nLab
strict n-category

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Higher category theory

higher category theory

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Extra properties and structure

1-categorical presentations

Contents

Definition

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

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

The category StrωCatStr\omega Cat of strict ω\omega-categories can then in turn be defined as a suitable colimit of the categories nCatn Cat.

Properties

Definition

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

Write

StrnCat gauntStrnCat Str n Cat_{gaunt} \hookrightarrow Str n Cat

for the full subcategory on the gaunt nn-categories, those nn-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 (,n)(\infty,n)-categories.

Example

For knk \leq n the kk-globe is gaunt, G kStrnCat gauntStrnCatG_k \in Str n Cat_{gaunt} \hookrightarrow \in Str n Cat.

Write

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

for the full subcategory of the globe category on the kk-globes for knk \leq n.

Being a subobject of a gaunt nn-category, also the boundary of a globe G kG k\partial G_k \hookrightarrow G_k is gaunt, i.e. the (k1)(k-1)-skeleton of G kG_k.

Definition

Write

σ k:Str(k)CatStr(k+1)Cat \sigma_k : Str (k) Cat \to Str (k+1) Cat

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

σ(C)(a,b)=C. \sigma(C)(a,b) = C \,.

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

Example

The kk-globe G kG_k is the kk-fold suspension of the 0-globe (the point)

G k=σ k(G 0). G_k = \sigma^k(G_0) \,.

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

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

Accordingly, the boundary inclusion G kG k\partial G_k \hookrightarrow G_k is the kk-fold suspension of the initial morphism G 0\emptyset \to G_0

(G kG k)=σ k(G 0). (\partial G_k \hookrightarrow G_k) = \sigma^k(\emptyset \to G_0) \,.
Proposition

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

(B-PS, lemma 3.5)

Observation

For A,BA,B two categories, a profunctor A op×BSetA^{op} \times B \to Set is equivalently a functor KG 1K \to G_1 equipped with an identification AK 0A \simeq K_0 and BK 1B \simeq K_1.

This motivates the following definition.

Definition

A kk-profunctor / kk-correspondence of strict nn-categories is a morphism KG kK \to G_k in StrnCatStr n Cat. The category of kk-correspondences is the slice category StrnCat/G kStr n Cat/ G_k.

Definition

The categories StrnCat gaunt/G kStr n Cat_{gaunt}/G_k of kk-correspondences between gaunt nn-categories are cartesian closed category.

(B-SP, cor. 5.4)

Remark

By standard facts, in a locally presentable category 𝒞\mathcal{C} with finite limits, a slice 𝒞/X\mathcal{C}/X is cartesian closed precisely if pullback along all morphisms f:YXf : Y \to X with codomain XX 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 ff in the above is a globe, the pullback f *f^* in StrnCatStr n Cat will in general fail to preserves colimits. For a simple example of this, consider the pushout diagram in Cat Cat (,1)\hookrightarrow Cat_{(\infty,1)} given by

Δ[0] δ 1 Δ[1] δ 0 δ 0 Δ[1] δ 2 Δ[2]. \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 Cat /Δ[2]Cat_{/\Delta[2]} and consider then the pullback δ 1 *:Cat /Δ[1]Cat /Δ[1]\delta_1^* : Cat_{/\Delta[1]} \to Cat_{/\Delta[1]} along the remaining face δ 1:Δ[1]Δ[2]\delta_1 : \Delta[1] \to \Delta[2]. This yields the diagram

Δ[1], \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

StrnCat genStrnCat gaunt Str n Cat_{gen} \hookrightarrow Str n Cat_{gaunt}

for the smallest full subcategory that

  1. contains the globe category 𝔾 n\mathbb{G}_{\leq n}, example 1;
  2. is closed under retracts in StrnCat gauntStr n Cat_{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 StrnCat genStr n Cat_{gen}

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 StrnCatStr n Cat we call the fundamental pushouts

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

    G k C k1G kG k+1G_k \coprod_{\partial C_{k-1}} G_k \simeq \partial G_{k+1}
  2. Gluing two kk-globes along an ii-face gives a pasting composition of the two globles

    G k G iG kG_k \coprod_{G_i} G_k
  3. The fiber product of globes along non-degenerate morphisms G i+jG iG_{i+j} \to G_i and G i+kG iG_{i+k} \to G_i is built from gluing of globes by

    G i+j× G iG i+k(G i+j G iG i+k) σ i+1(G j1×G k1)(G i+k G iG 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 (ab)(a \stackrel{\simeq}{\to} b) is obtained by forcing in Δ[3]\Delta[3] the morphisms (02)(0\to 2) and (13)(1 \to 3) to be identities and it is equivalent, as an nn-category, to the 0-globe

    Δ[3] {0,2}{1,3}(Δ[0]Δ[0])G 0 \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

    σ k(Δ[3]) σ k{0,2}σ k{1,3}(G kG k)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 ii on StrnCatStr n Cat preserves the fundamental pushouts if it preserves the first three classes of pushouts, and if for the last one the morphism i(σ k(Δ[3])) i(σ k{0,2})i(σ k{1,3})(i(G kG k))i(G k)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 n3n\ge 3, these two properties fail, so that strict nn-categories become less useful (though not useless). Instead, one needs to use (at least) semistrict categories.

References

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

and in section 2 of

Revised on May 21, 2012 11:42:37 by Urs Schreiber (89.204.138.230)