nLab final (infinity,1)-functor

Idea

The notion of final $(\infty,1)$-functor (also called a cofinal $(\infty,1)$-functor, but see Rem ) is the vertical categorification of the notion of final functor from category theory to (∞,1)-category-theory:

An (∞,1)-functor $p \colon K' \to K$ is final precisely if precomposition with $p$ preserves (∞,1)-colimits: if $p$ is final then for $F \colon K \to C$ 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.

Terminology

The terminology conventions surrounding the topic of final functors are a bit of a mess. Often

1. “final map” is used to refer to $\infty$-functors such that restriction along them preserves $\infty$-colimiting cocones.

2. “initial map” is used for functors for which restriction preserves $\infty$-limiting cones.

A mnemonic rule for this terminology is that final objects are picked out by final functors and likewise for initial objects.

But beware that this convention is not universal, for instance Lurie‘s work is a notable exception to this rule. Also beware of the following:

Remark

Beware that the term “cofinal (∞,1)-functor” can mean either a functor for which the precomposition functor preserves colimits (as in Lurie’s Higher Topos Theory) or limits (as in Cisinski’s Higher Categories and Homotopical Algebra).

Given that the two main sources for quasicategories assign opposite meanings to this term, it is best to avoid its usage altogether.

Further adding to the confusion is that some sources, like Borceux‘s Handbook of Categorical Algebra use the term “final functor” for a functor for which the precomposition functor preserves limits, in contrast to the majority of the literature. Such usage, fortunately, is marginal.

In more detail:

• Joyal‘s The Theory of Quasi-Categories and its Applications:

  colimit-preserving: final;

  limit-preserving: initial (page 171);

• Cisinski‘s Higher Categories and Homotopical Algebra:

  colimit-preserving: final (Definition .4.1.8);

  limit-preserving: cofinal (Definition 4.4.13);

• Riehl–Verity‘s Elements of ∞-Category Theory:

  colimit-preserving: final;

  limit-preserving: initial (Definition 2.4.5).

• Lurie‘s Higher Topos Theory:

  colimit-preserving: cofinal (Definition 4.1.1.1);

  limit-preserving: (no terminology introduced);

• Lurie‘s Higher Algebra:

  colimit-preserving: left cofinal;

  limit-preserving: right cofinal (page 13);

• Lurie‘s Kerodon:

  colimit-preserving: right cofinal (Tag 02N1);

  limit-preserving (initial (∞,1)-functors): left cofinal.

See also the warning in the article final functor (here).

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.

Remark

Suppose $F : C \to D$ is a functor between ordinary categories. If the induced morphism $N(F) : N(C) \to N(D)$ is a final (∞,1)-functor, then $F$ is a final functor in the sense of ordinary category theory.

However, the converse need not be true. For example, while the inclusion of the parallel pair $[1] \rightrightarrows [0]$ in $\Delta^{op}$ is final in the sense of ordinary category theory, it is not a final (∞,1)-functor.

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).

Related concepts

• final functor

• homotopy final functor

• initial (∞,1)-functor

basic properties of…

• functors

  • faithful functor

  • full functor

  • full and faithful functor

  • injective-on-objects functor

  • essentially injective functor

  • embedding of categories

  • essentially surjective functor

    • surjective-on-objects functor

    • bijective-on-objects functor

    • dominant functor

  • split essentially surjective functor

  • essentially surjective and full functor

  • equivalence of categories

  • conservative functor

  • localization functor

  • pseudomonic functor

  • (co)continuous functor

  • final functor, dense functor

• 2-functors

  • full and faithful 2-functor

  • locally fully faithful 2-functor

  • equivalence of 2-categories

• (∞,1)-functors:

  • fully faithful (∞,1)-functor

  • essentially surjective (∞,1)-functor

  • equivalence of (∞,1)-categories

  • homotopy final functor, final (∞,1)-functor

References

• Mark Hovey, Item 9 of: Errata to Model Categories (1999) [pdf, pdf]

• Philip S. Hirschhorn, Section 19.6 of Model Categories and Their Localizations, Mathematical Surveys and Monographs 99, AMS (2002) [ISBN:0-8218-3279-4, doi:10.1090/surv/099, pdf]

• Daniel Dugger, Section 6 of: A primer on homotopy colimits, (2008) [pdf, pdf, pdf]

• Jacob Lurie, Section 4.1 of: Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press (2009) [pup:8957, pdf]

• Denis-Charles Cisinski, Higher Categories and Homotopical Algebra, Cambridge Studies in Advanced Mathematics 180, Cambridge University Press (2019) [ISBN: 9781108588737, doi:10.1017/9781108588737, pdf]

• Jay Shah, Parametrized higher categories and higher algebra, Algebr. Geom. Topol. 23 (2023) 509-644 [arXiv:1809.05892, doi:10.2140/agt.2023.23.509]