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final (infinity,1)-functor

Contents

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 (∞,1)-colimit:

if pp is final then 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 (∞,1)-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 exhibits 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 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.

Example

A (weak) localization f:𝒞𝒟f: \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 tt:

(1){x b y}AAAA{ t x b y} \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 𝒞\mathcal{C} an (∞,1)-category and T𝒞T \in \mathcal{C} an object, (∞,1)-colimits in the under-(∞,1)-category

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

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

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

a small diagram, we have

U(limF)lim(T/U(F)) 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 𝒞 T/\mathcal{C}^{T/} are computed as \infty-colimits in 𝒞\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 𝒞\mathcal{C}.

Explicitly: Given

T = T = T ϕ X ϕ B ϕ Y X f B g Y \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 𝒞 T\mathcal{C}^T, hence with underlying objects

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

we have:

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

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

(X,ϕ X)(Y,ϕ Y)=(X,ϕ X)(T,id T)(Y,ϕ Y) (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((X,ϕ X)(Y,ϕ Y))XTY. U \Big( \; (X,\phi_X) \coprod (Y,\phi_Y) \; \Big) \;\;\;\simeq\;\;\; X \underset{T}{\coprod} Y \,.

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 non-full subcategory on the non-degenerate simplices.

Remark

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

Proposition

Suppose KK is a non-singular simplicial set. Then 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).

Remark

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

Proposition

For every simplicial set KK, evaluation at the initial vertex

N(Δ /K) opK. 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).

References

Section 4.1 of

Section 6 of

Jay Shah Parametrized higher categories and higher algebra: Expose II- Indexed homotopy limits and colimits.

Denis-Charles Cisinski Higher Categories and Homotopical Algebra.

Last revised on July 3, 2020 at 10:37:52. See the history of this page for a list of all contributions to it.