a unique isomorphism. Hence,

$\displaystyle(NI[\bullet]\otimes N\mathscr{B}_{*}\mathscr{O}_{bihh})_{1}$
$\displaystyle\cong
N(I[\bullet])_{1}\otimes(N\mathscr{B}_{*}\mathscr{O}_{bihh})_{1}$
$\displaystyle\cong(\Delta[\bullet]_{1}\coprod_{\Delta[\bullet]_{0}}\Delta[\bullet]_{1})\otimes(N\mathscr{B}_{*}\mathscr{O}_{bihh})_{1}$
$\displaystyle\cong(\Delta[\bullet]_{1}\otimes
N\mathscr{B}_{*}\mathscr{O}_{bihh})_{1}\coprod_{(\Delta[\bullet]_{0}\otimes
N\mathscr{B}_{*}\mathscr{O}_{bihh})_{1}}(\Delta[\bullet]_{1}\otimes
N\mathscr{B}_{*}\mathscr{O}_{bihh})_{1}$
$\displaystyle\cong(N\mathscr{B}_{*}\mathscr{O}_{bihh})_{11}\coprod_{(N\mathscr{B}_{*}\mathscr{O}_{bihh})_{01}}(N\mathscr{B}_{*}\mathscr{O}_{bihh})_{11},$

where the last step follows from 5.11.

Recall from 5.9 that
$(N\mathscr{B}_{*}\mathscr{O}_{bihh})_{11}=\coprod_{X,Y}\mathrm{Mor}_{\mathscr{B}(X,Y)\times\mathscr{B}(Y,X)}$,
i.e., its elements are pairs $(\alpha,\beta)$ with $\alpha:F\to G$ and
$\beta:H\to L$, with face maps made explicit in 5.9. It follows that the set
$(NI[\bullet]\otimes N\mathscr{B}_{*}\mathscr{O}_{bihh})_{1}$ consists of
pairs
$(\alpha,\beta):\mathrm{Dom}_{\alpha}\ast\mathrm{Dom}_{\beta}\to\mathrm{Cod}_{\beta}\ast\mathrm{Cod}_{\alpha}$
and
$(\alpha,\beta^{-1}):\mathrm{Dom}_{\beta}\ast\mathrm{Dom}_{\alpha}\to\mathrm{Cod}_{\alpha}\ast\mathrm{Cod}_{\beta}$,
such that for every $\alpha\in\mathrm{Mor}_{\mathscr{B}}(X,X)$ we have
$(\alpha,\mathrm{U}_{X})=(\alpha,\mathrm{U}_{X}^{-1}):\mathrm{Dom}_{\alpha}\to\mathrm{Cod}_{\alpha}$.

We now use the information in level $2$ to describe the various relations
between the morphisms. Notice that

$NI[\bullet]_{2}\cong NI[\bullet]_{1}\times_{NI[\bullet]_{0}}NI[\bullet]_{1}.$

Hence, the elements in the set $(NI[\bullet]\otimes N\mathscr{O}_{bihh})_{2}$
are of the form

  * •

$((\alpha_{0}\alpha_{1}),(\beta_{0},\beta_{1}),(\gamma_{0},\gamma_{1}))$,

  * •

$((\alpha_{0}\alpha_{1}),(\beta_{0}^{-1},\beta_{1}^{\pm 1}),(\gamma_{0}^{\pm
1},\gamma_{1}^{\pm 1}))$,

  * •

$((\alpha_{0}\alpha_{1}),(\beta_{0}^{\pm 1},\beta_{1}^{\pm
1}),(\gamma_{0}^{\pm 1},\gamma_{1}^{\pm 1}))$,

  * •

$((\alpha_{0}\alpha_{1}),(\beta_{0}^{\pm 1},\beta_{1}^{\pm
1}),(\gamma_{0}^{\pm 1},\gamma_{1}^{\pm 1}))$,

where $\alpha_{0}:F_{0}\to F_{1},\alpha_{1}:F_{1}\to
F_{2}\in\mathrm{Mor}_{\mathscr{B}}(X,Y)$, $\beta_{0}:G_{0}\to
G_{1},\beta_{1}:G_{1}\to G_{2}\in\mathrm{Mor}_{\mathscr{B}}(Y,Z)$ and
$\gamma_{0}:H_{0}\to H_{1},\gamma_{1}:H_{1}\to
H_{2}\in\mathrm{Mor}_{\mathscr{B}}(Z,X)$.

We will first focus on relations induced by elements of the form
$((\alpha_{0}\alpha_{1}),(\beta_{0},\beta_{1}),(\gamma_{0},\gamma_{1}))$.
Using the explicit description given in 5.9 we have the following boundaries:

  * •

$d_{0}((\alpha_{0},\alpha_{1}),(\beta_{0},\beta_{1}),(\gamma_{0},\gamma_{1}))=(\gamma_{1}\ast\alpha_{1},\beta_{1})$

  * •

$d_{1}((\alpha_{0},\alpha_{1}),(\beta_{0},\beta_{1}),(\gamma_{0},\gamma_{1}))=(\alpha_{1}\alpha_{0},\beta_{1}\beta_{0}\ast\gamma_{1}\gamma_{0})$

  * •

$d_{2}((\alpha_{0},\alpha_{1}),(\beta_{0},\beta_{1}),(\gamma_{0},\gamma_{1}))=(\alpha_{0}\ast\beta_{0},\gamma_{0})$

which means that we have the relation

(5.12)
$(\alpha_{1}\alpha_{0},\beta_{1}\beta_{0}\ast\gamma_{1}\gamma_{0})=(\gamma_{1}\ast\alpha_{1},\beta_{1})\circ(\alpha_{0}\ast\beta_{0},\gamma_{0}).$

Fix $\alpha:F\to G\in\mathrm{Mor}_{\mathscr{B}}(X,Y)$ and $\beta:H\to
L\in\mathrm{Mor}_{\mathscr{B}}(Y,X)$. Letting
$(\alpha_{0},\beta_{0},\gamma_{0})=(\alpha,\beta,\mathrm{U}_{X})$ and
$(\alpha_{1},\beta_{1},\gamma_{1})=(\mathrm{id}_{G},\mathrm{id}_{L},\mathrm{U}_{X})$
we get
$(\alpha,\beta)=(\mathrm{id}_{G},\mathrm{id}_{L})\circ(\alpha\ast\beta,\mathrm{U}_{X})$.
On the other side, letting
$(\alpha_{0},\beta_{0},\gamma_{0})=(\mathrm{id}_{F},\mathrm{id}_{H},\mathrm{U}_{X})$
and $(\alpha_{1},\beta_{1},\gamma_{1})=(\alpha,\beta,\mathrm{U}_{X})$ we get
$(\alpha,\beta)=(\beta\ast\alpha,\mathrm{U}_{X})\circ(\mathrm{id}_{F},\mathrm{id}_{H})$.
Hence we have

(5.13)
$(\alpha,\beta)=(\mathrm{id}_{G},\mathrm{id}_{L})\circ(\alpha\ast\beta,\mathrm{U}_{X})=(\beta\ast\alpha,\mathrm{U}_{X})\circ(\mathrm{id}_{F},\mathrm{id}_{H})$

Now, by symmetry, we can repeat the same arguments for $(\alpha,\beta^{-1})$
to conclude that

(5.14)
$(\alpha,\beta^{-1})=(\alpha\ast\beta,\mathrm{U}_{X})\circ(\mathrm{id}_{F},\mathrm{id}_{H}^{-1})=(\mathrm{id}_{G},\mathrm{id}_{L}^{-1})\circ(\beta\ast\alpha,\mathrm{U}_{X})$

We will denote the morphisms of the form $(\alpha,\mathrm{U}_{X})$ by the
symbol $\alpha$ and morphisms of the form $(\mathrm{id}_{F},\mathrm{id}_{G})$
by $(F,G)$ and morphisms of the form $(\mathrm{id}_{F},\mathrm{id}_{G}^{-1})$
by $(F,G)^{-1}$.

In 5.13 and 5.14 we have already established that every arbitrary morphism in
$\mathrm{THH}(\mathscr{B})$ is generated by these three classes of morphisms.
In order to finish the proof we need to understand how these morphisms
interact with each other and in particular confirm the relations from the
statement.

  1. (1)

The first relation follows from the definition of the fundamental category.

  2. (2)

Let $\alpha_{0}:F\to G,\alpha_{1}:G\to H\in\mathrm{Mor}_{\mathscr{B}}(X,X)$ be
two morphisms. The Equation 5.12 with
$(\alpha_{0},\beta_{0},\gamma_{0})=(\alpha_{0},\mathrm{U}_{X},\mathrm{U}_{X})$
and
$(\alpha_{1},\beta_{1},\gamma_{1})=(\alpha_{1},\mathrm{U}_{X},\mathrm{U}_{X})$
implies that

$(\alpha_{1}\alpha_{0},\mathrm{U}_{X})=(\alpha_{1},\mathrm{U}_{X})\circ(\alpha_{0},\mathrm{U}_{X}),$

which proves the second equation.

  3. (3)

Now for two objects $F:X\to Y$ and $G:Y\to X$ we have by definition of $I[2]$
a $2$-cell that we denote
$(\mathrm{id}_{F},\mathrm{id}_{G},\mathrm{id}_{G}^{-1})$ that witnesses the
composition
$(\mathrm{id}_{F},\mathrm{id}_{G}^{-1})\circ(\mathrm{id}_{F},\mathrm{id}_{G})=(\mathrm{id}_{FG},\mathrm{U}_{X})$.
We can similarly deduce that
$(\mathrm{id}_{F},\mathrm{id}_{G})\circ(\mathrm{id}_{F},\mathrm{id}_{G}^{-1})=(\mathrm{id}_{GF},\mathrm{U}_{Y})$.
This proves that $(F,G)$ is in fact an isomorphism with inverse $(F,G)^{-1}$

  4. (4)

We already confirmed the fourth condition in 5.13.

  5. (5)

Finally, we want to understand when two morphisms of the form $(F,G)$ commute.
Plugging in
$(\alpha_{0},\beta_{0},\gamma_{0})=(\alpha_{1},\beta_{1},\gamma_{1})=(\mathrm{id}_{F},\mathrm{id}_{G},\mathrm{id}_{H})$
into Equation 5.12 we get

$(\mathrm{id}_{F},\mathrm{id}_{G}\ast\mathrm{id}_{H})=(\mathrm{id}_{H}\ast\mathrm{id}_{F},\mathrm{id}_{G})\circ(\mathrm{id}_{F}\ast\mathrm{id}_{G},\mathrm{id}_{H}),$

which gives us the desired relation $(F,G\ast H)=(H\ast F,G)\circ(F\ast G,H)$.

As we have checked all possible relations between all generating morphisms we
have a complete characterization of $\mathrm{biHH}(\mathscr{B})$ and hence we
are done. ∎

In certain cases we can simplify the result in 5.10 further. For a given
strict monoidal category $\mathscr{C}$, let $B\mathscr{C}$ be the category
with on object $*$ and $B\mathscr{C}(*,*)=\mathscr{C}$.

######  Corollary 5.15.

Let $(\mathscr{C},\otimes,\mathrm{U})$ be a strict monoidal category. Then
$\mathrm{biHH}(B\mathscr{C})$ permits the following presentation:

  * •

$\mathrm{Obj}_{\mathrm{biHH}(\mathscr{B})}=\mathrm{Obj}_{\mathscr{C}}$

  * •

Generating Morphisms

    * –

Symbols $\alpha:\mathrm{Dom}_{\alpha}\to\mathrm{Cod}_{\alpha}$ where
$\alpha\in\mathrm{Mor}_{\mathscr{C}}$.

    * –

Symbols $(X,Y):X\otimes Y\to Y\otimes X$ where
$X,Y\in\mathrm{Obj}_{\mathscr{C}}$

  * •

Subject to the Relations

    * –

$(X,\mathrm{U}):X\to X$ is the identity morphism of the object
$X\in\mathrm{Obj}_{\mathscr{C}}$.

    * –

For two symbols $\alpha,\beta$ coming from $\mathrm{Mor}_{\mathscr{C}}$ such
that $\mathrm{Dom}_{\beta}=\mathrm{Cod}_{\alpha}$, the composition is given by
the composition $\beta\circ\alpha$ given in $\mathrm{Mor}_{\mathscr{C}}$.

    * –

The symbols $(X,Y)$ are isomorphisms.

    * –

For symbols $\alpha:X\to Y$ and $\beta:Z\to W$ in
$\mathrm{Mor}_{\mathscr{C}}$, we have
$(Y,W)\circ\alpha\otimes\beta=\beta\otimes\alpha\circ(X,Z)$.

    * –

For three objects $X,Y,Z\in\mathrm{Obj}_{\mathscr{C}}$ we have the equality
$(X,Y\otimes Z)=(Z\otimes X,Y)\circ(X\otimes Y,Z)$.

###### Proof.

Follows directly from applying 5.10 to the $2$-category $B\mathscr{C}$ with
only one object. ∎

We now want to apply this result to some key examples, which is the goal of
the remainder of this section.

######  Proposition 5.16.

$\displaystyle\mathrm{biHH}(B\mathbb{N})\simeq\\{0\\}\coprod_{n\in\\{1,2,3,...\\}}\\{n\\}\times
B\mathbb{Z}.$

###### Proof.

We apply 5.15 to symmetric monoidal category $\mathbb{N}$ with only identity
morphisms. It has objects $\\{0,1,2,...\\}$ and no non-trivial morphisms.
Moreover, for every $n,m\in\mathbb{N}$ there is an isomorphism $(n,m):n+m\to
n+m$.

Now, the last relation in 5.15 implies that for all $m>1$

$(n,m)=(n,(m-1)+1)=(1+n,m)\circ(n+m,1).$

By induction, this implies that $(n,m)=(n,1)^{m}$ and so every object $n>0$
has a unique automorphism $(n-1,1)$. This finishes the proof. ∎

######  Proposition 5.17.

$\displaystyle\mathrm{biHH}(B(\mathbb{N}*\mathbb{N}))\simeq\\{(0,0)\\}\coprod_{(n,m)\in\mathbb{N}\times\mathbb{N}\backslash\\{(0,0)\\}}\\{(n,m)\\}\times
B\mathbb{Z}.$

###### Proof.

Again, we want to use 5.15 for the symmetric monoidal category
$B(\mathbb{N}\ast\mathbb{N})$. According to the result the objects are
isomorphic to the elements in $\mathbb{N}*\mathbb{N}$. However, for two
elements $x,y\in\mathbb{N}*\mathbb{N}$ we have $xy\cong yx$ in the category
$\mathrm{biHH}(B(\mathbb{N}\ast\mathbb{N}))$ and so it suffices to take one
object from each isomorphism class, which corresponds to the commutator
classes and are precisely $\mathbb{N}\times\mathbb{N}$.

Now for a given object $(n,m)$ an automorphism is given by a tuple
$((n_{1},m_{1}),(n_{2},m_{2}))$ such that $n=n_{1}+n_{2}$ and
$m=m_{1}+m_{2}=m_{2}+m_{1}$. If $n=0$ or $m=0$, then this reduces to 5.16 and
it follows that there is a unique generating automorphism.

If $n,m\neq 0$, then the only elements that commute with $(n,m)$ are
$((n,m),(0,0))$ and $((0,0),(n,m))$. The first is the identity, by 5.15, and
so we again have one free automorphism and so the desired result follows. ∎

######  Remark 5.18.

As $\mathbb{N}$ is a monoid, we could not have used 3.11. However, the group
completion of $\mathbb{N}$ is given by $\mathbb{Z}$ and so we can ask
ourselves how the previous two results compare to the computation of
$\mathrm{biHH}(B\mathbb{Z})$ and $\mathrm{biHH}(B(\mathbb{Z}*\mathbb{Z}))$.

First of all the free loop space of $B\mathbb{Z}$ is equivalent to
$\mathbb{Z}\times B\mathbb{Z}$, which has an inclusion
$\\{0\\}\coprod_{n\in\\{1,2,3,...\\}}\\{n\\}\times
B\mathbb{Z}\to\mathbb{Z}\times B\mathbb{Z}$.

On the other hand the free loop space on $B(\mathbb{Z}*\mathbb{Z})$ is given
by the groupoid $\mathrm{Fun}(B\mathbb{Z},B(\mathbb{Z}*\mathbb{Z}))$, which
has objects automorphisms of $B(\mathbb{Z}*\mathbb{Z})$ i.e.
$\mathbb{Z}*\mathbb{Z}$ and morphisms natural transformations, which are given
by conjugation. Isomorphism classes of objects are given by conjugacy classes,
which correspond to $\mathbb{Z}\times\mathbb{Z}$ and automorphism group is
given by the centralizer, which in case of $(0,0)$ is given by
$\mathbb{Z}\times\mathbb{Z}$ and for any other object is given by
$\mathbb{Z}$. Again we have an inclusion

$\displaystyle\\{(0,0)\\}\coprod_{(n,m)\in\mathbb{N}\times\mathbb{N}\backslash\\{(0,0)\\}}\\{(n,m)\\}\times
B\mathbb{Z}\to
B(\mathbb{Z}*\mathbb{Z})\coprod_{(n,m)\in\mathbb{Z}\times\mathbb{Z}\backslash\\{(0,0)\\}}B\mathbb{Z}.$

Notice, neither inclusion are full, making the result non-trivial and in
particular challenging to deduce $\mathrm{biHH}(B\mathbb{N})$ from
$\mathrm{biHH}(B\mathbb{Z})$.

We now move on to a more complicated example. Let ${\mathbbe{\Delta}}_{+}$ be
the category of finite ordinals and order-preserving morphisms, which can also
be characterized as the category ${\mathbbe{\Delta}}$ together with one
additional initial object corresponding to the empty ordinal. Notice
${\mathbbe{\Delta}}_{+}$ is a strictly monoidal category with monoidal
structure given by disjoint union and unit given by the empty set. We want to
compute $\mathrm{biHH}(B{\mathbbe{\Delta}}_{+})$. This requires us to review
the paracyclic category, as defined in [DK15, Example I.22]

######  Definition 5.19.

Let $\Lambda_{\infty}$ be the paracyclic category, with objects
$\\{0,1,2,...\\}$ and morphisms $n\to m$ given by linear functions
$f:\mathbb{Z}\to\mathbb{Z}$ such that $f(l+m+1)=f(l)+n+1$.

The paracyclic category was introduced in [FL91], but the name was introduced
in [GJ93]. We need the following concrete characterization of the category as
described in [DK15, Example I.28].

######  Remark 5.20.

The paracyclic category can be characterized via the following generators and
relations:

  * •

Objects $\\{0,1,2,...\\}$

  * •

Generating morphisms $d^{i}$, $s^{i}$ $t^{i}$

  * •

$s^{i}$ and $d^{i}$ satisfy the cosimplicial relations and additionally we
have for $n\geq 1$

    * –

$t^{n}d^{i}=d^{i-1}t^{n-1}$ where $i>0$

    * –

$t^{n}d^{0}=d^{n}$

    * –

$t^{n}s^{i}=s^{i-1}t^{n+1}$, where $i>0$

    * –

$t^{n}s^{0}=s^{n}(t^{n+1})^{2}$

######  Theorem 5.21.

$\mathrm{biHH}(B{\mathbbe{\Delta}}_{+})\simeq(\Lambda_{\infty})^{\lhd}.$

###### Proof.

It suffices to prove that $\mathrm{biHH}(B{\mathbbe{\Delta}}_{+})$ has the
same presentation as $(\Lambda_{\infty})^{\lhd}$ described in 5.20. By 5.15 we
know that $\mathrm{biHH}(B{\mathbbe{\Delta}}_{+})$ has objects $\mathbb{N}$.
The monoidal structure is the same as for $B\mathbb{N}$ and so, following
5.16, there is a unique isomorphism $t^{n}:n\to n$ for every $n>0$.

By 5.15 the morphisms are generated by $d^{i},s^{i},t^{i}$, where the
interaction of $s^{i},d^{i}$ is given by the cosimplicial relations. What
remains is to check how the $s^{i},d^{i}$ interact with the $t^{i}$.

We start with $s^{0}:[n+1]\to[n]$. We have

$t^{n}s^{0}=([n-1],[0])\circ
s^{0}=s^{0}([n-1],[1])=s^{0}([n],[0])\circ([n],[0])=s^{0}(t^{n+1})^{2}$

Now, let $s^{i}:[n+1]\to[n]$, where $i>0$. Then
$s^{i}=\mathrm{id}_{[0]}\coprod s^{i-1}:[0]\coprod[n-1]\to[0]\coprod[n]$. Then
we have

$t^{n}s^{i}=([n-1],[0])\circ(\mathrm{id}_{[0]}\coprod
s^{i-1})=(\mathrm{id}_{[0]}\coprod s^{i-1}([n],[0])=s^{i-1}t^{n+1}$

We move on to $d^{i}$. Let $i_{0}:\emptyset\to[0]$ be the unique map. Then
$d^{0}=i_{0}\coprod\mathrm{id}_{[n-1]}:[n-1]\to[n]$. Hence

$t^{n}d^{0}=([n-1],[0])\circ
i_{0}\coprod\mathrm{id}_{[n-1]}=\mathrm{id}_{[n-1]}\coprod
i_{0}([n],\emptyset)=d^{n}.$

Next, for $i>0$, $d^{i}=\mathrm{id}_{[0]}\coprod
d^{i-1}:[0]\coprod[n-1]\to[0]\coprod[n]$ and so

$t^{n}d^{i}=([n-1],[0])\mathrm{id}_{[0]}\coprod
d^{i-1}=d^{i-1}\coprod\mathrm{id}_{[0]}([n-2],[0])=d^{i-1}t^{n-1}.$

Finally we need to confirm that $\emptyset$ is still the initial object and we
have

$t^{0}i_{0}=(\emptyset,[0])i_{0}=i_{0}(\emptyset,\emptyset)=i_{0},$

which proves that $\emptyset$ is still initial. This confirms all relations
and hence we are done. ∎

We can now use this result to get to the main result of interest, which again
requires us to review some concepts regarding the free adjunction category
$\mathscr{A}\mathrm{dj}$.

######  Remark 5.22.

The free adjunction bicategory $\mathscr{A}\mathrm{dj}$ can be described as
follows. It has two objects $0,1$, and the following morphisms:

  * •

$\mathscr{A}\mathrm{dj}(0,0)={\mathbbe{\Delta}}_{+}$, where we think of $[n]$
as the morphism $(GF)^{n}$.

  * •

$\mathscr{A}\mathrm{dj}(1,1)=({\mathbbe{\Delta}}_{+})^{op}$, the elements of
which we denote by $[n]^{op}$ to distinguish them from the previous item and
think of as $(FG)^{n}$.

  * •

$\mathscr{A}\mathrm{dj}(0,1)={\mathbbe{\Delta}}_{max}$, the wide subcategory
of ${\mathbbe{\Delta}}$ consisting of morphisms in ${\mathbbe{\Delta}}$ that
preserve the maximum, the elements of which we denote by $[n]_{max}$ and think
of as $(FG)^{n}F$

  * •

$\mathscr{A}\mathrm{dj}(0,1)={\mathbbe{\Delta}}_{min}$, the wide subcategory
of ${\mathbbe{\Delta}}$ consisting of morphisms in ${\mathbbe{\Delta}}$ that
preserve the minimum, with elements denoted $[n]_{min}$ and think of as
$(GF)^{n}G$.

For more details, see the original description of the free adjunction in
[SS86].

Notice the full subcategory of $\mathscr{A}\mathrm{dj}$ consisting of $0$ is
the free monad $2$-category and denoted by $\mathscr{M}\mathrm{on}$, whereas
the full subcategory of $\mathscr{A}\mathrm{dj}$ with object $1$ is the free
comonad $2$-category and denoted
$\mathscr{C}\mathrm{o}\mathscr{M}\mathrm{on}$. We now have the following
result.

######  Theorem 5.23.

There is a diagram of equivalences of categories

${\mathrm{biHH}(\mathscr{M}\mathrm{on})}$${\mathrm{biHH}(\mathscr{A}\mathrm{dj})}$${\mathrm{biHH}(\mathscr{C}\mathrm{o}\mathscr{M}\mathrm{on})}$${(\Lambda_{\infty})^{\lhd}}$${(\Lambda_{\infty})^{\lhd\rhd}}$${(\Lambda_{\infty}^{op})^{\rhd}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

###### Proof.

The two equivalences on the left and right side already follows from 5.21.
Hence we will prove the middle map is an equivalence and the diagrams commute.

Following 5.10 and the explicit description of the $2$-category
$\mathscr{A}\mathrm{dj}$ given above in 5.22, the category
$\mathrm{biHH}(\mathscr{A}\mathrm{dj})$ permits the following general
description:

  * •

Objects two copies of $\mathbb{N}$, which we denote $\emptyset,[0],[1],\cdots$
and $\emptyset^{op},[0]^{op},[1]^{op},\cdots$.

  * •

Morphisms $\sigma:[m]\to[n]$, $\sigma^{op}:[n]^{op}\to[m]^{op}$ for all
$\sigma\in{\mathbbe{\Delta}}([m],[n])$.

  * •

Isomorphisms $([n],[m]):[n+m+1]\to[n+m+1]$,
$([n]^{op},[m]^{op}):[n+m+1]^{op}\to[n+m+1]^{op}$ for objects
$n,m\in\mathbb{N}$.

  * •

Isomorphisms $([n]_{min},[m]_{max}):[n+m+1]\to[n+m+1]^{op}$ and
$([m]_{max},[n]_{min}):[n+m+1]^{op}\to[n+m+1]$, where $n,m\in\mathbb{N}$.

  * •

Relations between the morphisms described in 5.10.

We will now use the various relations to reduce the structure and obtain the
desired result.

First of all, if we restrict to the objects $[0],[1],\cdots$, then we
precisely have the generating morphisms and relations of
${\mathbbe{\Delta}}_{+}$ and hence, by 5.21, obtain a copy of
$(\Lambda_{\infty})^{\lhd}$. Similarly, the generating morphisms and
isomorphisms restricted to $[0]^{op},[1]^{op},\cdots$ gives us a copy of
$(\Lambda_{\infty}^{op})^{\rhd}$. What remains is to explain how the
additional isomorphisms $([n]_{min},[m]_{max})$ and $([m]_{max},[n]_{min})$
influence $\mathrm{biHH}(\mathscr{A}\mathrm{dj})$.

Now, for a given $n,m\in\mathbb{N}$, we, by the last relation in 5.10, have

$\displaystyle([n]_{min},[m]_{max})$
$\displaystyle=([n]_{min},[m-1]_{max}[0]^{op})$
$\displaystyle=([0]^{op}[n]_{min},[m-1]_{max})\circ([n+m],[0])$
$\displaystyle=([n+1]_{min},[m-1]_{max})\circ([n+m],[0])$

and so, by induction, we we have the equality

$([n]_{min},[m]_{max})=([n+m]_{min},[0]_{max})\circ([n+m],[0])^{m}$

and similarly

$([n]_{max},[m]_{min})=([n+m],[0])^{m}\circ([n+m]_{max},[0]_{min})$

Hence, for every $n\in\mathbb{N}$ we can reduce the additional isomorphisms of
the form $([n]_{min},[0]_{max}):[n+1]\to[n+1]^{op}$ and
$([n]_{max},[0]_{min}):[n+1]^{op}\to[n+1]$.

Finally, again by the last relation in 5.10, we have

$([n],[0])^{2}=([n-1],[1])=([n]_{max},[0]_{min})\circ([n]_{min},[0]_{max}).$

Hence, we can give the following simplified explicit description of
$\mathrm{biHH}(\mathscr{A}\mathrm{dj})$:

  * •

Objects $[0],[1],...$ and $[0]^{op},[1]^{op},...$

  * •

Generating morphisms

    * –

$d^{i}:[n]\to[n+1],s^{i}:[n]\to[n-1],t^{n}:[n]\to[n]$

    * –

$(d^{i})^{op}:[n]^{op}\to[n+1]^{op},(s^{i})^{op}:[n]^{op}\to[n-1]^{op},(t^{n})^{op}:[n]^{op}\to[n]^{op}$

    * –

$c^{n}:[n]\to[n]^{op}$

  * •

Subject to relations

    * –

$d^{i},s^{i},t^{n}$ satisfy the paracyclic relations (5.20)

    * –

$(d^{i})^{op},(s^{i})^{op},(t^{n})^{op}$ satisfy the opposite paracyclic
relations

    * –

$c^{n}t^{n}=(t^{n})^{op}c^{n}$ as

$([n]_{min},[0]_{max})\circ([n],[0])=([n]_{min},[1]_{max})=([n]^{op},[0]^{op})\circ([n]_{min},[0]_{max})$

    * –

$c^{n}d^{i}=(d^{i})^{op}c^{n}$, which follows directly from the fourth
relation in 5.10,

    * –

$c^{n}s^{i}=(d^{i})^{op}c^{n}$.

The description of $\mathrm{THH}(\mathscr{A}\mathrm{dj})$ as two copies of
$\Lambda_{\infty}$ and $(\Lambda_{\infty})^{op}$ and an additional isomorphism
$c_{n}$ is just an explicit description of the following pseudo-pushout:

${\Lambda_{\infty}}$${(\Lambda_{\infty})^{\lhd}}$${((\Lambda_{\infty})^{op})^{\rhd}}$${\mathrm{biHH}(\mathscr{A}\mathrm{dj})}$

which is equivalent to $(\Lambda_{\infty})^{\lhd\rhd}$ and by construction the
diagrams above commute. ∎

######  Remark 5.24.

In recent work Ayala and Francis have (independently) computed
$\mathrm{biHH}(\mathscr{A}\mathrm{dj})$ [AF21] using methods from
factorization homology [AFR17].

Having this computation we can now deduce the following canonical
strengthening of 4.9.

######  Corollary 5.25.

The functor $\widehat{\chi}$ is fact given by restricting to the unique map
from the initial to terminal object in
$\mathrm{biHH}(\mathscr{A}\mathrm{dj})\simeq(\Lambda_{\infty})^{\lhd\rhd}$.
This means the definition of the Euler characteristic is independent of any
choice and is canonical.

We now move on to the final computation.

######  Theorem 5.26.

$\displaystyle\mathrm{biHH}(\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd})\simeq(\Lambda_{\infty})^{\lhd\rhd}\times\\{0\\}\coprod_{n\in\\{1,2,3,...\\}}B\mathbb{Z}\times{\mathbbe{\Delta}}_{+}\times\\{n\\}$

###### Proof.

We want to use the various results in Section 5 to characterize
$\mathrm{biHH}(\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd})$. We first start
by reducing the set of objects so that no two objects are isomorphic.
Following 4.14 and 5.10 the objects in
$\mathrm{biHH}(\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd})$ are bijective to

$\displaystyle\mathrm{Obj}_{\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd}(0,0)}\coprod\mathrm{Obj}_{\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd}(1,1)}$
$\displaystyle\cong\mathbb{N}\ast\mathbb{N}\coprod\mathbb{N}\ast\mathbb{N}$
$\displaystyle=\\{\emptyset,[0],...\\}\ast\\{0,1,...\\}\coprod\\{\emptyset^{op},[0]^{op},...\\}\ast\\{0,1,...\\},$

where we are using 5.22.

Now, if $x,y\in\\{\emptyset,[0],...\\}\ast\\{0,1,...\\}$, then 5.10 implies
that we have an isomorphism $(x,y):xy\to yx$. This means the isomorphism
classes are given by the commutators classes of the free words (as already
explained in 5.17). Hence isomorphism classes of objects of
$\mathrm{biHH}(\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd})$ can in our first
step be reduced to

$\displaystyle\mathbb{N}\times\mathbb{N}\coprod\mathbb{N}\times\mathbb{N}$
$\displaystyle=\\{\emptyset,[0],...\\}\times\\{0,1,...\\}\coprod\\{\emptyset^{op},[0]^{op},...\\}\times\\{0,1,...\\}$
$\displaystyle\cong\\{[n]k,[n]^{op}k:[n]\in\mathrm{Obj}_{{\mathbbe{\Delta}}},k\in\mathbb{N}\\}$

We will make one further reduction of our objects. Indeed, using the same
argument as in 5.23 (and again notation 5.22) it follows that

$([n-1]_{min},[0]_{max})k:[n]k\to[n]^{op}k$

is an isomorphism. Hence, we can conclude that the isomorphism classes of
objects are precisely given by

$\mathbb{N}\times\mathbb{N}\cong\\{[n]k:[n]\in\mathrm{Obj}_{{\mathbbe{\Delta}}_{+}},k\in\mathbb{N}\\}.$

Now that we have determined the objects, we can move on to precisely the
determine the generating morphisms and isomorphisms, again using 5.10.

By the explanation in 4.14, morphisms in
$\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd}(0,0)$ are given by words of
morphisms in ${\mathbbe{\Delta}}_{+}$ and so there is only a morphism from
$[n_{1}]m_{1}$ to $[n_{2}]m_{2}$ if and only if $m_{1}=m_{2}$. Hence

(5.27)
$\mathrm{Mor}_{\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd}(0,0)}=\\{\sigma
k:[n]k\to[m]k:\sigma\in{\mathbbe{\Delta}}_{+}([n],[m]),k\in\mathbb{N}\\}$

On the other side, the monoidal structure on
$\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd}(0,0)$ coincides with the one on
$B(\mathbb{N}\ast\mathbb{N})$ and so, as explained in 5.17, every object
$[n]k$ has a unique generating non-trivial automorphism i.e.
$\mathrm{Aut}_{\mathrm{biHH}(\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd})}([n]k)=\mathbb{Z}$.

In order to finish the proof we need to understand the interaction between the
automorphisms and morphisms, using the relations given in 5.10. Let
$\mathrm{biHH}(\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd})_{k}$ be the full
subcategory of $\mathrm{biHH}(\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd})$
consisting of objects of the form $[n]k$. The explanation in 5.27 implies that

$\mathrm{biHH}(\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd})\cong\coprod_{k\in\mathbb{N}}\mathrm{biHH}(\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd})_{k}$

Hence, we can break down our analysis into the different
$\mathrm{biHH}(\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd})_{k}$.

Let us start with
$\mathrm{biHH}(\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd})_{0}$. In that
case we have morphisms $\sigma 0:[n]0\to[m]0$, for $\sigma:[n]\to[m]$ in
${\mathbbe{\Delta}}_{+}$ and automorphisms $t^{n}:[n]0\to[n]0$, which, by the
relations given in 5.23 interact precisely as given in 5.23 and so we have the
equivalence

$\mathrm{biHH}(\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd})\simeq(\Lambda_{\infty})^{\lhd\rhd}.$

Now, let us assume $k>0$. Then the generating isomorphism of $[n]k$ is given
by the symbol $(\emptyset 0,[n]k)$, which, by 5.10, satisfies the equality

$((\emptyset 0,[n]k)=([n]0,\emptyset k)\circ(\emptyset k,[n]0),$

where $([n]0,\emptyset k):[n]k\to k[n]$ is a twisting isomorphism and
$(\emptyset k,[n]0)$ is defined similarly. Now, for an arbitrary morphism
$\sigma k:[n]k\to[m]k$ in
$\mathrm{biHH}(\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd})_{k}$, where
$\sigma:[n]\to[m]$, we have

$\displaystyle((\emptyset 0,[m]k)\circ\sigma k$ $\displaystyle=([m]0,\emptyset
k)\circ(\emptyset k,[m]0)\circ\sigma$ $\displaystyle=([m]0,\emptyset k)\circ
k\sigma\circ(\emptyset k,[n]0)$ $\displaystyle=\sigma k\circ([n]0,\emptyset
k)\circ(\emptyset k,[n]0)$ $\displaystyle=\sigma k\circ(\emptyset 0,[n]k)$

Here, for the third equality we used the fact that $m\neq 0$. This shows that
$(\emptyset 0,[n]k)$ commutes will all morphism, proving the desired
equivalence

$\mathrm{biHH}(\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd})_{m}\simeq
B\mathbb{Z}\times{\mathbbe{\Delta}}_{+},$

finishing the proof. ∎

Now combining this computation with 4.16 we obtain the following key
observation.

######  Corollary 5.28.

The functor $\widehat{\mathrm{tr}}$ is obtained by restricting to the morphism
$\emptyset 1\to[0]1$ in
$\mathrm{biHH}(\mathscr{A}\mathrm{dj}\mathscr{E}\mathrm{nd})$. As
$\mathrm{Aut}(\emptyset 1)=\mathbb{Z}$, this means $\widehat{\mathrm{tr}}$ is
in fact not canonically determined and is only fixed up to a choice of
automorphism.

Notice the non-canonicity does not influence the result in 4.17, however it is
not clear whether the choice of automorphism would influence other future
results.

##  6\. Towards shadows on $(\infty,2)$-categories

We are finally in a position to generalize shadows to $(\infty,2)$-categories.
To motivate our definition, we first analyze the nature of shadows on the
homotopy bicategory of a $(\infty,2)$-category with values in homotopy
category of an $(\infty,1)$-category.

Let $\mathscr{B}$ be an $(\infty,2)$-category specified by a symmetric
monoidal functor

$\mathscr{B}:\mathscr{B}\mathrm{ypass}_{\mathscr{B}_{0}}\to\mathscr{C}\mathrm{at}_{\infty}.$

Its _homotopy bicategory_ $\mathrm{Ho}(\mathscr{B})$ is the bicategory
specified by the composite

$\mathscr{B}\mathrm{ypass}_{\mathscr{B}_{0}}\xrightarrow{\mathscr{B}}\mathscr{C}\mathrm{at}_{\infty}\xrightarrow{\mathrm{Ho}}\mathscr{C}\mathrm{at}.$

This composite does indeed give rise to a bicategory (i.e., enriched over
$\mathscr{C}\mathrm{at}$ in the sense of 2.8), since $\mathrm{Ho}$ preserves
finite products and thus is symmetric monoidal.

######  Proposition 6.1.

Let $\mathscr{B}$ be an $(\infty,2)$-category and $\mathscr{D}$ an
$(\infty,1)$-category. There is an equivalence

$\mathscr{S}\mathrm{ha}\big{(}\mathrm{Ho}(\mathscr{B}),\mathrm{Ho}(\mathscr{D})\big{)}\simeq\mathrm{Fun}_{\infty}\big{(}\mathrm{biHH}_{\infty}(\mathscr{B}),\mathrm{Ho}(\mathscr{D})\big{)},$

where on the right-hand side we implicitly view $\mathrm{Ho}(\mathscr{D})$ as
an $(\infty,1)$-category via the nerve functor.

###### Proof.

There are equivalences

$\mathscr{S}\mathrm{ha}\big{(}\mathrm{Ho}(\mathscr{B}),\mathrm{Ho}(\mathscr{D})\big{)}\simeq\mathrm{Fun}\big{(}\mathrm{biHH}(\mathrm{Ho}(\mathscr{B})),\mathrm{Ho}(\mathscr{D})\big{)}\simeq\mathrm{Fun}_{\infty}\big{(}\mathrm{biHH}_{\infty}(\mathscr{B}),\mathrm{Ho}(\mathscr{D})\big{)}.$

The first equivalence is an immediate consequence of 3.19. The second follows
from the fact that $\mathrm{Ho}$ commutes with colimits and so

$\mathrm{biHH}(\mathrm{Ho}(\mathscr{B}))\simeq\mathrm{Ho}(\mathrm{biHH}_{\infty}(\mathscr{B})),$

together with the fact that $\mathrm{Ho}$ is left adjoint to the nerve functor
(Subsection 1.6). ∎

In particular, there is a localization functor

$\mathrm{Fun}(\mathrm{biHH}_{\infty}(\mathscr{B}),\mathscr{D})\to\mathscr{S}\mathrm{ha}\big{(}\mathrm{Ho}(\mathscr{B}),\mathrm{Ho}(\mathscr{D})\big{)}$

that takes a trace of $(\infty,2)$-categories to a shadow.

Motivated by 3.19 and the analysis above, we define a shadow on an
$(\infty,2)$-category $\mathscr{B}$ with values in an $(\infty,1)$-category
$\mathscr{D}$ to be a trace (2.17), i.e., an $\infty$-functor

$\mathrm{biHH}_{\infty}(\mathscr{B})\to\mathscr{D},$

where $\mathrm{biHH}_{\infty}(\mathscr{B})$ is constructed as in 2.10. This
definition can be viewed as a homotopy coherent lift of the definition of a
shadow.

It is natural to ask when $(\infty,2)$-categorical shadows exist and how to
construct them. In the important special case of $\mathscr{V}$-enriched
categories and their bimodules, we conjecture that there should an exist an
analogue of the Hochschild shadow for spectral categories.

######  Conjecture 6.2.

Let $(\mathscr{V},\otimes,I)$ be a presentably symmetric monoidal
$(\infty,1)$-category. Let $\mathscr{M}\mathrm{od}_{\mathscr{V}}$ be the
$(\infty,2)$-category with as objects $\mathscr{V}$-enriched categories and
morphisms given by bimodules (cf. [Hau16]). There exists a functor of
$\infty$-categories

$\mathrm{HH}_{\mathscr{V}}:\mathrm{biHH}_{\infty}(\mathscr{M}\mathrm{od}_{\mathscr{V}})\to\mathscr{V}.$

Moreover, if $\mathscr{V}$ is closed symmetric monoidal, then
$\mathrm{HH}_{\mathscr{V}}$ is a $\mathscr{V}$-enriched functor of
$\mathscr{V}$-enriched $\infty$-categories

We cannot yet prove this conjecture, but take an important first step and
construct a functor to $\mathrm{Ho}\mathscr{V}$ (6.11), then discuss some
technical challenges that arise when trying to lift to an
$(\infty,2)$-categorical shadow (6.14). We begin by relativizing Berman’s
approach to $\mathscr{V}$-enriched $\infty$-categories [Ber22] in order to
describe $\mathscr{V}$-enriched bimodules.

######  Definition 6.3.

Let $S,T$ be two sets. A directed graph from $S$ to $T$ is a directed graph
with vertex set $S\coprod T$, set of edges $E$, and source and target maps
$s,t:E\to S\coprod T$ such that if $t(e)\in S$, then $s(e)\in S$, i.e., there
are no edges starting in $T$ and ending in $S$

######  Definition 6.4.

Let $\mathscr{B}\mathrm{ypass}_{S,T}$ be the full subcategory of
$\mathscr{B}\mathrm{ypass}_{S\coprod T}$ with objects the directed graphs from
$S$ to $T$.

This full subcategory is still symmetric monoidal, as graphs from $S$ to $T$
are closed under disjoint union, and the empty graph on $S\coprod T$ is in
particular a graph from $S$ to $T$.

The following notation for objects in $\mathscr{B}\mathrm{ypass}_{S,T}$ proves
useful below.

######  Notation 6.5.

Let $X,Y\in S\coprod T$, where if $Y\in S$, then $X\in S$. There exists a
graph with a unique edge from $X$ to $Y$, which we denote $(X,Y)$. More
generally, let $(X_{0},X_{1},...,X_{n})$ be the graph with a single path
$X_{0}\to X_{1}\to...\to X_{n}$, where if $X_{i}\in S$, then $X_{j}\in S$ for
all $j<i$.

As $\mathscr{B}\mathrm{ypass}_{S,T}$ is defined as a full symmetric monoidal
subcategory of $\mathscr{B}\mathrm{ypass}_{S\coprod T}$, the explanation after
[Ber22, Definition 2.3] provides an explicit description of the symmetric
monoidal category via generators and relations as follows.

  * •

Objects are pairs $(X,Y)$ of elements of $S\coprod T$, where if $Y\in S$, then
$X\in S$.

  * •

Morphisms are generated by $(X,Y,Z)=(X,Y)\otimes(Y,Z)\to(X,Z)$, which exists
for every triple $X,Y,Z\in S\coprod T$, where $Z\in S$ implies $X,Y\in S$, and
$Y\in S$ implies $X\in S$.

  * •

For every $X\in S\coprod T$, there is an identity morphism $\emptyset\to(X,X)$

  * •

The associativity and unitality relations of [Ber22, Definition 2.3].

There is an evident inclusion functor
$\mathscr{B}\mathrm{ypass}_{S}\coprod\mathscr{B}\mathrm{ypass}_{T}\to\mathscr{B}\mathrm{ypass}_{S,T}$.
We can use this to define our desired modules.

######  Definition 6.6.

Let $\mathscr{C}:\mathscr{B}\mathrm{ypass}_{S}\to\mathscr{V}$ and
$\mathscr{D}:\mathscr{B}\mathrm{ypass}_{T}\to\mathscr{V}$ be two
$\mathscr{V}$-enriched $\infty$-categories. A $\mathscr{V}$-enriched
$(\mathscr{C},\mathscr{D})$-bimodule is a symmetric monoidal functor
$\mathscr{M}:\mathscr{B}\mathrm{ypass}_{S,T}\to\mathscr{V}$ that makes the
following diagram commute.

${\mathscr{B}\mathrm{ypass}_{S}}$${\mathscr{B}\mathrm{ypass}_{S,T}}$${\mathscr{V}}$${\mathscr{B}\mathrm{ypass}_{T}}$$\scriptstyle{\mathscr{C}}$$\scriptstyle{\mathscr{M}}$$\scriptstyle{\mathscr{D}}$

It is helpful to unpack this definition somewhat. The objects in
$\mathscr{B}\mathrm{ypass}_{S,T}$ can be classified into three distinct types:
graphs whose edges all start and end at elements of $S$, graphs whose edges
all start and end at elements of $T$, and graphs that have an edge that starts
in $S$ and ends in $T$. The first two types of graph lie in the essential
image of the inclusion functor
$\mathscr{B}\mathrm{ypass}_{S}\coprod\mathscr{B}\mathrm{ypass}_{T}\to\mathscr{B}\mathrm{ypass}_{S,T}$,
and so the value of $\mathscr{M}$ on those graphs is predetermined. In
particular for every $X_{0},...,X_{n}\in S$,

$\mathscr{M}(X_{0},...,X_{n})=\mathscr{C}(X_{0},...,X_{n}),$

and for every $X_{0},...,X_{n}\in T$,

$\mathscr{M}(X_{0},...,X_{n})=\mathscr{D}(X_{0},...,X_{n}).$

On the other hand if $X_{0}\in S$ and $X_{1}\in T$, then there are no
constraints on the object $\mathscr{M}(X_{0},X_{1})$ in $\mathscr{V}$.

In $\mathscr{B}\mathrm{ypass}_{S,T}$, for every $X_{0},...,X_{n}\in S$ and
$X_{n+1}\in T$, there a bypass morphism
$(X_{0},...,X_{n+1})\to(X_{0},X_{n+1})$, the image of which under
$\mathscr{M}$ is a map

$\mathscr{C}(X_{0},X_{1})\otimes...\otimes\mathscr{C}(X_{n-1},X_{n})\otimes\mathscr{M}(X_{n},X_{n+1})\to\mathscr{M}(X_{0},X_{n+1}).$

Similarly for $X_{0}\in S$ and $X_{1},...,X_{n+1}\in T$, there is a map

$\mathscr{M}(X_{0},X_{1})\otimes\mathscr{D}(X_{0},X_{1})\otimes...\otimes\mathscr{D}(X_{n},X_{n+1})\to\mathscr{M}(X_{0},X_{n+1}).$

This is precisely the expected structural data of a bimodule. The composition
rules in $\mathscr{B}\mathrm{ypass}_{S,T}$ guarantees that these bimodule
actions satisfy the appropriate coherence conditions.

Before we proceed, it is important to confirm that the definition given here
matches with the existing literature. Bimodules of $\mathscr{V}$-enriched
$\infty$-categories have been studied extensively by Haugseng [Hau16]. We can
confirm that the definition above does indeed coincide with that of Haugseng,
by a slight generalization of the argument given by Berman in [Ber22,
Proposition 2.7].

######  Proposition 6.7.

The notion of bimodules formulated in 6.6 coincides with that of Haugseng
[Hau16, Definition 4.3].

###### Proof.

We follow the lines of the proof of [Ber22, Proposition 2.7]. First recall
that a bimodule is an algebra on the non-symmetric $\infty$-operad
${\mathbbe{\Delta}}^{op}_{S,T}$ (described explicitly in [Hau16, Definition
4.1]). Because $\mathscr{V}$ is symmetric monoidal, we can use the symmetric
monoidal envelope of ${\mathbbe{\Delta}}^{op}_{S,T}$ which is given by the
active morphisms [Lur17, Construction 2.2.4.1].

As in the proof of [Ber22, Proposition 2.7], the active morphisms to a pair
$(A,B)$ of elements of $S\coprod T$ are of the form

$(X_{1},Y_{1})\otimes...\otimes(X_{n},Y_{n})\to(A,B).$

It follows that the symmetric monoidal envelope of
${\mathbbe{\Delta}}_{S,T}^{op}$ is given by $\mathscr{B}\mathrm{ypass}_{S,T}$
and the result then follows by the universal property of symmetric monoidal
envelopes [Lur17, Proposition 2.2.4.9]. ∎

We are now almost ready to define $\mathrm{THH}$ for $\infty$-categorical
bimodules. This requires one last definition.

######  Definition 6.8.

For $S,T$ two sets, define the symmetric monoidal category
$\mathscr{B}\mathrm{ypass}_{S\circlearrowleft T}$ as the pushout of symmetric
monoidal categories

${\mathscr{B}\mathrm{ypass}_{S}\coprod\mathscr{B}\mathrm{ypass}_{T}}$${\mathscr{B}\mathrm{ypass}_{S,T}}$${\mathscr{B}\mathrm{ypass}_{T,S}}$${\mathscr{B}\mathrm{ypass}_{S\circlearrowleft
T}}$

Now, let $\mathscr{C}:\mathscr{B}\mathrm{ypass}_{S}\to\mathscr{V}$ be a
$\mathscr{V}$-enriched $\infty$-category and
$\mathscr{M}:\mathscr{B}\mathrm{ypass}_{S,S}\to\mathscr{V}$ a
$(\mathscr{C},\mathscr{C})$-bimodule. To distinguish between the two copies of
$S$, we denote an element $X\in S$ as $X^{\prime}$ if it is in the second
copy. By universal property of pushouts, $\mathscr{M}$ lifts to a functors
$\mathscr{B}\mathrm{ypass}_{S\circlearrowleft S}\to\mathscr{V}$, which we also
denote by $\mathscr{M}$, to simplify notation.

Let $\mathscr{M}^{*}:\mathcal{P}(\mathscr{B}\mathrm{ypass}_{S\circlearrowleft
S})\to\mathscr{V}$ be the left Kan extension of $\mathscr{M}$. Let
$(\mathscr{O}_{hh_{\mathscr{V}}})_{\bullet}:{\mathbbe{\Delta}}^{op}\to\mathcal{P}(\mathscr{B}\mathrm{ypass}_{S\circlearrowleft
S})$ be the functor defined by

$(\mathscr{O}_{hh_{\mathscr{V}}})_{n}=\coprod_{X_{0},...,,X_{n}\in
S}(X_{0},X_{1},...,X_{n},X_{0}^{\prime}),$

where we are using 6.5.

######  Definition 6.9.

For any $\mathscr{V}$-enriched $\infty$-category $\mathscr{C}$ and
$(\mathscr{C},\mathscr{C})$-bimodule $\mathscr{M}$, the $\mathscr{V}$-enriched
Hochschild homology of $\mathscr{C}$ with coefficients in $\mathscr{M}$,
denoted $\mathrm{HH}_{\mathscr{V}}(\mathscr{C},\mathscr{M})$, is the object in
$\mathscr{V}$ that is the colimit of
$\mathscr{M}^{*}\circ(\mathscr{O}_{hh_{\mathscr{V}}})_{\bullet}$.

Since $\mathscr{M}^{*}$ preserves colimits,
$\mathrm{HH}_{\mathscr{V}}(\mathscr{C},\mathscr{M})$ is equivalent to the
colimit of the simplicial diagram

$\coprod_{X_{0},...,X_{n}}\mathscr{C}(X_{0},X_{1})\otimes...\otimes\mathscr{C}_{(}X_{n-1},X_{n})\otimes\mathscr{M}(X_{n},X_{0})$

Our goal is to show that the collection of the
$\mathrm{THH}(\mathscr{C},\mathscr{M})$ underlies a shadow. To do so, we need
an appropriate bicategory. We begin by defining morphisms of bimodules.

######  Definition 6.10.

Let $\mathscr{C}:\mathscr{B}\mathrm{ypass}_{S}\to\mathscr{V}$ and
$\mathscr{D}:\mathscr{B}\mathrm{ypass}_{T}\to\mathscr{V}$ be
$\mathscr{V}$-enriched $\infty$-categories, and let $\mathscr{M}$ and
$\mathscr{N}$ be $(\mathscr{C},\mathscr{D})$-bimodules. A morphism of
bimodules is a natural transformation $\alpha:\mathscr{M}\to\mathscr{N}$ such
that $\alpha$ restricts to the identity on $\mathscr{B}\mathrm{ypass}_{S}$ and
$\mathscr{B}\mathrm{ypass}_{T}$.

In [Hau16, Theorem 1.2] Haugseng proves that there is an $(\infty,2)$-category
$\mathscr{M}\mathrm{od}_{\mathscr{V}}$ with

  * •

$\mathscr{V}$-enriched $\infty$-categories as objects,

  * •

bimodules as morphisms,

  * •

morphisms of bimodules as $2$-morphisms.

We are now ready to generalize 3.29.

######  Theorem 6.11.

There is a functor of $\infty$-categories

$\mathrm{HH}_{\mathscr{V}}:\mathrm{biHH}_{\infty}(\mathscr{M}\mathrm{od}_{\mathscr{V}})\to\mathrm{Ho}\mathscr{V}$

such that that the corresponding shadow on
$\mathscr{M}\mathrm{od}_{\mathscr{V}}$ (cf. 6.1) sends any
$\mathscr{C}$-bimodule $\mathscr{M}$ to
$\mathrm{HH}_{\mathscr{V}}(\mathscr{C},\mathscr{M})$.

###### Proof.

We first need to show that the construction of
$\mathrm{HH}_{\mathscr{V}}(\mathscr{C},\mathscr{M})$ is natural in the
coefficient bimodule, i.e., that it extends to a functor
$\mathrm{HH}_{\mathscr{V}}(\mathscr{C},-):\mathrm{Ho}\mathscr{M}\mathrm{od}_{\mathscr{V}}(\mathscr{C},\mathscr{C})\to\mathrm{Ho}\mathscr{V}$
for every $\mathscr{V}$-enriched category $\mathscr{C}$. For any
$\mathscr{C}$-bimodule morphism $\alpha:\mathscr{M}\to\mathscr{N}$, we define
$\mathrm{HH}_{\mathscr{V}}(\mathscr{C},\alpha)$ by

$\mathrm{HH}_{\mathscr{V}}(\mathscr{C},\alpha)=\mathrm{colim}_{{\mathbbe{\Delta}}^{op}}\alpha^{*}\circ\mathscr{O}_{hh_{\mathscr{V}}}:\mathrm{colim}_{{\mathbbe{\Delta}}^{op}}\mathscr{M}^{*}\circ\mathscr{O}_{hh_{\mathscr{V}}}\to\mathrm{colim}_{{\mathbbe{\Delta}}^{op}}\mathscr{N}^{*}\circ\mathscr{O}_{hh_{\mathscr{V}}}.$

This construction is functorial, as it can be described as the following
composite of functors.

$\mathrm{Fun}(\mathscr{B}\mathrm{ypass}_{S\circlearrowleft
S},\mathscr{V})\xrightarrow{Lan}\mathrm{Fun}\big{(}\mathrm{Fun}(\mathscr{B}\mathrm{ypass}_{S\circlearrowleft
S}^{op},\mathscr{S}),\mathscr{V}\big{)}\xrightarrow{(\mathscr{O}_{hh_{\mathscr{V}}})^{*}}\mathrm{Fun}({\mathbbe{\Delta}}^{op},\mathscr{V})\xrightarrow{\mathrm{colim}}\mathscr{V}$

It remains to define the twisting isomorphism for this shadow. To do so, we
adapt the Dennis-Morita-Waldhausen argument [BM12, Proposition 6.2], which is
also used by Campbell and Ponto [CP19, Theorem 2.17], to the
$\infty$-categorical setting.

The first step towards defining the twisting isomorphism consists of the
following bisimplicial construction. Let $(\mathscr{O}^{S\circlearrowleft
T}_{thh})_{\bullet\bullet}:{\mathbbe{\Delta}}^{op}\times{\mathbbe{\Delta}}^{op}\to\mathcal{P}(\mathscr{B}\mathrm{ypass}_{S\circlearrowleft
T})$ be the functor that takes $([n],[m])$ to

$\coprod_{X_{0},...,X_{n}\in S,Y_{0},...,Y_{m}\in
T}(X_{0},...,X_{n},Y_{0},...,Y_{m},X_{0}),$

and let $\mathscr{O}^{S\circlearrowleft T}_{thh}$ be the colimit of this
bisimplicial diagram in
$\mathcal{P}(\mathscr{B}\mathrm{ypass}_{S\circlearrowleft T})$. Similarly,
define $(\mathscr{O}^{T\circlearrowleft
S}_{thh})_{\bullet\bullet}:{\mathbbe{\Delta}}^{op}\times{\mathbbe{\Delta}}^{op}\to\mathcal{P}(\mathscr{B}\mathrm{ypass}_{S\circlearrowleft
T})$ as the functor with value

$\coprod_{X_{0},...,X_{n}\in S,Y_{0},...,Y_{m}\in
T}(Y_{0},...,Y_{m},X_{0},...,X_{n},Y_{0}),$

on $([n],[m])$ and $\mathscr{O}^{T\circlearrowleft S}_{thh}$ as its colimit in
$\mathcal{P}(\mathscr{B}\mathrm{ypass}_{S\circlearrowleft T})$.

The symmetric monoidal structure of the bypass categories provides us with a
canonical isomorphism

$(X_{0},...,X_{n},Y_{0},...,Y_{m},X_{0})\overset{\cong}{\longrightarrow}(Y_{0},...,Y_{m},X_{0},...,X_{n},Y_{0}),$

which implies that there is an equivalence

(6.12) $\mathscr{O}^{S\circlearrowleft
T}_{thh}\overset{\cong}{\Longrightarrow}\mathscr{O}^{T\circlearrowleft
S}_{thh}.$

in $\mathcal{P}(\mathscr{B}\mathrm{ypass}_{S\circlearrowleft T})$.

We are now ready to define the desired twisting isomorphism. Let

$\mathscr{C}:\mathscr{B}\mathrm{ypass}_{S}\to\mathscr{V}\quad\text{and}\quad\mathscr{D}:\mathscr{B}\mathrm{ypass}_{T}\to\mathscr{V}$

be two $\mathscr{V}$-enriched $\infty$-categories, and let
$\mathscr{M}:\mathscr{B}\mathrm{ypass}_{S,T}\to\mathscr{V}$ be a
$(\mathscr{C},\mathscr{D})$-bimodule and
$\mathscr{N}:\mathscr{B}\mathrm{ypass}_{T,S}\to\mathscr{V}$ a
$(\mathscr{D},\mathscr{C})$-bimodule. The universal property of the pushout
implies that these data give rise to functor

$\mathscr{M}\otimes\mathscr{N}:\mathscr{B}\mathrm{ypass}_{S\circlearrowleft
T}\to\mathscr{V},$

which we can left Kan extend to a functor

$(\mathscr{M}\otimes\mathscr{N})^{*}:\mathcal{P}(\mathscr{B}\mathrm{ypass}_{S\circlearrowleft
T}^{op})\to\mathscr{V}.$

Note that $(\mathscr{M}\otimes\mathscr{N})^{*}(\mathscr{O}^{S\circlearrowleft
T}_{thh})$ is the colimit of the bisimplicial object

$(\mathscr{M}\otimes\mathscr{N})^{*}\circ(\mathscr{O}^{S\circlearrowleft
T}_{thh})_{\bullet\bullet}:{\mathbbe{\Delta}}^{op}\times{\mathbbe{\Delta}}^{op}\to\mathscr{V}$

that takes $([n],[m])$ to

$\coprod_{X_{0},...,X_{n}\in S,Y_{0},...,Y_{m}\in
T}\mathscr{C}(X_{0},X_{1})\otimes...\otimes\mathscr{C}(X_{n-1},X_{n})\otimes\mathscr{M}(X_{n},Y_{0})\otimes\mathscr{D}(Y_{0},Y_{1})\otimes...\otimes\mathscr{D}(Y_{m-1},Y_{m})\otimes\mathscr{N}(Y_{m},X_{0}).$

Fix an object $[n]$ in ${\mathbbe{\Delta}}$. By [Hau16, End of Section 5], the
colimit of the simplicial diagram

$\displaystyle(\mathscr{M}\otimes\mathscr{N})^{*}\circ(\mathscr{O}^{S\circlearrowleft
T}_{thh})_{n\bullet}=$ $\displaystyle\coprod_{X_{0},...,X_{n}\in
S,Y_{0},...,Y_{\bullet}\in
T}\mathscr{C}(X_{0},X_{1})\otimes...\otimes\mathscr{C}(X_{n-1},X_{n})\otimes\mathscr{M}(X_{n},Y_{0})\otimes\mathscr{D}(Y_{0},Y_{1})\otimes...\otimes\mathscr{N}(Y_{\bullet},X_{0})$

is

$\coprod_{X_{0},...,X_{n}\in
S}\mathscr{C}(X_{0},X_{1})\otimes...\otimes\mathscr{C}(X_{n-1},X_{n})\otimes(\mathscr{M}\otimes\mathscr{N})(X_{n},X_{0}),$

where $\mathscr{M}\otimes\mathscr{N}$ is the
$(\mathscr{C},\mathscr{C})$-bimodule obtained via tensor product of
$\mathscr{M}$ and $\mathscr{N}$ as defined in [Hau16, Remark 5.4]. By 6.9 the
colimit of this simplicial object is precisely
$\mathrm{HH}_{\mathscr{V}}(\mathscr{C},\mathscr{M}\otimes\mathscr{N})$.

Repeating the same argument with the roles of $\mathscr{M}$ and $\mathscr{N}$
reversed, we deduce that
$\mathrm{HH}_{\mathscr{V}}(\mathscr{D},\mathscr{N}\otimes\mathscr{M})$ can be
obtained as the colimit of the bisimplicial diagram of the form

$\coprod_{X_{0},...,X_{n}\in S,Y_{0},...,Y_{m}\in
T}\mathscr{D}(Y_{0},Y_{1})\otimes...\otimes\mathscr{D}(Y_{n-1},Y_{m})\otimes\mathscr{N}(Y_{m},X_{0})\otimes\mathscr{C}(X_{0},X_{1})\otimes...\otimes\mathscr{M}(X_{n},Y_{0}),$

i.e., $(\mathscr{M}\otimes\mathscr{N})^{*}(\mathscr{O}^{T\circlearrowleft
S}_{thh})$. Equivalence (6.12) then implies that

$\mathrm{HH}_{\mathscr{V}}(\mathscr{D},\mathscr{N}\otimes\mathscr{M})\simeq\mathrm{HH}_{\mathscr{V}}(\mathscr{C},\mathscr{M}\otimes\mathscr{N}),$

as desired.

Finally, as mentioned in the proof of [CP19, Theorem 2.17], it is
straightforward to check that this isomorphism satisfies the two compatibility
conditions of 2.3, and hence we can conclude. ∎

We can apply LABEL:{thm:thh_shadow} to the case of spectrally enriched
$\infty$-categories and thus, in particular, to stable $\infty$-categories,
since every stable $\infty$-category is in fact enriched over $\mathrm{Sp}$
[GH15, Example 7.4.14].

######  Corollary 6.13.

There is a functor of $\infty$-categories

$\mathrm{THH}:\mathrm{biHH}_{\infty}(\mathscr{M}\mathrm{od}_{\mathrm{Sp}})\to\mathrm{Ho}\mathrm{Sp}$

such that that the corresponding shadow on
$\mathscr{M}\mathrm{od}_{\mathrm{Sp}}$ (cf. 6.1) sends any spectral bimodule
$\mathscr{M}$ to $\mathrm{THH}(\mathscr{C},\mathscr{M})$.

We end this section with a discussion of possible obstructions to further
generalizations.

######  Remark 6.14.

To establish 6.2, we need to prove that if $\mathscr{V}$ is closed monoidal
(i.e., if $\mathscr{V}=\mathrm{Sp}$), then the functor
$\mathrm{HH}_{\mathscr{V}}:\mathrm{biHH}_{\infty}(\mathscr{M}\mathrm{od}_{\mathscr{V}})\to\mathscr{V}$
is a $\mathscr{V}$-enriched functor of $\mathscr{V}$-enriched
$\infty$-categories. While it is clear that $\mathscr{V}$ is
$\mathscr{V}$-enriched [GH15, Corollary 7.4.10], proving that
$\mathrm{biHH}_{\infty}(\mathscr{M}\mathrm{od}_{\mathscr{V}})$ is
$\mathscr{V}$-enriched requires showing that
$\mathscr{M}\mathrm{od}_{\mathscr{V}}$ is a $\mathscr{V}$-enriched
$(\infty,2)$-category i.e., enriched in $\mathscr{V}$-enriched
$\infty$-categories rather than just $\infty$-categories.

If $\mathscr{V}$ is presentable, then this is expected to be true and should
be established in future work333Based on private conversation with Rune
Haugseng..The existing literature, such as [Hau16], does not include such
results, however, and makes the further study of $\mathrm{HH}_{\mathscr{V}}$
more challenging.

##  Appendix A Truncated Simplicial Pseudo-Diagrams

In this appendix we perform a detailed analysis of pseudofunctors out of the
truncated simplex category $({\mathbbe{\Delta}}_{\leq 2})^{op}$, which is
essential for our proof of 3.19.

A $2$-category is by definition a category enriched over categories. Hence,
there is a default notion of morphisms between $2$-categories, called _$2$
-functors_, which are strict functors that respect the categorical enrichment.
On the other hand, we can also consider pseudofunctors, for which the
composition and unit hold only up to chosen natural isomorphisms. For a more
detailed discussion of pseudofunctors see [Gro71, Bén67].

We want to characterize pseudofunctors out of $({\mathbbe{\Delta}}_{\leq
2})^{op}$ and out of $(({\mathbbe{\Delta}}_{\leq 2})^{op})^{\rhd}$ into a
$2$-category $\mathscr{B}$. Constructing a pseudofunctor by hand can be quite
challenging, however, since we need to specify a natural isomorphism for every
pair of composable morphisms in the domain. Fortunately, there is a way to
simplify the task. In [Lac02, Lac04] Lack constructs a model structure on the
category of $2$-categories that has the property that any pseudofunctor
between $2$-categories with cofibrant domain is equivalent to a $2$-functor
[Lac02, Remark 4.10].

We start therefore by recalling the characterization of the cofibrant objects
in Lack’s model structure, which requires the notion of free categories.
Recall that a graph is a specified by a pair of functions with the same domain
and the same codomain. Every small category has an underlying graph, given by
the source and target functions on the set of morphisms, i.e., there is a
forgetful functor $U:\mathscr{C}\mathrm{at}\to\mathscr{G}\mathrm{rph}$. The
forgetful functor admits a left adjoint
$\mathscr{F}_{1}:\mathscr{G}\mathrm{rph}\to\mathscr{C}\mathrm{at}$, which
takes a graph to its free category. For more details regarding free categories
and graphs see [ML98, Section II.7]. The following result regarding cofibrant
objects summarizes [Lac02, Theorem 4.8].

######  Theorem A.1.

A $2$-category is cofibrant in Lack’s model structure if and only its
underlying $1$-category is a free category on a graph.

Unfortunately, the categories $({\mathbbe{\Delta}}_{\leq 2})^{op}$ and
$(({\mathbbe{\Delta}}_{\leq 2})^{op})^{\rhd}$ are not cofibrant in Lack’s
model structure on $2$-categories. Indeed the generating morphisms satisfy
non-trivial relations (cf. Lemma A.2). Hence our first task is to find
appropriate cofibrant replacements, for which we apply the theory of computads
introduced in [Str76]. A computad consists of a graph $\mathscr{G}$ together
with a set of $2$-arrows between parallel morphisms in the free category on
$\mathscr{G}$. We refer to the original source [Str76, Section 2] for a more
complete description, as we need only certain computads.

Before we proceed, it is helpful to describe fully the category
$({\mathbbe{\Delta}}_{\leq 2})^{op}$, in particular its morphisms. As it is a
full subcategory of ${\mathbbe{\Delta}}^{op}$, we can state the following
result in terms of the characterization of ${\mathbbe{\Delta}}$ via generators
and relations found in [GJ09, Section I.1] and [ML98, Section VII.5]

######  Lemma A.2.

The category $({\mathbbe{\Delta}}_{\leq 2})^{op}$ can be described as follows.

  * •

It has three objects: $0,1,2$.

  * •

It has eight generating morphisms.

    * –

$s_{0}:0\to 1$

    * –

$d_{0},d_{1}:1\to 0$

    * –

$s_{0},s_{1}:1\to 2$

    * –

$d_{0},d_{1},d_{2}:2\to 1$

  * •

The generating morphisms satisfy the following relations.

    * –

$d_{0}d_{0}=d_{0}d_{1},d_{0}d_{2}=d_{1}d_{0},d_{1}d_{1}=d_{1}d_{2}$

    * –

$d_{1}s_{0}=\mathrm{id}=d_{0}s_{0}$

    * –

$d_{0}s_{0}=d_{1}s_{1}=\mathrm{id}=d_{1}s_{0}=d_{2}s_{1}$

    * –

$s_{0}s_{0}=s_{1}s_{0}$

Motivated by this description, we now construct the desired computad, starting
with its underlying graph.

######  Definition A.3.

Let $\mathcal{D}_{2}$ be the graph with three objects $\\{0,1,2\\}$ and eight
arrows $\\{s_{0}:0\to 1;d_{0},d_{1}:1\to 0;s_{0},s_{1}:1\to
2;d_{0},d_{1},d_{2}:2\to 1\\}$.

Next we add the relevant $2$-arrows based on the relations in Lemma A.2.

######  Definition A.4.

Let $\mathcal{G}_{2}$ be the computad with underlying graph $\mathcal{D}_{2}$,
equipped with following $2$-arrows in $\mathscr{F}_{1}(\mathcal{D}_{2})$.

  * •

$d_{0}d_{0}\Rightarrow d_{0}d_{1},d_{0}d_{2}\Rightarrow
d_{1}d_{0},d_{1}d_{1}\Rightarrow d_{1}d_{2}$

  * •

$d_{1}s_{0}\Rightarrow\mathrm{id}\Leftarrow d_{0}s_{0}$

  * •

$d_{0}s_{0}\Rightarrow d_{1}s_{1}\Rightarrow\mathrm{id}\Leftarrow
d_{1}s_{0}\Leftarrow d_{2}s_{1}$

  * •

$s_{0}s_{0}\Rightarrow s_{1}s_{0}$

Applying [Str76, Theorem 2] to the computad $\mathcal{G}_{2}$ gives us
following result.

######  Lemma A.5.

There is a $2$-category $\mathscr{F}_{2}(\mathcal{G}_{2})$ satisfying the
universal property that every $2$-functor
$\mathscr{F}_{2}(\mathcal{G}_{2})\to\mathscr{B}$ is specified by the following
data.

  * •

A functor from the free category $\mathscr{F}_{1}(\mathcal{D}_{2})$ on the
graph of $\mathcal{D}_{2}$ to the underlying category of $\mathscr{B}$.

  * •

A choice of $2$-morphism in $\mathscr{B}$ with the appropriate source and
target for every $2$-arrow of the computad $\mathcal{G}_{2}$.

We refer to [Str76, Page 155] for a detailed explanation of what “appropriate”
means in this context.

Applying the universal property of the free category functor, let

$\pi_{1}:\mathscr{F}_{1}(\mathcal{D}_{2})\to({\mathbbe{\Delta}}_{\leq
2})^{op}$

denote the functor that is the identity on objects and generating
$1$-morphisms. Note that by the universal property of
$\mathscr{F}_{2}(\mathcal{G}_{2})$, there is a $2$-functor

$\pi_{2}:\mathscr{F}_{2}(\mathcal{D}_{2})\to({\mathbbe{\Delta}}_{\leq
2})^{op}$

with underlying functor $\pi_{1}$ and sending every 2-arrow of the computad
$\mathcal{D}_{2}$ to an identity.

Before proceeding further, we analyze the structure of the $2$-category
$\mathscr{F}_{2}(\mathcal{G}_{2})$ and the relations satisfied by its
$2$-morphisms.

######  Lemma A.6.

The $2$-functor
$\pi_{2}:\mathscr{F}_{2}(\mathcal{G}_{2})\to({\mathbbe{\Delta}}_{\leq
2})^{op}$ is a cofibrant replacement in Lack’s model structure on
bicategories.

###### Proof.

By A.1, the $2$-category $\mathscr{F}_{2}(\mathcal{G}_{2})$ is evidently
cofibrant as its underlying category is just
$\mathscr{F}_{1}(\mathcal{D}_{2})$, which is free by definition.

We therefore need only to show that the projection map

$\pi_{2}:\mathscr{F}_{2}(\mathcal{D}_{2})\to({\mathbbe{\Delta}}_{\leq
2})^{op}$

is a weak equivalence in Lack’s model structure. However, as follows from the
proof of [Lac02, Proposition 4.2] and the explanation immediately thereafter,
this functor is actually a trivial fibration. ∎

We apply this result to establish a useful characterization of the
$2$-category $\mathscr{F}_{2}(\mathcal{G}_{2})$. To do so, it is helpful to
review contractible groupoids.

######  Lemma A.7.

The forgetful functor $\mathscr{G}\mathrm{rpd}\to\mathscr{S}\text{et}$, which
takes a groupoid to its set of objects, admits a right adjoint
$I:\mathscr{S}\text{et}\to\mathscr{G}\mathrm{rpd}$, which takes a set $S$ to
the groupoid $I(S)$ with the same set of objects and a unique isomorphism
between any two objects. Moreover, a category with object set $S$ is
contractible if and only if it is isomorphic to $I(S)$.

###### Proof.

Only the last sentence requires an argument. If $\mathscr{C}$ is a
contractible category with object set $S$, then for any object $s\in S$, the
functor $s:[0]\to\mathscr{C}$ is an equivalence and thus fully faithful, which
implies that $\mathscr{C}(s,s)=\\{\mathrm{id}_{s}\\}$.

Let $s,s^{\prime}\in S$. Since the map $s:[0]\to\mathscr{C}$ is an
equivalence, it is essentially surjective, whence there exists an isomorphism
$s\to s^{\prime}$ in $\mathscr{C}$, which must be unique. Indeed, the
existence of two distinct isomorphisms $f,g:s\to s^{\prime}$ would imply the
existence of a non-trivial automorphism $g^{-1}f:s\to s$, in contradiction
with the conclusion of the previous paragraph. We conclude that $\mathscr{C}$
is isomorphic to $I(S)$, as desired. ∎

We call $I(S)$ the _contractible groupoid based on the set $S$_, as it is
equivalent to the terminal category. We can formulate an alternative
characterization of the morphism categories in
$\mathscr{F}_{2}(\mathcal{G}_{2})$ in terms of contractible groupoids.

######  Lemma A.8.

The $2$-category $\mathscr{F}_{2}(\mathcal{G}_{2})$ has three objects $0,1,2$.
Moreover, for any $i,j\in\\{0,1,2\\}$, the category
$\mathscr{F}_{2}(\mathcal{G}_{2})(i,j)$ has as objects the set
$\mathscr{F}_{1}(\mathcal{D}_{2})(i,j)$, while for two objects $f,g$, there is
a unique morphism from $f$ to $g$ if and only and only if
$\pi_{1}(f)=\pi_{1}(g)$ in $({\mathbbe{\Delta}}_{\leq 2})^{op}$. In
particular, $\mathscr{F}_{2}(\mathcal{G}_{2})(i,j)$ is a groupoid, and there
is an isomorphism of groupoids

$\mathscr{F}_{2}(\mathcal{G}_{2})(i,j)\cong\coprod_{f\in({\mathbbe{\Delta}}_{\leq
2})^{op}(i,j)}I\big{(}(\pi_{1})^{-1}(f)\big{)}.$

###### Proof.

The characterization of the underlying category of the $2$-category
$\mathscr{F}_{2}(\mathcal{G}_{2})$ follows from Lemma A.5. For all
$i,j\in\\{0,1,2\\}$, Lemma A.6 implies that

$\mathscr{F}_{2}(\mathcal{G}_{2})(i,j)\to({\mathbbe{\Delta}}_{\leq
2})^{op}(i,j)$

is an equivalence of categories. Since the right hand side is discrete,
$\mathscr{F}_{2}(\mathcal{G}_{2})(i,j)$ is a disjoint union of contractible
categories. The desired isomorphism now follows from the characterization of
contractible categories given in Lemma A.7. ∎

We next use this cofibrant replacement to characterize pseudofunctors out of
$({\mathbbe{\Delta}}_{\leq 2})^{op}$. To simplify notation, we henceforth
denote the cofibrant replacement above by

$\pi_{Q}:Q({\mathbbe{\Delta}}_{\leq 2})^{op}\to({\mathbbe{\Delta}}_{\leq
2})^{op}.$

The next result is an immediate consequence of Lemma A.8 and [Lac02, Remark
4.10].

######  Corollary A.9.

Let $\mathscr{B}$ be a $2$-category. Precomposing with the cofibrant
replacement functor $\pi_{Q}:Q({\mathbbe{\Delta}}_{\leq
2})^{op}\to({\mathbbe{\Delta}}_{\leq 2})^{op}$ induces an equivalence between
the category of pseudofunctors $({\mathbbe{\Delta}}_{\leq
2})^{op}\to\mathscr{B}$ and that of $2$-functors $Q({\mathbbe{\Delta}}_{\leq
2})^{op}\to\mathscr{B}$. In other words, for every pseudofunctor
$F:({\mathbbe{\Delta}}_{\leq 2})^{op}\to\mathscr{B}$, there exists a
$2$-functor $\widehat{F}:Q({\mathbbe{\Delta}}_{\leq 2})^{op}\to\mathscr{B}$
and an equivalence $F\circ\pi_{Q}\simeq\widehat{F}$.

Combining the results above, we can now characterize pseudofunctors out of
$({\mathbbe{\Delta}}_{\leq 2})^{op}$.

######  Lemma A.10.

For any $2$-category $\mathscr{B}$, a pseudofunctor $({\mathbbe{\Delta}}_{\leq
2})^{op}\to\mathscr{B}$ is determined up to equivalence by a choice of the
following data.

  * •

Three objects $B_{0},B_{1},B_{2}$

  * •

Eight $1$-morphisms

    * –

$s_{0}:B_{0}\to B_{1}$

    * –

$d_{0},d_{1}:B_{1}\to B_{0}$

    * –

$s_{0},s_{1}:B_{1}\to B_{2}$

    * –

$d_{0},d_{1},d_{2}:B_{2}\to B_{1}$

  * •

Ten invertible $2$-morphisms

    * –

$d_{0}d_{0}\overset{\alpha_{0}}{\Rightarrow}d_{0}d_{1},d_{0}d_{2}\overset{\alpha_{1}}{\Rightarrow}d_{1}d_{0},d_{1}d_{1}\overset{\alpha_{2}}{\Rightarrow}d_{1}d_{2}$
in $\mathscr{B}(B_{2},B_{0})$

    * –

$d_{1}s_{0}\overset{\beta_{0}}{\Rightarrow}\mathrm{id}\overset{\beta_{1}}{\Leftarrow}d_{0}s_{0}$
in $\mathscr{B}(B_{0},B_{0})$

    * –

$d_{0}s_{0}\overset{\gamma_{0}}{\Rightarrow}d_{1}s_{1}\overset{\gamma_{1}}{\Rightarrow}\mathrm{id}\overset{\gamma_{2}}{\Leftarrow}d_{1}s_{0}\overset{\gamma_{3}}{\Leftarrow}d_{2}s_{1}$
in $\mathscr{B}(B_{1},B_{1})$

    * –

$s_{0}s_{0}\overset{\delta_{0}}{\Rightarrow}s_{1}s_{0}$ in
$\mathscr{B}(B_{0},B_{2})$

Moreover, any two $2$-morphisms generated by the $2$-morphisms above with
equal domain and codomain are equal.

###### Proof.

By A.9 a pseudofunctor $({\mathbbe{\Delta}}_{\leq 2})^{op}\to\mathscr{B}$ is
determined up to equivalence by a $2$-functor $Q({\mathbbe{\Delta}}_{\leq
2})^{op}\to\mathscr{B}$. The universal property of $Q({\mathbbe{\Delta}}_{\leq
2})^{op}$ as formulated in Lemma A.5 implies that a $2$-functor
$Q({\mathbbe{\Delta}}_{\leq 2})^{op}\to\mathscr{B}$ is specified by a choice
of $0,1,2$-morphisms and relations as above. ∎

Now that we have a useful description of pseudofunctors out of
$({\mathbbe{\Delta}}_{\leq 2})^{op}$, the next step is to study
pseudofunctorial lifts to $(({\mathbbe{\Delta}}_{\leq 2})^{op})^{\rhd}$.
Recall that the category $(({\mathbbe{\Delta}}_{\leq 2})^{op})^{\rhd}$ is the
join of $({\mathbbe{\Delta}}_{\leq 2})^{op}$ and a final object (3.12). To
characterize pseudofunctors out of $(({\mathbbe{\Delta}}_{\leq
2})^{op})^{\rhd}$ directly is challenging, so we prefer instead to study
$2$-functors out of its cofibrant replacement, which requires proving results
analogous to Lemma A.5 and Lemma A.10. First we describe
$(({\mathbbe{\Delta}}_{\leq 2})^{op})^{\rhd}$ in terms generators and
relations, analogously to Lemma A.2.

######  Lemma A.11.

Let $(({\mathbbe{\Delta}}_{\leq 2})^{op})^{\rhd}$ be the category constructed
by joining a terminal object to $({\mathbbe{\Delta}}_{\leq 2})^{op}$. More
precisely, $(({\mathbbe{\Delta}}_{\leq 2})^{op})^{\rhd}$ is specified by
following generators and relations.

  * •

Objects $0,1,2,f$

  * •

Morphisms

    * –

The generating morphisms in $({\mathbbe{\Delta}}_{\leq 2})^{op}$: $s_{0}:0\to
1$, $d_{0},d_{1}:1\to 0$, $s_{0},s_{1}:1\to 2$, $d_{0},d_{1},d_{2}:2\to 1$,

    * –

Three additional morphisms: $t_{0}:0\to f$, $t_{1}:1\to f$, $t_{2}:2\to f$.

  * •

Relations

    * –

The relations that already hold in $({\mathbbe{\Delta}}_{\leq 2})^{op}$:

      * *

$d_{0}d_{0}=d_{0}d_{1}$, $d_{0}d_{2}=d_{1}d_{0}$, $d_{1}d_{1}=d_{1}d_{2}:2\to
0$,

      * *

$d_{1}s_{0}=\mathrm{id}=d_{0}s_{0}:0\to 0$,

      * *

$d_{0}s_{0}=d_{1}s_{1}=\mathrm{id}=d_{1}s_{0}=d_{2}s_{1}:1\to 1$,

      * *

$s_{0}s_{0}=s_{1}s_{0}:0\to 2$,

    * –

The additional relations for the terminal object:

      * *

$t_{0}=t_{1}s_{0}=t_{2}s_{0}s_{0}=t_{2}s_{1}s_{0}:0\to f$,

      * *

$t_{1}=t_{0}d_{0}=t_{0}d_{1}=t_{2}s_{0}=t_{2}s_{1}:1\to f$,

      * *

$t_{2}=t_{1}d_{0}=t_{1}d_{1}=t_{1}d_{2}=t_{0}d_{0}d_{0}=t_{0}d_{0}d_{1}=t_{0}d_{0}d_{2}=t_{0}d_{1}d_{0}=t_{0}d_{1}d_{2}=t_{0}d_{1}d_{1}:2\to
f$.

There is an obvious fully faithful inclusion $({\mathbbe{\Delta}}_{\leq
2})^{op}\to(({\mathbbe{\Delta}}_{\leq 2})^{op})^{\rhd}$. The list of relations
for $(({\mathbbe{\Delta}}_{\leq 2})^{op})^{\rhd}$ given above contains many
redundancies, which we can reduce as follows.

######  Lemma A.12.

The category $(({\mathbbe{\Delta}}_{\leq 2})^{op})^{\rhd}$ can be specified by
the following data.

  * •

The same objects as above.

  * •

The generating morphisms in $({\mathbbe{\Delta}}_{\leq 2})^{op}$ together with
one additional morphism $t_{0}:0\to f$

  * •

The relations that hold in $({\mathbbe{\Delta}}_{\leq 2})^{op}$ together with
one additional relation $t_{0}d_{0}=t_{0}d_{1}:1\to f$.

###### Proof.

We define $t_{1}=t_{0}d_{0}$ and $t_{2}=t_{0}d_{0}d_{0}$. Since the relations
listed in this lemma are a subset of the relations in Lemma A.11, the
necessity is evident. We prove that these relations suffice by recovering the
remaining relations from them, as follows.

  1. (1)

$t_{2}s_{0}s_{0}\overset{s_{0}s_{0}=s_{1}s_{0}}{=}t_{2}s_{1}s_{0}$.

  2. (2)

$t_{2}\overset{def}{=}t_{0}d_{0}d_{0}\overset{t_{1}=t_{0}d_{0}}{=}t_{1}d_{0}$

  3. (3)

$t_{2}\overset{def}{=}t_{0}d_{0}d_{0}\overset{d_{0}d_{0}=d_{0}d_{1}}{=}t_{0}d_{0}d_{1}\overset{t_{0}d_{0}=t_{1}}{=}t_{1}d_{1}$

  4. (4)

$t_{2}\overset{def}{=}t_{0}d_{0}d_{0}\overset{t_{1}=t_{0}d_{0}}{=}t_{1}d_{0}\overset{t_{1}=t_{0}d_{1}}{=}t_{0}d_{1}d_{0}\overset{d_{1}d_{0}=d_{0}d_{2}}{=}t_{0}d_{0}d_{2}\overset{t_{0}d_{0}=t_{1}}{=}t_{1}d_{2}$

  5. (5)

$t_{2}\overset{def}{=}t_{0}d_{0}d_{0}\overset{t_{0}d_{0}=t_{0}d_{1}}{=}t_{0}d_{1}d_{0}\overset{d_{1}d_{0}=d_{0}d_{2}}{=}t_{0}d_{0}d_{2}\overset{t_{0}d_{0}=t_{0}d_{1}}{=}t_{0}d_{1}d_{2}\overset{d_{1}d_{2}=d_{1}d_{1}}{=}t_{0}d_{1}d_{1}$

  6. (6)

$t_{2}s_{0}\overset{t_{2}=t_{0}d_{0}d_{0}}{=}t_{0}d_{0}d_{0}s_{0}\overset{t_{1}=t_{0}d_{0}}{=}t_{1}d_{0}s_{0}\overset{d_{0}s_{0}=\mathrm{id}}{=}t_{1}$

  7. (7)

$t_{2}s_{1}\overset{t_{2}=t_{0}d_{0}d_{0}}{=}t_{0}d_{0}d_{0}s_{1}\overset{d_{0}d_{0}=d_{0}d_{1}}{=}t_{0}d_{0}d_{1}s_{1}\overset{t_{0}d_{0}=t_{1}}{=}t_{1}d_{1}s_{1}\overset{d_{1}s_{1}=\mathrm{id}}{=}t_{1}$

  8. (8)

$t_{2}s_{0}s_{0}\overset{t_{2}=t_{0}d_{0}d_{0}}{=}t_{0}d_{0}d_{0}s_{0}s_{0}\overset{\mathrm{id}=d_{0}s_{0}}{=}t_{0}d_{0}s_{0}\overset{\mathrm{id}=d_{0}s_{0}}{=}t_{0}$

  9. (9)

$t_{1}s_{0}\overset{t_{1}=t_{0}d_{0}}{=}t_{0}d_{0}s_{0}\overset{d_{0}s_{0}=\mathrm{id}}{=}t_{0}$

As we have recovered all the equalities in Lemma A.11, the result follows. ∎

Given this lemma, we can easily modify the proofs of Lemma A.5 and Lemma A.10
to establish the following analogous result.

######  Definition A.13.

Let $(\mathcal{D}_{2})^{\rhd}$ denote the graph with

  * •

vertices $0,1,2,f$, and

  * •

the same edges as in $\mathcal{D}_{2}$ (A.3), together with one additional
edge $0\to f$.

######  Lemma A.14.

The $2$-category $\big{(}({\mathbbe{\Delta}}_{\leq 2})^{op}\big{)}^{\rhd}$
admits a cofibrant replacement given by the $2$-category
$Q\Big{(}\big{(}({\mathbbe{\Delta}}_{\leq 2})^{op}\big{)}^{\rhd}\Big{)}$,
which is the $2$-category determined by the computad with

  * •

underlying graph is $(\mathcal{D}_{2})^{\rhd}$, and

  * •

the same $2$-arrows as $\mathcal{G}_{2}$ (A.4), along with one additional
$2$-arrow $\theta:t_{0}d_{0}\Rightarrow t_{0}d_{1}$.

The morphism categories of $Q\Big{(}\big{(}({\mathbbe{\Delta}}_{\leq
2})^{op}\big{)}^{\rhd}\Big{)}$ can be characterized as in Lemma A.8, leading
in particular to the following observation.

######  Corollary A.15.

The category $Q\Big{(}\big{(}({\mathbbe{\Delta}}_{\leq
2})^{op}\big{)}^{\rhd}\Big{)}(i,f)$ is a contractible groupoid for all
$i\in\\{0,1,2\\}$.

###### Proof.

The proof is essentially the same as that of Lemma A.8, given that there are
unique maps $i\to f$ in $Q\Big{(}\big{(}({\mathbbe{\Delta}}_{\leq
2})^{op}\big{)}^{\rhd}\Big{)}$. ∎

######  Remark A.16.

A.15 enables us to better understand $Q\Big{(}\big{(}({\mathbbe{\Delta}}_{\leq
2})^{op}\big{)}^{\rhd}\Big{)}$. There are four objects $0,1,2,f$, and
morphisms are words with letters

$s_{0},d_{0},d_{1},s_{0},s_{1},d_{0},d_{1},d_{2},t_{0}.$

In particular, the only generating morphism that has codomain $f$ is
$t_{0}:0\to f$, whence every object in the groupoid
$Q\Big{(}\big{(}({\mathbbe{\Delta}}_{\leq 2})^{op}\big{)}^{\rhd}\Big{)}(i,f)$
is necessarily of the form $t_{0}g$ for some morphism $g:i\to 0$, where
$i\in\\{0,1,2\\}$. Contractibility of
$Q\Big{(}\big{(}({\mathbbe{\Delta}}_{\leq 2})^{op}\big{)}^{\rhd}\Big{)}(i,f)$
implies that there is precisely one isomorphism between $t_{0}g_{0}$ and
$t_{0}g_{1}$ for any $g_{0},g_{1}:i\to 0$.

The equivalence below of $2$-functors and pseudofunctors follows from Lemma
A.14 and [Lac02, Remark 4.10] .

######  Corollary A.17.

Let $\mathscr{B}$ be a $2$-category. Precomposing with the cofibrant
replacement functor

$\pi_{Q}^{\rhd}:Q\Big{(}\big{(}({\mathbbe{\Delta}}_{\leq
2})^{op}\big{)}^{\rhd}\Big{)}\to\big{(}({\mathbbe{\Delta}}_{\leq
2})^{op}\big{)}^{\rhd}$

induces an equivalence between the category of pseudofunctors
$\big{(}({\mathbbe{\Delta}}_{\leq 2})^{op}\big{)}^{\rhd}\to\mathscr{B}$ and
that of $2$-functors $Q\Big{(}\big{(}({\mathbbe{\Delta}}_{\leq
2})^{op}\big{)}^{\rhd}\Big{)}\to\mathscr{B}$. In other words, for every
pseudofunctor $F:\big{(}({\mathbbe{\Delta}}_{\leq
2})^{op}\big{)}^{\rhd}\to\mathscr{B}$, there exists a $2$-functor
$\widehat{F}:Q\Big{(}\big{(}({\mathbbe{\Delta}}_{\leq
2})^{op}\big{)}^{\rhd}\Big{)}\to\mathscr{B}$ and an equivalence
$F\circ\pi_{Q}^{\rhd}\simeq\widehat{F}$.

We can use this result to characterize pseudofunctors out of
$(({\mathbbe{\Delta}}_{\leq 2})^{op})^{\rhd}$, using the inclusion functor
$Q({\mathbbe{\Delta}}_{\leq 2})^{op}\to
Q\Big{(}\big{(}({\mathbbe{\Delta}}_{\leq 2})^{op}\big{)}^{\rhd}\Big{)}$
arising from the inclusion of computads. In the statement and the proof below,
we use the notation of Lemma A.10.

######  Lemma A.18.

Let $\mathscr{B}$ be a $2$-category, and let $F:({\mathbbe{\Delta}}_{\leq
2})^{op}\to\mathscr{B}$ be a pseudofunctor. An extension of $F$ to
$\big{(}({\mathbbe{\Delta}}_{\leq 2})^{op}\big{)}^{\rhd}$ is specified by

${({\mathbbe{\Delta}}_{\leq
2})^{op}}$${\mathscr{B}}$${(({\mathbbe{\Delta}}_{\leq
2})^{op})^{\rhd}}$$\scriptstyle{F}$$\scriptstyle{\widehat{F}}$

  * •

an object $B_{f}$ in $\mathscr{B}$,

  * •

a $1$-morphism $t_{0}:B_{0}\to B_{f}$, and

  * •

an invertible $2$-morphism $t_{0}d_{0}\overset{\theta}{\Rightarrow}t_{0}d_{1}$

such that the following equalities hold, where $*$ denotes whiskering of
2-cells by 1-cells, and $\circ$ denotes vertical composition of 2-cells.

  * •

$\theta*s_{0}=t_{0}*((\beta_{0})^{-1}\circ\beta_{1}):t_{0}d_{0}s_{0}\to
t_{0}d_{1}s_{0}$

  * •

$(t_{0}(\alpha_{1})^{-1})\circ(\theta*d_{0})=(\theta^{-1}*d_{2})\circ(t_{0}*\alpha_{2})\circ(\theta*d_{1})\circ(t_{0}*\alpha_{0}):t_{0}d_{0}d_{0}\to
t_{0}d_{0}d_{2}$.

###### Proof.

By A.9 and A.17, we can instead start with a $2$-functor
$F:Q(({\mathbbe{\Delta}}_{\leq 2})^{op})\to\mathscr{B}$ and define a lift
$\widehat{F}:Q(({\mathbbe{\Delta}}_{\leq 2})^{op})^{\rhd}\to\mathscr{B}$. The
nature of the data required to specify the image of $0,1$,and $2$-morphisms
under $\widehat{F}$ follows from the characterization of
$Q(({\mathbbe{\Delta}}_{\leq 2})^{op})^{\rhd}$ in Lemma A.14.

In order to complete the proof we make explicit the relations that guarantee
that the groupoids $Q\Big{(}\big{(}({\mathbbe{\Delta}}_{\leq
2})^{op}\big{)}^{\rhd}\Big{)}(i,f)$ are in fact contractible, i.e., that for
any two morphisms $g_{1},g_{2}:i\to f$, there is a unique natural isomorphism
$g_{1}\cong g_{2}$. Any other natural isomorphism generated by
$\theta:t_{0}d_{0}\cong t_{0}d_{1}$ must therefore coincide with the original
one.

We start by analyzing $Q\Big{(}\big{(}({\mathbbe{\Delta}}_{\leq
2})^{op}\big{)}^{\rhd}\Big{)}(1,f)$. As explained in A.16, every object in
this category is of the form $t_{0}g^{\prime}$ for some $g^{\prime}:1\to 0$.
By construction, every object $g^{\prime}$ in
$Q\Big{(}\big{(}({\mathbbe{\Delta}}_{\leq 2})^{op}\big{)}^{\rhd}\Big{)}(1,0)$
is uniquely  isomorphic to either $d_{0}:1\to 0$ or $d_{1}:1\to 0$. For any
two 1-cells $g_{1},g_{2}:1\to t$, with $g_{1}=t_{0}g_{1}^{\prime}$,
$g_{2}=t_{0}g_{2}^{\prime}$, there are thus four possible scenarios.

  1. (1)

There are unique isomorphisms $g_{1}^{\prime}\overset{\kappa_{1}}{\cong}d_{0}$
and $g_{2}^{\prime}\overset{\kappa_{2}}{\cong}d_{0}$, whence

$g_{1}=t_{0}g_{1}^{\prime}\overset{t_{0}\kappa_{1}}{\cong}t_{0}d_{0}\overset{t_{0}\kappa_{2}}{\cong}t_{0}g_{2}^{\prime}=g_{2}$

is the unique isomorphism between $g_{1}$ and $g_{2}$.

  2. (2)

There are unique isomorphisms $g_{1}^{\prime}\overset{\kappa_{1}}{\cong}d_{1}$
and $g_{2}^{\prime}\overset{\kappa_{2}}{\cong}d_{1}$. This case is essentially
identical to the previous one.

  3. (3)

There are unique isomorphisms $g_{1}^{\prime}\overset{\kappa_{1}}{\cong}d_{0}$
and $g_{2}^{\prime}\overset{\kappa_{2}}{\cong}d_{1}$. In this case the unique
isomorphism between $g_{1}$ and $g_{2}$ is obtained as follows.

$g_{1}=t_{0}g_{1}^{\prime}\overset{t_{0}\kappa_{1}}{\cong}t_{0}d_{0}\overset{\theta}{\cong}t_{0}d_{1}\overset{t_{0}\kappa_{2}}{\cong}t_{0}g_{2}^{\prime}=g_{2}$

  4. (4)

There are unique isomorphisms $g_{1}^{\prime}\overset{\kappa_{1}}{\cong}d_{1}$
and $g_{2}^{\prime}\overset{\kappa_{2}}{\cong}d_{0}$. This case is the same as
the previous one, up to permutation of the roles of $g_{1}$ and $g_{2}$.

Consider now the case of $Q\Big{(}\big{(}({\mathbbe{\Delta}}_{\leq
2})^{op}\big{)}^{\rhd}\Big{)}(0,f)$. There are three types of objects in this
category.

  1. (1)

$t_{0}g$, where $g:0\to 0$ is uniquely isomorphic to the identity in
$Q\Big{(}\big{(}({\mathbbe{\Delta}}_{\leq 2})^{op}\big{)}^{\rhd}\Big{)}(0,0)$.

  2. (2)

$t_{0}d_{0}g$, where $g:0\to 1$ is uniquely isomorphic to $s_{0}$ in
$Q\Big{(}\big{(}({\mathbbe{\Delta}}_{\leq 2})^{op}\big{)}^{\rhd}\Big{)}(0,1)$.

  3. (3)

$t_{0}d_{1}g$, where $g:0\to 1$ is uniquely isomorphic to $s_{0}$ in
$Q\Big{(}\big{(}({\mathbbe{\Delta}}_{\leq 2})^{op}\big{)}^{\rhd}\Big{)}(0,1)$.

As in the previous case, there is a unique isomorphism $t_{0}d_{0}g\cong
t_{0}d_{1}g$. We need therefore only to compare a 1-cell of the form
$t_{0}g_{1}$, where $g_{1}\cong\mathrm{id}_{0}$, and a 1-cell of the form
$t_{0}d_{0}g_{2}$, where $g_{2}\cong s_{0}$, and to find the necessary and
sufficient conditions to guarantee that there is a unique isomorphism between
$t_{0}g_{1}$ and $t_{0}d_{0}g_{2}$.

By assumption, $t_{0}g_{1}\cong t_{0}$ and $t_{0}d_{0}g_{2}\cong
t_{0}d_{0}s_{0}$, via unique isomorphisms. There are two potential
isomorphisms between $t_{0}$ and $t_{0}d_{0}s_{0}$:

  1. (1)

$t_{0}d_{0}s_{0}\overset{t_{0}*\beta_{1}}{\Rightarrow}t_{0}$, and

  2. (2)

$t_{0}d_{0}s_{0}\overset{\theta*s_{0}}{\Rightarrow}t_{0}d_{1}s_{0}\overset{t_{0}*\beta_{0}}{\Rightarrow}t_{0}$.

Hence, there is a unique isomorphism $t_{0}g_{1}\cong t_{0}d_{0}g_{2}$ if and
only if $\theta*s_{0}=t_{0}*((\beta_{0})^{-1}\circ\beta_{1})$.

The final case is that of $Q\Big{(}\big{(}({\mathbbe{\Delta}}_{\leq
2})^{op}\big{)}^{\rhd}\Big{)}(2,f)$. If $g:2\to f$ is any 1-cell, then
$g=t_{0}g^{\prime}$, for some $g^{\prime}:2\to 0$, which is necessarily of one
of the following three types.

  1. (1)

$g^{\prime}\cong d_{0}d_{0}$ via unique isomorphism

  2. (2)

$g^{\prime}\cong d_{0}d_{2}$ via unique isomorphism

  3. (3)

$g^{\prime}\cong d_{1}d_{2}$ via unique isomorphism

It suffices therefore to find the necessary and sufficient conditions such
that for a choice of $g_{1}^{\prime},g_{2}^{\prime}$ among the three types
above, there is a unique isomorphism $t_{0}g_{1}^{\prime}\cong
t_{0}g_{2}^{\prime}$. There are three types of pairs of 1-cells to consider.

  1. (1)

In the case where $g_{1}^{\prime}\cong g_{2}^{\prime}\cong d_{0}d_{0}$,
observe that the diagram of unique isomorphisms between 1-cells

${t_{0}d_{0}d_{0}}$${t_{0}d_{1}d_{0}}$${t_{0}d_{0}d_{2}}$${t_{0}d_{0}d_{1}}$${t_{0}d_{1}d_{1}}$${t_{0}d_{1}d_{2},}$$\scriptstyle{\theta*d_{0}}$$\scriptstyle{\cong}$$\scriptstyle{t_{0}*\alpha_{0}}$$\scriptstyle{\cong}$$\scriptstyle{t_{0}*(\alpha_{1})^{-1}}$$\scriptstyle{\cong}$$\scriptstyle{\theta*d_{1}}$$\scriptstyle{\cong}$$\scriptstyle{t_{0}*\alpha_{2}}$$\scriptstyle{\cong}$$\scriptstyle{\theta^{-1}*d_{2}}$$\scriptstyle{\cong}$

.

commutes by uniqueness of isomorphisms between 1-cells. It follows that there
is a unique isomorphism $t_{0}g_{1}^{\prime}\cong t_{0}g_{2}^{\prime}$ if and
only if

$(t_{0}(\alpha_{1})^{-1})\circ(\theta*d_{0})=(\theta^{-1}*d_{2})\circ(t_{0}*\alpha_{2})\circ(\theta*d_{1})\circ(t_{0}*\alpha_{0})$

  2. (2)

When $g_{1}^{\prime}\cong d_{0}d_{0},g_{2}^{\prime}\cong d_{0}d_{2}$, the same
commuting diagram as in the previous case implies that the same necessary and
sufficient conditions apply in this case.

  3. (3)

When $g_{1}^{\prime}=d_{0}d_{2}$, $g_{2}^{\prime}=d_{1}d_{2}$, the same
commuting diagram again leads to the same necessary and sufficient conditions.

It follows that $Q\Big{(}\big{(}({\mathbbe{\Delta}}_{\leq
2})^{op}\big{)}^{\rhd}\Big{)}(i,f)$ is contractible for $i=0,1,2$ if and only
if

$\displaystyle\theta*s_{0}$
$\displaystyle=t_{0}*((\beta_{0})^{-1}\circ\beta_{1})$
$\displaystyle(t_{0}(\alpha_{1})^{-1})\circ(\theta*d_{0})$
$\displaystyle=(\theta^{-1}*d_{2})\circ(t_{0}*\alpha_{2})\circ(\theta*d_{1})\circ(t_{0}*\alpha_{0}),$

which finishes the proof. ∎

######  Remark A.19.

The results of this section (and in particular Lemma A.18) focused on the case
where $\mathscr{B}$ is a $2$-category, as we apply these results only to the
$2$-category of categories (in 3.19). However, following [Lac04, Lemma 9], an
analogous result should certainly hold for bicategories and pseudofunctors of
bicategories (called homomorphisms of bicategories in [Lac04]).

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