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final (infinity,1)-functor

Idea

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

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

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

$$\underrightarrow{\lim }(K\stackrel{F}{\to }C)\simeq \underrightarrow{\lim }(K\prime \stackrel{p}{\to }K\stackrel{F}{\to }C)$$\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 $p:S\to T$ of simplicial sets is final if for every right fibration $X\to T$ the induced morphism of simplicial sets

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

is a homotopy equivalence.

So in the overcategory $\mathrm{sSet}/T$ 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.

$$\left\{\begin{array}{ccccc}T& & \to & & X\\ & {}_{=}\searrow & & {\swarrow }_{}\\ & & T\end{array}\right\}\simeq \left\{\begin{array}{ccccc}S& & \to & & X\\ & {}_p\searrow & & {\swarrow }_{}\\ & & T\end{array}\right\}.$$\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 $p:S\to T$ is final precisely if the terminal morphism $(p\to *)=\left(\begin{array}{ccccc}S& & \to & & T\\ & {}_{}\searrow & & {\swarrow }_{=}\\ & & T\end{array}\right)$ in the overcategory ${\mathrm{sSet}}_T$ is a weak equivalence in the model structure for right fibrations on ${\mathrm{sSet}}_T$.

Proof

This is HTT, prop. 4.1.2.5.

Corollary

If $T$ is a Kan complex then $p:S\to T$ 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 $p:K\prime \to K$ of simplicial sets is final precisely if for every quasicategory $C$

• and for every morphism $\overline{F}:K^{▹}\to C$ that exibits a colimit co-cone in $C$, also $(K\prime {)}^{▹}\stackrel{p}{\to }{K}^{▹}\stackrel{\overline{F}}{\to }C$ is a colimit co-cone.

and equivalently precisely if

• … and for every $F:K\to C$ the morphism

  $$F/C\to (F\circ p)/C$$F/C \to (F\circ p)/C

  of under quasi-categories induced by composition with $p$ is an equivalence of (∞,1)-categories.

Proof

This is HTT, prop. 4.1.1.8.

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

Theorem

(recognition theorem for final $(\mathrm{\infty },1)$-functors)

A morphism $p:K\to C$ of simplicial sets with $C$ a quasi-category is final precisely if for each object $c\in C$ the comma-object $c/p:=c/C{\times }_{C}K$ is weakly contractible.

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

$$\begin{array}{ccc}c/p& \to & c/C\\ \downarrow & & \downarrow \\ K& \stackrel{p}{\to }& C\end{array},$$\array{ c/p &\to& c/C \\ \downarrow && \downarrow \\ K &\stackrel{p}{\to}& C } \,,

where $c/C$ is the under quasi-category under $c$.

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 $*\to 𝒞$ of a terminal object is final.

Proof

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

On categories of simplices

Definition

For $K\in$ sSet a simplicial set, write ${\Delta }_{/K}$ for its category of elements, called here the category of simplices of the simplicial set:

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

$$\begin{array}{ccccc}{\Delta }^{n_1}& & \to & & {\Delta }^{n_2}\\ & \searrow & & \swarrow \\ & & K\end{array}.$$\array{ \Delta^{n_1}&&\to&& \Delta^{n_2} \\ & \searrow && \swarrow \\ && K } \,.

Moreover, write

$${\Delta }_{/K}^{\mathrm{nd}}\hookrightarrow {\Delta }_{/K}$$\Delta_{/K}^{nd} \hookrightarrow \Delta_{/K}

for the full subcategory on the non-degenerate simplices.

Remark

The category ${\Delta }_{/K}^{\mathrm{nd}}$ is also known as the barycentric subdivision of $K$.

Proposition

The inclusion

$$N({\Delta }_{/K}^{\mathrm{nd}})\hookrightarrow N({\Delta }_{/K})$$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 $K$ there exists a cofinal map

$$N({\Delta }_{/K})\to K.$$N(\Delta_{/K}) \to K \,.

This is (Lurie, prop. 4.2.3.14).

Related concepts

• final functor

• homotopy final functor

• final $(\mathrm{\infty },1)$-functor

References

Section 4.1 of

• Jacob Lurie, Higher Topos Theory

Section 6 of

• Dan Dugger, A primer on homotopy colimits (pdf)