nLab strict n-category

Context

Higher category theory

higher category theory

Contents

Definition

A strict $n$-category is a strict omega-category all whose k-morphisms for $k \gt n$ are identities.

The category $n Cat$ of strict $n$-categories and n-functors between them can also be defined inductively by

• starting by setting $0 Cat :=$ Set;

• noticing that Set is canonically a (symmetric, in fact cartesian) closed monoidal category such that one can consider categories enriched over it;

• noticing that for $V$ any complete and cocomplete closed monoidal category, also $V Cat$ (the category of $V$-enriched categories) has these same properties;

• finally setting, recursively,

$(n+1)Cat := (n Cat) Cat \,.$

The category $Str\omega Cat$ of strict $\omega$-categories can then in turn be defined as a suitable colimit of the categories $n Cat$.

Properties

Definition

Write $Str n Cat$ for the 1-category of strict n-categories.

Write

$Str n Cat_{gaunt} \hookrightarrow Str n Cat$

for the full subcategory on the gaunt $n$-categories, those $n$-categories whose only invertible k-morphisms are the identities.

This subcategory was considered in (Rezk). The term “gaunt” is due to (Barwick, Schommer-Pries). See prop. below for a characterization intrinsic to $(\infty,n)$-categories.

Example

For $k \leq n$ the $k$-globe is gaunt, $G_k \in Str n Cat_{gaunt} \hookrightarrow \in Str n Cat$.

Write

$\mathbb{G}_{\leq n} \hookrightarrow Str n Cat_{gaunt}$

for the full subcategory of the globe category on the $k$-globes for $k \leq n$.

Being a subobject of a gaunt $n$-category, also the boundary of a globe $\partial G_k \hookrightarrow G_k$ is gaunt, i.e. the $(k-1)$-skeleton of $G_k$.

Definition

Write

$\sigma_k : Str (k) Cat \to Str (k+1) Cat$

for the “categorical suspension” functor which sends a strict $k$-category to the object $\sigma(X) \in Str (k+1) Cat \simeq (Str k Cat)Cat$ which has precisely two objects $a$ and $b$, has $\sigma(C)(a,a) = \{id_a\}$, $\sigma(C)(b,b) = \{id_b\}$, $\sigma(C)(b,a) = \emptyset$ and

$\sigma(C)(a,b) = C \,.$

We usually suppress the subscript $k$ and write $\sigma^i = \sigma_{k+i} \circ \cdots \circ \sigma_{k+1} \circ \sigma_k$, etc.

Example

The $k$-globe $G_k$ is the $k$-fold suspension of the 0-globe (the point)

$G_k = \sigma^k(G_0) \,.$

The boundary $\partial G_k$ of the $k$-globe is the $k$-fold suspension of the empty category

$\partial G_k = \sigma^k(\emptyset) \,.$

Accordingly, the boundary inclusion $\partial G_k \hookrightarrow G_k$ is the $k$-fold suspension of the initial morphism $\emptyset \to G_0$

$(\partial G_k \hookrightarrow G_k) = \sigma^k(\emptyset \to G_0) \,.$
Proposition

The category $Str n Cat_{gaunt}$ is a locally presentable category and in fact a locally finitely presentable category.

Observation

For $A,B$ two categories, a profunctor $A^{op} \times B \to Set$ is equivalently a functor $K \to G_1$ equipped with an identification $A \simeq K_0$ and $B \simeq K_1$.

This motivates the following definition.

Definition

A $k$-profunctor / $k$-correspondence of strict $n$-categories is a morphism $K \to G_k$ in $Str n Cat$. The category of $k$-correspondences is the slice category $Str n Cat/ G_k$.

Definition

The categories $Str n Cat_{gaunt}/G_k$ of $k$-correspondences between gaunt $n$-categories are cartesian closed category.

Remark

By standard facts, in a locally presentable category $\mathcal{C}$ with finite limits, a slice $\mathcal{C}/X$ is cartesian closed precisely if pullback along all morphisms $f : Y \to X$ with codomain $X$ preserves colimits (see at locally cartesian closed category the section Cartesian closure in terms of base change and dependent product).

Example

Without the restriction that the codomain of $f$ in the above is a globe, the pullback $f^*$ in $Str n Cat$ will in general fail to preserves colimits. For a simple example of this, consider the pushout diagram in Cat $\hookrightarrow Cat_{(\infty,1)}$ given by

$\array{ \Delta &\stackrel{\delta_1}{\to}& \Delta \\ {}^{\mathllap{\delta_0}}\downarrow && \downarrow^{\mathrlap{\delta_0}} \\ \Delta &\stackrel{\delta_2}{\to}& \Delta } \,.$

Notice that this is indeed also a homotopy pushout/(∞,1)-pushout since, by remark , all objects involved are 0-truncated.

Regard this canonically as a pushout diagram in the slice category $Cat_{/\Delta}$ and consider then the pullback $\delta_1^* : Cat_{/\Delta} \to Cat_{/\Delta}$ along the remaining face $\delta_1 : \Delta \to \Delta$. This yields the diagram

$\array{ \emptyset &\stackrel{}{\to}& \emptyset \\ {}^{}\downarrow && \downarrow^{} \\ \emptyset &\stackrel{}{\to}& \Delta } \,,$

which evidently no longer is a pushout.

Definition

Write

$Str n Cat_{gen} \hookrightarrow Str n Cat_{gaunt}$

for the smallest full subcategory that

1. contains the globe category $\mathbb{G}_{\leq n}$, example ;
2. is closed under retracts in $Str n Cat_{gaunt}$;
3. has all fiber products over globes (equivalently: such that all slice categories over globes have products).
Example

The following categories are naturally full subcategories of $Str n Cat_{gen}$

• the $n$-fold simplex category $\Delta^{\times n}$;

• the $n$th Theta-category.

This is discussed in more detail in (infinity,n)-category in Presentation by Theta-spaces and by n-fold Segal spaces-category#PresentationByThetaSpaces).

Definition

The following pushouts in $Str n Cat$ we call the fundamental pushouts

1. Gluing two $k$-globes along their boundary gives the boundary of the $(k+1)$-globle

$G_k \coprod_{\partial C_{k-1}} G_k \simeq \partial G_{k+1}$
2. Gluing two $k$-globes along an $i$-face gives a pasting composition of the two globles

$G_k \coprod_{G_i} G_k$
3. The fiber product of globes along non-degenerate morphisms $G_{i+j} \to G_i$ and $G_{i+k} \to G_i$ is built from gluing of globes by

$G_{i+j} \times_{G_i} G_{i+k} \simeq (G_{i+j} \coprod_{G_i} G_{i+k}) \coprod_{\sigma^{i+1}(G_{j-1} \times G_{k-1})} (G_{i+k} \coprod_{G_i} G_{i+j})$
4. The interval groupoid $(a \stackrel{\simeq}{\to} b)$ is obtained by forcing in $\Delta$ the morphisms $(0\to 2)$ and $(1 \to 3)$ to be identities and it is equivalent, as an $n$-category, to the 0-globe

$\Delta \coprod_{\{0,2\} \coprod \{1,3\}} (\Delta \coprod \Delta) \stackrel{\sim}{\to} G_0$

and the analog is true for all suspensions of this relation

$\sigma^k(\Delta) \coprod_{\sigma^k\{0,2\} \coprod \sigma^k\{1,3\}} (G_k\coprod G_k) \stackrel{\sim}{\to} G_k \,.$

We say a functor $i$ on $Str n Cat$ preserves the fundamental pushouts if it preserves the first three classes of pushouts, and if for the last one the morphism $i(\sigma^k(\Delta)) \coprod_{i(\sigma^k\{0,2\}) \coprod i(\sigma^k\{1,3\})} (i(G_k \coprod G_k)) \to i(G_k)$ is an equivalence.

Examples

A strict 1-category is just a category.

Strict 2-categories are important, because the coherence theorem for bicategories states that every (“weak”) 2-category is equivalent to a strict one, and also because many 2-categories, such as Cat, are naturally strict. However, for $n\ge 3$, these two properties fail, so that strict $n$-categories become less useful (though not useless). Instead, one needs to use (at least) semistrict categories.

With an eye towards the generalization to (∞,n)-categories, strict $n$-categories are discussed in

and in section 2 of