nLab
strict n-category

Definition

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

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

• starting by setting $0\mathrm{Cat}:=$ 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 $V\mathrm{Cat}$ (the category of $V$-enriched categories) has these same properties;

• finally setting, recursively,

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

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

Properties

Definition

Write $\mathrm{Str}n\mathrm{Cat}$ for the 1-category of strict n-categories.

Write

$$\mathrm{Str}n{\mathrm{Cat}}_{\mathrm{gaunt}}\hookrightarrow \mathrm{Str}n\mathrm{Cat}$$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. \ref{GauntIs0Truncated} below for a characterization intrinsic to $(\mathrm{\infty },n)$-categories.

Example

For $k\le n$ the $k$-globe is gaunt, $G_k\in \mathrm{Str}n{\mathrm{Cat}}_{\mathrm{gaunt}}\hookrightarrow \in \mathrm{Str}n\mathrm{Cat}$.

Write

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

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

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

$${\sigma }_k:\mathrm{Str}(k)\mathrm{Cat}\to \mathrm{Str}(k+1)\mathrm{Cat}$$\sigma_k : Str (k) Cat \to Str (k+1) Cat

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

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

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

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

$$\partial G_k={\sigma }^k(\varnothing ).$$\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 $\varnothing \to G_0$

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

Proposition

The category $\mathrm{Str}n{\mathrm{Cat}}_{\mathrm{gaunt}}$ is a locally presentable category and in fact a locally finitely presentable category.

(B-PS, lemma 3.5)

Observation

For $A,B$ two categories, a profunctor $A^{\mathrm{op}}\times B\to \mathrm{Set}$ 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 $\mathrm{Str}n\mathrm{Cat}$. The category of $k$-correspondences is the slice category $\mathrm{Str}n\mathrm{Cat}/G_k$.

Definition

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

(B-SP, cor. 5.4)

Remark

By standard facts, in a locally presentable category $𝒞$ with finite limits, a slice $𝒞/X$ 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 $\mathrm{Str}n\mathrm{Cat}$ will in general fail to preserves colimits. For a simple example of this, consider the pushout diagram in Cat $\hookrightarrow {\mathrm{Cat}}_{(\mathrm{\infty },1)}$ given by

$$\begin{array}{ccc}\Delta [0]& \stackrel{{\delta }_1}{\to }& \Delta [1]\\ {}^{{\delta }_0}\downarrow & & {\downarrow }^{{\delta }_0}\\ \Delta [1]& \stackrel{{\delta }_2}{\to }& \Delta [2]\end{array}.$$\array{ \Delta[0] &\stackrel{\delta_1}{\to}& \Delta[1] \\ {}^{\mathllap{\delta_0}}\downarrow && \downarrow^{\mathrlap{\delta_0}} \\ \Delta[1] &\stackrel{\delta_2}{\to}& \Delta[2] } \,.

Notice that this is indeed also a homotopy pushout/(∞,1)-pushout since, by remark \ref{GauntIs0Truncted}, all objects involved are 0-truncated.

Regard this canonically as a pushout diagram in the slice category ${\mathrm{Cat}}_{/\Delta [2]}$ and consider then the pullback ${\delta }_1^{*}:{\mathrm{Cat}}_{/\Delta [1]}\to {\mathrm{Cat}}_{/\Delta [1]}$ along the remaining face ${\delta }_1:\Delta [1]\to \Delta [2]$. This yields the diagram

$$\begin{array}{ccc}\varnothing & \stackrel{}{\to }& \varnothing \\ {}^{}\downarrow & & {\downarrow }^{}\\ \varnothing & \stackrel{}{\to }& \Delta [1]\end{array},$$\array{ \emptyset &\stackrel{}{\to}& \emptyset \\ {}^{}\downarrow && \downarrow^{} \\ \emptyset &\stackrel{}{\to}& \Delta[1] } \,,

which evidently no longer is a pushout.

(See also the discussion here).

Definition

Write

$$\mathrm{Str}n{\mathrm{Cat}}_{\mathrm{gen}}\hookrightarrow \mathrm{Str}n{\mathrm{Cat}}_{\mathrm{gaunt}}$$Str n Cat_{gen} \hookrightarrow Str n Cat_{gaunt}

for the smallest full subcategory that

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

(B-SP, def. 5.6)

Example

The following categories are naturally full subcategories of $\mathrm{Str}n{\mathrm{Cat}}_{\mathrm{gen}}$

• 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 $\mathrm{Str}n\mathrm{Cat}$ we call the fundamental pushouts

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

   $$G_k\coprod _{\partial C_{k-1}}{G}_k\simeq \partial G_{k+1}$$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

   $$G_k\coprod _{G_i}{G}_k$$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

   $$G_{i+j}{\times }_{G_i}{G}_{i+k}\simeq (G_{i+j}\coprod _{G_i}{G}_{i+k})\coprod _{{\sigma }^{i+1}(G_{j-1}\times G_{k-1})}(G_{i+k}\coprod _{G_i}{G}_{i+j})$$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 $\Delta [3]$ the morphisms $(0\to 2)$ and $(1\to 3)$ to be identities and it is equivalent, as an $n$-category, to the 0-globe

   $\Delta [3]{\coprod }_{\{0,2\}\coprod \{1,3\}}(\Delta [0]\coprod \Delta [0])\stackrel{\sim }{\to }{G}_0$

   and the analog is true for all suspensions of this relation

   $${\sigma }^k(\Delta [3])\coprod _{{\sigma }^k\{0,2\}\coprod {\sigma }^k\{1,3\}}(G_k\coprod G_k)\stackrel{\sim }{\to }{G}_k.$$\sigma^k(\Delta[3]) \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 $\mathrm{Str}n\mathrm{Cat}$ preserves the fundamental pushouts if it preserves the first three classes of pushouts, and if for the last one the morphism $i({\sigma }^k(\Delta [3])){\coprod }_{i({\sigma }^k\{0,2\})\coprod i({\sigma }^k\{1,3\})}(i(G_k\coprod G_k))\to i(G_k)$ 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.

Related concepts

• (∞,n)-category

References

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

• Charles Rezk, A cartesian presentation of weak n-categories (arXiv:0901.3602)

and in section 2 of

• Clark Barwick, Chris Schommer-Pries, On the Unicity of the Homotopy Theory of Higher Categories (pdf, unusual slides)