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final (infinity,1)-functor

Contents

Idea

The notion of final (,1)(\infty,1)-functor (also called a cofinal (,1)(\infty,1)-functor) is the generalization of the notion of final functor from category theory to (∞,1)-category-theory.

An (∞,1)-functor p:KKp : K' \to K is final precisely if precomposition with pp preserves colimits:

if pp is final then for for F:KCF : K \to C any (∞,1)-functor we have

lim (KFC)lim (KpKFC) \lim_\to (K \stackrel{F}{\to} C) \simeq \lim_{\to} ( K' \stackrel{p}{\to} K \stackrel{F}{\to} C)

when either of these colimits exist.

Definition

Definition

(final morphism of simplicial set)

A morphism p:STp : S \to T of simplicial sets is final if for every right fibration XTX \to T the induced morphism of simplicial sets

sSet /T(T,X)sSet /T(S,X) sSet_{/T}(T,X) \to sSet_{/T}(S,X)

is a homotopy equivalence.

So in the overcategory sSet/TsSet/T a final morphism is an object such that morphisms out of it into any right fibration are the same as morphisms out of the terminal object into that right fibration.

{T X = T}{S X p T}. \left\{ \array{ T &&\to&& X \\ & {}_{\mathllap{=}}\searrow && \swarrow_{} \\ && T } \right\} \;\; \simeq \left\{ \array{ S &&\to&& X \\ & {}_{\mathllap{p}}\searrow && \swarrow_{} \\ && T } \right\} \,.

This definition is originally due to Andre Joyal. It appears as HTT, def 4.1.1.1.

This is equivalent to the following definition, in terms of the model structure for right fibrations:

Proposition

The morphism p:STp : S \to T is final precisely if the terminal morphism (p*)=(S T = T)(p \to *) = \left( \array{ S &&\to&& T \\ & {}_{\mathllap{}}\searrow && \swarrow_{=} \\ && T } \right) in the overcategory sSet TsSet_T is a weak equivalence in the model structure for right fibrations on sSet TsSet_T.

Proof

This is HTT, prop. 4.1.2.5.

Corollary

If TT is a Kan complex then p:STp : S \to T is final precisely if it is a weak equivalence in the standard model structure on simplicial sets.

Proof

This is HTT, cor. 4.1.2.6.

Properties

Proposition

(preservation of undercategories and colimits)

A morphism p:KKp : K' \to K of simplicial sets is final precisely if for every quasicategory CC

  • and for every morphism F¯:K C\bar F : K^{\triangleright} \to C that exibits a colimit co-cone in CC, also (K) pK F¯C(K')^\triangleright \stackrel{p}{\to} K^{\triangleright} \stackrel{\bar F}{\to} C is a colimit co-cone.

and equivalently precisely if

Proof

This is HTT, prop. 4.1.1.8.

The following result is the (,1)(\infty,1)-categorical analog of what is known as Quillen’s Theorem A.

Theorem

(recognition theorem for final (,1)(\infty,1)-functors)

A morphism p:KCp : K \to C of simplicial sets with CC a quasi-category is final precisely if for each object cCc \in C the comma-object c/p:=c/C× CKc/p := c/C \times_C K is weakly contractible.

More explicitly, the comma object in question here is the pullback

c/p c/C K p C, \array{ c/p &\to& c/C \\ \downarrow && \downarrow \\ K &\stackrel{p}{\to}& C } \,,

where c/Cc/C is the under quasi-category under cc.

Proof

This is due to Andre Joyal. A proof appears as HTT, prop. 4.1.3.1.

The following says that up to equivalence, the cofinal maps of simplicial sets are the right anodyne morphisms

Proposition

A map of simplicial sets is cofinal precisely if it factors as a right anodyne map followed by a trivial fibration.

This is (Lurie, cor. 4.1.1.12).

Examples

General

Example

The inclusion *𝒞\ast \to \mathcal{C} of a terminal object is final.

Proof

By theorem 1 the inclusion of the point is final precisely if for all c𝒞c \in \mathcal{C}, the (∞,1)-categorical hom-space 𝒞(c,*)\mathcal{C}(c,\ast) is contractible. This is the definition of *\ast being terminal.

On categories of simplices

Definition

For KK \in sSet a simplicial set, write Δ /K\Delta_{/K} for its category of elements, called here the category of simplices of the simplicial set:

an object of Δ /K\Delta_{/K} is a morphism of simplicial sets of the form Δ nK\Delta^n \to K for some nn \in \mathbb{N} (hence an nn-simplex of KK) and a morphism is a commuting diagram

Δ n 1 Δ n 2 K. \array{ \Delta^{n_1}&&\to&& \Delta^{n_2} \\ & \searrow && \swarrow \\ && K } \,.

Moreover, write

Δ /K ndΔ /K \Delta_{/K}^{nd} \hookrightarrow \Delta_{/K}

for the full subcategory on the non-degenerate simplices.

Remark

The category Δ /K nd\Delta_{/K}^{nd} is also known as the barycentric subdivision of KK.

Proposition

The inclusion

N(Δ /K nd)N(Δ /K) N(\Delta_{/K}^{nd}) \hookrightarrow N(\Delta_{/K})

is a cofinal morphism of quasi-categories.

This appears as (Lurie, variant 4.2.3.15).

Proposition

For every simplicial set KK there exists a cofinal map

N(Δ /K)K. N(\Delta_{/K}) \to K \,.

This is (Lurie, prop. 4.2.3.14).

References

Section 4.1 of

Section 6 of

Revised on April 18, 2013 12:30:05 by Urs Schreiber (89.204.130.31)