Section 4. (0MLS) · ScalingStacks

4. Endomorphisms and Automorphisms of $\gaunt_{n}$ [0MLS]

We now demonstrate that the full subcategory of $\Fun(\gaunt_{n},\gaunt_{n})$ spanned by the autoequivalences is discrete, and in particular it is the set $(\mathbb{Z}/2)^{n}$ . We will also give conditions ensuring an endomorphism is an equivalence.

[0MHP]

Definition 4.1. The globular category $\mathbb{G}_{n}$ consists of the full subcategory of $\gaunt_{n}$ consisting of the $i$ -cells $C_{i}$ for $i\leq n$ .

An $n$ -globular set is a presheaf of sets on $\mathbb{G}_{n}$ . The $k$ -cells $X_{k}$ of a globular set is the set obtained by evaluating the presheaf $X$ on $C_{k}$ .

[0MHQ]

Example 4.2. Any strict $n$ -category $X$ gives rise to an $n$ -globular set, which we also denote by $X$ , such that $X_{k}\mathrel{\mathop{:}}=\cat_{n}(C_{k},X)$ . In other words, $X_{k}$ is the set of $k$ -morphisms of $X$ .

[0MHR]

Remark 4.3. The globular sets considered here are sometimes called reflexive globular sets, in order to emphasize the fact that our globular category $\mathbb{G}_{n}$ includes degeneracies $C_{k}\to C_{k-1}$ . The non-reflexive globular category $\mathbb{G}_{n}^{\textrm{nr}}$ consists of the subcategory of $\mathbb{G}_{n}$ with the same objects but only with the morphisms which are injective on $n$ -morphisms. As a category it is generated by object $C_{k}$ $0\leq k\leq n$ with morphisms

s_{k},t_{k}:C_{k-1}\to C_{k}

satisfying $s_{k}t_{k-1}=t_{k}t_{k-1}$ and $s_{k}s_{k-1}=t_{k}s_{k-1}$ . We will have only cursory use for non-reflexive globular sets in this paper.

[0MHS]

Remark 4.4. In this language, an alternative, noninductive definition of strict $n$ -category is possible: a strict $n$ -category $X$ is an $n$ -globular set together with a family of operations $n\geq k\geq j$ :

\ast_{j}:X_{k}\times_{X_{j-1}}X_{k}\to X_{k},

which are associative, unital, and suitably compatible.

[0MHT]

Lemma 4.5. There is a unique natural transformation from the identity functor on $\gaunt_{n}$ to itself.

[0MHU]

Proof. Such a natural transformation consists of component maps (i.e., functors) $\eta_{X}\colon X\to X$ for each gaunt $n$ -category $X$ . We will show that $\eta_{X}=\id_{X}$ for all $X$ . The functor $\eta_{X}$ induces, for each $0\leq k\leq n$ , a map on sets of $k$ -cells,

(\eta_{X})_{k}\colon X_{k}\to X_{k}.

Since a functor is completely determined by the map on $k$ -cells for each $k$ , it is enough to show that each $(\eta_{X})_{k}$ is the identity. By naturality of $\eta$ it is enough to show that the single functor $\eta_{C_{n}}=\id_{C_{n}}$ .

One my now show that $\eta_{C_{k}}=\id_{C_{k}}$ by inducting on $k$ . When $k=0$ the claim is obvious. Now the inductive hypothesis asserts $\eta_{C_{k}}$ is a functor which restricts to the identity functor on $\partial C_{k}$ . There is only one functor with this property, namely $\eta_{C_{k}}=\id_{C_{k}}$ . ∎

[0MHV]

Construction 4.6. For any category $C$ enriched in a symmetric monoidal category $V$ , one may of course form the opposite enriched category $X^{\mathrm{op}}$ . This is an involution on the category of $V$ -enriched categories.

Inducting this yields a free action $\rho$ of $(\mathbb{Z}/2)^{n}$ on $\cat_{n}$ . We will show in a moment that in fact this action produces an equivalence between $(\mathbb{Z}/2)^{n}$ and the monoidal full subacategory of $\Fun(\gaunt_{n},\gaunt_{n})$ spanned by the autoequivalences.

For now, let us restrict this action: we note that $\rho$ restricts to an action on the globular category $\mathbb{G}_{n}$ .

[0MHW]

Proposition 4.7. Every autoequivalence of $\mathbb{G}_{n}$ is isomorphic to $\rho(g)$ for some element $g\in(\mathbb{Z}/2)^{n}$ .

[0MHX]

Proof. First we observe that the $k$ -cell $C_{k}\in\mathbb{G}_{n}$ is the unique object such that, up to isomorphism, there exists precisely $k$ other objects of $\mathbb{G}_{n}$ which occur as proper retracts (namely all the cells $C_{i}$ with $0\leq i<k$ ). Consequently, every autoequivalence $F$ of $\mathbb{G}_{n}$ must fix the objects.

Next we observe that for each $i<j$ , there exists precisely one epimorphism $C_{j}\twoheadrightarrow C_{i}$ . Since epimorphisms are preserved by any equivalence of categories, this unique epimorphism is also preserved by $F$ . Similarly, for each $0\leq i<n$ , there exist precisely two monomorphisms $C_{i}\hookrightarrow C_{n}$ ; these are either preserved by $F$ or else they are permuted. Thus every autoequivalence determines an element $\gamma(F)\in(\mathbb{Z}/2)^{n}$ such that $\gamma(F)_{i}=0$ just in case the pair of monomorphisms $C_{i}\hookrightarrow C_{n}$ is preserved by $F$ , and $\gamma(F)_{i}=1$ just in case the pair of monomorphisms $C_{i}\hookrightarrow C_{n}$ is permuted by $F$ . Note that of course $\gamma(\rho(g))=g$ .

To conclude the proof, we observe that the symbol $\gamma(F)$ determines $F$ . Indeed, every morphisms $C_{i}\to C_{j}$ in $\mathbb{G}_{n}$ admits a factorization

C_{i}\twoheadrightarrow C_{k}\hookrightarrow C_{n}\twoheadrightarrow C_{j}

for some $C_{k}\in\mathbb{G}_{n}$ . ∎

[0MHY]

Lemma 4.8. Let $F$ be an autoequivalence of the category $\gaunt_{n}$ . Then $F$ restricts to an equivalence between $\mathbb{G}_{n}$ and its essential image in $\gaunt_{n}$ ; that is, $F(C_{k})\cong C_{k}$ for all $0\leq k\leq n$ .

[0MHZ]

Proof. The proper retracts of $C_{n}$ are precisely the cells $C_{k}$ for $0\leq k<n$ , and, as before, each $C_{k}$ is distinguished as the unique such retract such that, up to isomorphism, there exists precisely $k$ other objects which occur as further proper retracts (the cells $C_{s}$ for $s<k$ ). Thus it is enough to show that $F(C_{n})\cong C_{n}$ for the $n$ -cell alone, as this implies the analogous statement $F(C_{k})\cong C_{k}$ for all $0\leq k\leq n$ .

Recall that a generator of a category $\mathcal{C}$ is an object $X$ such that the corepresentable functor $\mathcal{C}(X,-):\mathcal{C}\to\set$ is faithful [30, pg. 127]. The collection of generators is preserved under any autoequivalence. The $n$ -cell $C_{n}$ is a generator for $\cat_{n}$ and hence also $\gaunt_{n}$ , however the $k$ -cells $C_{k}$ for $k<n$ are not generators. Thus no proper retract of $C_{n}$ is a generator. We claim that in fact $C_{n}$ is the unique generator such that every proper retract is not a generator. If this characterization holds, then any autoequivalence necessarily preserves the $n$ -cell up to automorphism and the lemma follows.

In fact we will prove a stronger statement: we claim that the $n$ -cell $C_{n}$ is a retract of every generator of $\gaunt_{n}$ . Now consider the gaunt $n$ -category $\partial C_{n+1}$ . This may be written as

\partial C_{n+1}=C_{n}\cup^{\partial C_{n}}C_{n}.

There are exactly two non-identity $n$ -morphisms in $\partial C_{n+1}$ ; call them $a,b:C_{n}\to\partial C_{n+1}$ . Observe that these two functors differ only on the unique nontrivial $n$ -morphism of $C_{n}$ . The unique non-trivial $n$ -morphism of $C_{n}$ , viewed as a map $C_{n}\to C_{n}$ , corresponds to the element $\id\in\gaunt_{n}(C_{n},C_{n})$ .

Suppose now that $X$ is a generator of $\gaunt_{n}$ . Then there must exist a functor $X\to C_{n}$ such that the induced map $X_{n}\to\gaunt_{n}(C_{n},C_{n})$ contains the element $\id\in\gaunt_{n}(C_{n},C_{n})$ in its image, for otherwise $\mathcal{C}(X,-)$ would not be able to distinguish $a$ and $b$ , contradicting the fact that $X$ is a generator. Thus there exists an $n$ -morphism $f$ of $X$ which maps via this functor to $\id\in\gaunt_{n}(C_{n},C_{n})$ . Corresponding to $f$ is a section $C_{n}\to X$ that carries the unique nontrivial $n$ -morphism of $C_{n}$ to $f$ . This exhibits $C_{n}$ as a retract of $X$ , as desired. ∎

[0MI0]

Remark 4.9. We thank Dimitri Ara who pointed out an error in an earlier version of the above lemma. An alternative proof of this lemma has appeared in work of Dimitri Ara, Moritz Groth, and Javier Gutiérrez [2].

The composition law $\ast_{j}$ (for $k$ -morphisms, $n\geq k\geq j$ ) is corepresented by a map

w_{j}^{k}\colon C_{k}\to C_{k}\cup^{C_{j-1}}C_{k}.

Let $E\colon\gaunt_{n}\to\gaunt_{n}$ be an endofunctor. We will say that $E$ commutes with compositional pushouts if for all $k\geq j\geq 1$ the natural map is a isomorphism:

E(C_{k}\cup^{C_{j-1}}C_{k})\cong E(C_{k})\cup^{E(C_{j-1})}E(C_{k}).

Note that it is sufficient to consider only the case $k=n$ . The other cases, being retracts of the $n=k$ case, follow automatically.

[0MI1]

Lemma 4.10. Any endofunctor $E$ of the category $\gaunt_{n}$ that commutes with compositional pushouts and restricts to an automorphism of the category $\mathbb{G}_{n}$ is an autoequivalence and isomorphic to a functor of the form $\rho(g)$ for some $g\in(\mathbb{Z}/2)^{n}$ .

[0MI2]

Proof. By Proposition 4.7 the restriction of $E$ to $\mathbb{G}_{n}$ is necessarily of the form $\rho(g)$ for some $g\in(\mathbb{Z}/2)^{n}$ . It suffices to prove that $E\circ\rho(g)\simeq\rho(g)\circ\rho(g)$ , which is in turn simply the identity, so without loss of generality we may assume $\rho(g)=\id$ , that is, that $E$ restricts to the identity functor on $\mathbb{G}_{n}$ .

Now in this case, the isomorphisms

\gaunt_{n}(C_{k},X)\cong\gaunt_{n}(F(C_{k}),F(X))=\gaunt_{n}(C_{k},F(X))

are natural in both $C_{k}$ and $X$ , whence one obtains a natural isomorphism $\gamma\colon U\cong U\circ F$ , where $U\colon\gaunt_{n}\to\Fun(\mathbb{G}_{n}^{\mathrm{op}},\set)$ is the forgetful functor from gaunt $n$ -categories to globular sets. It thus remains to show that this natural isomorphism is compatible with the compositions $\ast_{j}$ .

For this, consider

w_{j}^{k}\colon C_{k}\to C_{k}\cup^{C_{j-1}}C_{k},

the morphism that corepresents the composition law $\ast_{j}$ , as above. Since $E$ preserves compositional pushouts, one obtains a commutative diagram

Hence for any gaunt $n$ -category $X$ , we obtain a commutative diagram

in which the top and bottom morphisms are exactly the composition functors. Hence the natural isomorphism $\gamma$ is compatible with compositions, whence it lifts to a natural isomorphism $\id\cong E$ , as desired. ∎

[0MI3]

Remark 4.11. The above lemma is also valid for $\gaunt_{n}^{\omega}$ in place of $\gaunt_{n}$ .

[0MI4]

Corollary 4.12. Any autoequivalence of the category $\gaunt_{n}$ is isomorphic to a functor of the form $\rho(g)$ for some $g\in(\mathbb{Z}/2)^{n}$ .

[0MI5]

Proof. Any autoequivalence preserves all colimits, in particular the compositional pushouts. Moreover by Lemma 4.8, every autoequivalence $F$ restricts to an autoequivalence of $\mathbb{G}_{n}$ , and hence the assumptions of the previous lemma are met.

∎

[0MI6]

Theorem 4.13. The full subcategory

\Aut(\gaunt_{n})\subset\Fun(\gaunt_{n},\gaunt_{n})

of the category spanned by the autoequivalences is equivalent to the discrete set $(\mathbb{Z}/2)^{n}$ .

[0MI7]

Proof. Indeed, it follows from Corollary 4.12 that it is essentially surjective, and it follows from Lemma 4.5 that the action functor $\rho:(\mathbb{Z}/2)^{n}\to\Aut(\gaunt_{n})$ is fully faithful. ∎

Combining these results with Rm. 3.6 we also obtain:

[0MI8]

Corollary 4.14. The full subcategory

\Aut(\gaunt_{n}^{\omega})\subset\Fun(\gaunt_{n}^{\omega},\gaunt_{n}^{\omega})

spanned by the autoequivalences is equivalent to the discrete set $(\mathbb{Z}/2)^{n}$ .

Clark Barwick, Christopher Schommer-Pries

Original source: arXiv:1112.0040v6

Original source · 1112.0040v6

4. Endomorphisms and Automorphisms of Gauntn\gaunt_{n}[0MLS]

4. Endomorphisms and Automorphisms of $\gaunt_{n}$ [0MLS]