# nLab final (infinity,1)-functor

Contents

### Context

#### Limits and colimits

limits and colimits

# Contents

## Idea

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

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

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

$\lim_\to (K \stackrel{F}{\to} C) \simeq \lim_{\to} ( K' \stackrel{p}{\to} K \stackrel{F}{\to} C)$

when either of these (∞,1)-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

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

is a homotopy equivalence.

So in the overcategory $sSet/T$ 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.

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

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 $(p \to *) = \left( \array{ S &&\to&& T \\ & {}_{\mathllap{}}\searrow && \swarrow_{=} \\ && T } \right)$ in the overcategory $sSet_T$ is a weak equivalence in the model structure for right fibrations on $sSet_T$.

###### 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 $p : K' \to K$ of simplicial sets is final precisely if for every quasicategory $C$

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

and equivalently precisely if

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

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

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

###### Theorem

(recognition theorem for final $(\infty,1)$-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 $c/p \coloneqq c/C \times_C K$ is weakly contractible.

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

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

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 $\ast \to \mathcal{C}$ of a terminal object is final.

###### Proof

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

###### Example

A (weak) localization $f: \mathcal{C} \to \mathcal{D}$ is both initial and final.

This appears, for example, as (Cisinski, 7.1.10).

### Cofiber products in co-slice categories

###### Example

Consider the inclusion of the walking span-category, into the result of adjoining an initial object $t$:

(1)$\Big\{ \array{ x &\longleftarrow& b &\longrightarrow& y } \Big\} \;\; \overset{\phantom{AAAA}}{\hookrightarrow} \;\; \left\{ \array{ && t \\ & \swarrow & \downarrow & \searrow \\ x &\longleftarrow& b &\longrightarrow& y } \right\}$

One readily sees that for each object on the right, its comma category over this inclusion has contractible nerve, whence Theorem implies that this inclusion is a final $\infty$-functor.

As an application of the finality of (1), observe that for $\mathcal{C}$ an (∞,1)-category and $T \in \mathcal{C}$ an object, (∞,1)-colimits in the under-(∞,1)-category

$\mathcal{C}^{T/} \overset{\;\;U\;\;}{\longrightarrow} \mathcal{C}$

are given by the $\infty$-colimit in $\mathcal{C}$ itself of the given cone of the original diagram, with tip $X$ (by this Prop.): For

$F \;\colon\; \mathcal{I} \longrightarrow \mathcal{C}^{T/}$

a small diagram, we have

$U \big( \underset{\longrightarrow}{\lim}\, F \big) \;\simeq\; \underset{\longrightarrow}{\lim}\, \big( T/U(F) \big)$

(when either $\infty$-colimit exists).

Now for $\mathcal{I}$ the walking span diagram on the left of (1), this means that homotopy cofiber products in $\mathcal{C}^{T/}$ are computed as $\infty$-colimits in $\mathcal{C}$ of diagrams of the shape on the right of (1). But since the inclusion in (1) is final, these are just homotopy cofiber products in $\mathcal{C}$.

Explicitly: Given

$\array{ T &=& T &=& T \\ {}^{\mathllap{ \phi_X }} \big\downarrow && {}^{\mathllap{ \phi_B }} \big\downarrow && {}^{\mathllap{ \phi_Y }} \big\downarrow \\ X & \underset{ f }{\longleftarrow} & B & \underset{ g }{ \longrightarrow } & Y }$

regarded as a span in $\mathcal{C}^T$, hence with underlying objects

$U\big( (X,\phi_X) \big) \;=\; X \,, \;\;\;\;\;\; U\big( (B,\phi_B) \big) \;=\; B \,, \;\;\;\;\;\; U\big( (Y,\phi_Y) \big) \;=\; Y \,,$

we have:

$U \Big( \; (X,\phi_X) \underset{ (B,\phi_B) }{\coprod} (Y,\phi_Y) \; \Big) \;\;\;\simeq\;\;\; X \underset{B}{\coprod} Y \,.$

In particular, if $(B,\phi_B) \;\coloneqq\; (T,id_T)$ is the initial object in $\mathcal{C}^{T/}$, in which case the cofiber product is just the coproduct

$(X,\phi_X) \coprod (Y,\phi_Y) \;\;=\;\; (X,\phi_X) \underset{ (T,id_T) }{\coprod} (Y,\phi_Y)$

we find that the coproduct in the co-slice category is the co-fiber product under the given tip object in the underlying category

$U \Big( \; (X,\phi_X) \coprod (Y,\phi_Y) \; \Big) \;\;\;\simeq\;\;\; X \underset{T}{\coprod} Y \,.$

### 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

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

Moreover, write

$\Delta_{/K}^{nd} \hookrightarrow \Delta_{/K}$

for the non-full subcategory on the non-degenerate simplices.

###### Remark

When the simplicial set $K$ is non-singular, i.e. every face of a non-degenerate simplex is still non-degenerate, then the category $\Delta_{/K}^{nd}$ is a poset and it coincides with the barycentric subdivision of $K$.

###### Proposition

Suppose $K$ is a non-singular simplicial set. Then the inclusion

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

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

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

This is (Lurie, prop. 4.2.3.14).

###### Remark

If the simplicial set $K$ is singular, it is not in general true that the inclusion $N(\Delta_{/K}^{nd}) \hookrightarrow N(\Delta_{/K})$ is final. For a counter-example, see (Hovey, 9).

###### Proposition

For every simplicial set $K$, evaluation at the initial vertex

$N(\Delta_{/K})^{op} \to K \,.$

is both initial and final.

This appears as (Shah, 12.2) and also follows from the fact that this map is a weak localization (Cisinski, 7.3.15).

###### Remark

This can be used to establish a Bousfield-Kan formula for homotopy colimits; see (Shah, 12.3).