nLab
limit in a quasi-category

Context

(,1)(\infty,1)-Category theory

Limits and colimits

Contents

Idea

The notion of limit and colimit generalize from category theory to (∞,1)-category theory. One model for (∞,1)-categories are quasi-categories. This entry discusses limits and colimits in quasi-categories.

Definition

Definition

For KK and CC two quasi-categories and F:KCF : K \to C an (∞,1)-functor (a morphism of the underlying simplicial sets) , the limit over FF is, if it exists, the quasi-categorical terminal object in the over quasi-category C /FC_{/F}:

limFTerminalObj(C /F) \underset{\leftarrow}{\lim} F \coloneqq TerminalObj(C_{/F})

(well defined up to a ontractible space of choices).

A colimit in a quasi-category is accordingly an limit in the opposite quasi-category.

Remark

Notice from the discussion at join of quasi-categories that there are two definitions – denoted \star and \diamondsuit – of join, which yield results that differ as simplicial sets, though are equivalent as quasi-categories.

The notation C /FC_{/F} denotes the definition of over quasi-category induced from **, while the notation C /FC^{/F} denotes that induced from \diamondsuit. Either can be used for the computation of limits in a quasi-category, as for quasi-categorical purposes they are weakly equivalent.

So we also have

limFTerminalObj(C /F). \underset{\leftarrow}{\lim} F \coloneqq TerminalObj(C^{/F}) \,.

See HTT, prop 4.2.1.5.

Properties

In terms of slice \infty-categories

Proposition

Let 𝒞\mathcal{C} be a quasi-category and let f:K𝒞f \colon K \to \mathcal{C} be a diagram with (,1)(\infty,1)-colimiting cocone f˜:KΔ 0𝒞\tilde f \colon K \star \Delta^0 \to \mathcal{C}. Then the induced map of slice quasi-categories

𝒞 /f˜𝒞 f \mathcal{C}_{/\tilde f} \to \mathcal{C}_{f}

is an acyclic Kan fibration.

Proposition

For F:𝒦𝒞F \colon \mathcal{K} \to \mathcal{C} a diagram in an (,1)(\infty,1)-category and limF\underset{\leftarrow}{\lim} F its limit, there is a natural equivalence of (∞,1)-categories

𝒞 /F𝒞 /limF \mathcal{C}_{/F} \simeq \mathcal{C}_{/\underset{\leftarrow}{\lim} F}

between the slice (∞,1)-categories over FF (the (,1)(\infty,1)-category of \infty-cones over FF) and over just limF\underset{\leftarrow}{\lim}F.

Proof

Let F˜:Δ 0𝒦𝒞\tilde F \colon \Delta^0 \star \mathcal{K} \to \mathcal{C} be the limiting cone. The canonical cospan of \infty-functors

*Δ 0𝒦𝒦 \ast \to \Delta^0 \star \mathcal{K} \leftarrow \mathcal{K}

induces a span of slice \infty-categories

𝒞 /limF𝒞 /F˜𝒞 /F. \mathcal{C}_{/\underset{\leftarrow}{\lim}F} \leftarrow \mathcal{C}_{/\tilde F} \rightarrow \mathcal{C}_{/F} \,.

The right functor is an equivalence by prop. 2. The left functor is induced by restriction along an op-final (∞,1)-functor (by the Examples discussed there) and hence is an equivalence by the discussion at slice (∞,1)-category (Lurie, prop. 4.1.1.8).

This appears for instance in (Lurie, proof of prop. 1.2.13.8).

In terms of \infty-Hom adjunction

The definition of the limit in a quasi-category in terms of terminal objects in the corresponding over quasi-category is well adapted to the particular nature the incarnation of (,1)(\infty,1)-categories by quasi-categories. But more intrinsically in (,1)(\infty,1)-category theory, it should be true that there is an adjunction characterization of (,1)(\infty,1)-limits : limit and colimit, should be (pointwise or global) right and left adjoint (infinity,1)-functor of the constant diagram (,1)(\infinity,1)-functor, const:KFunc(K,C)const : K \to Func(K,C).

(colimconstlim):Func(K,C)colimconstlimFunc(*,C)C. (colim \dashv const \dashv lim) : Func(K,C) \stackrel{\overset{lim}{\to}}{\stackrel{\overset{const}{\leftarrow}} {\underset{colim}{\to}}} Func(*,C) \simeq C \,.

By the discussion at adjoint (∞,1)-functor (HTT, prop. 5.2.2.8) this requires exhibiitng a morphism η:Id Func(K,C)constcolim\eta : Id_{Func(K,C)} \to const colim in Func(Func(K,C),Func(K,C))Func(Func(K,C),Func(K,C)) such that for every fFunc(K,C)f \in Func(K,C) and YCY \in C the induced morphism

Hom C(colim(f),Y)Hom Func(K,C)(constcolim(f),constY)Hom(η,constY)Hom Func(K,Y)(f,constY) Hom_{C}(colim(f),Y) \to Hom_{Func(K,C)}(const colim(f), const Y) \stackrel{Hom(\eta, const Y)}{\to} Hom_{Func(K,Y)}(f, const Y)

is a weak equivalence in sSet QuillensSet_{Quillen}.

But first consider the following pointwise characterization.

Proposition

Let CC be a quasi-category, KK a simplicial set. A co-cone diagram p¯:KΔ[0]C\bar p : K \star \Delta[0] \to C with cone point XCX \in C is a colimiting diagram (an initial object in C p/C_{p/}) precisely if for every object YCY \in C the morphism

ϕ Y:Hom C(X,Y)Hom Func(K,C)(p,constY) \phi_Y : Hom_C(X,Y) \to Hom_{Func(K,C)}(p, const Y)

induced by the morpism pconstX p \to const X that is encoded by p¯\bar p is an equivalence (i.e. a homotopy equivalence of Kan complexes).

Proof

This is HTT, lemma 4.2.4.3.

The key step is to realize that Hom Func(K,C)(p,constY)Hom_{Func(K,C)}(p, const Y) is given (up to equivalence) by the pullback C p/× C{Y}C^{p/} \times_C \{Y\} in sSet.

Here is a detailed way to see this, using the discussion at hom-object in a quasi-category.

We have that Hom Func(K,C)(p,constY)Hom_{Func(K,C)}(p, const Y) is given by (C K) p/× C KconstY(C^K)^{p/} \times_{C^K} const Y. We compute

((C K) p/× C KconstY) n =Hom Δ[0]/sSet(Δ[0]Δ[n],C K)× (C K) n{constY} =Hom Δ[0]/sSet(Δ[0] Δ[n]Δ[n]×Δ[1],C K)× (C K) n{constY} ={p}× Hom(Δ[0],C K)Hom(Δ[0],C K)× Hom(Δ[n],C K)Hom(Δ[n]×Δ[1],C KUnknown character)× Hom(Δ[n],C K){constY} ={p}× Hom(K,C)Hom(K,C)× Hom(Δ[n]×K,C)Hom(Δ[n]×K×Δ[1],C)× Hom(Δ[n]×K,C)Hom(Δ[n],C)× Δ[n],C{Y} ={p}× Hom(K,C)Hom(KΔ[n],C)× Hom(Δ[n],C){Y} =(C p/× C{Y}) n \begin{aligned} ((C^K)^{p/} \times_{C^K} const Y)_n & = Hom_{{\Delta[0]}/sSet}( \Delta[0] \diamondsuit \Delta[n] , C^K ) \times_{(C^K)_n} \{const Y\} \\ & = Hom_{{\Delta[0]}/sSet}( \Delta[0] \coprod_{\Delta[n]} \Delta[n] \times \Delta[1] , C^K ) \times_{(C^K)_n} \{const Y\} & = \{p\} \times_{Hom(\Delta[0],C^K)} Hom(\Delta[0], C^K) \times_{Hom(\Delta[n], C^K)} Hom(\Delta[n] \times \Delta[1], C^K \) \times_{Hom(\Delta[n], C^K)} \{const Y\} \\ & = \{p\} \times_{Hom(K,C)} Hom(K,C) \times_{Hom(\Delta[n]\times K,C)} Hom(\Delta[n]\times K \times \Delta[1], C) \times_{Hom(\Delta[n]\times K, C)} Hom(\Delta[n],C) \times_{\Delta[n],C} \{Y\} \\ &= \{p\} \times_{Hom(K,C)} Hom(K \diamondsuit \Delta[n], C) \times_{Hom(\Delta[n],C)} \{Y\} \\ &= (C^{p/}\times_C \{Y\})_n \end{aligned}

Under this identification, ϕ Y\phi_Y is the morphism

(C X/ϕC p¯/ϕC p/)× C{Y}, \left( C^{X/} \stackrel{\phi'}{\to} C^{\bar p/} \stackrel{\phi''}{\to} C^{p/} \right) \times_C \{Y\} \,,

in sSet where ϕ\phi' is a section of the map C p¯/C X/C^{\bar p/} \to C^{X/}, (which one checks is an acyclic Kan fibration) obtained by choosing composites of the co-cone components with a given morphism XYX \to Y.

The morphism ϕ\phi'' is a left fibration (using HTT, prop. 4.2.1.6)

One finds that the morphism ϕ\phi'' is a left fibration.

The strategy for the completion of the proof is: realize that the first condition of the proposition is equivalent to ϕ\phi'' being an acyclic Kan fibration, and the second statement equivalent to ϕ Y\phi''_Y being an acyclic Kan fibration, then show that these two conditions in turn are equivalent.

In terms of products and equalizers

A central theorem in ordinary category theory asserts that a category has limits already if it has products and equalizers. The analog statement is true here:

Proposition

Let κ\kappa be a regular cardinal. An (∞,1)-category CC has all κ\kappa-small limits precisely if it has equalizers and κ\kappa-small products.

This is HTT, prop. 4.4.3.2.

In terms of homotopy limits

The notion of homotopy limit, which exists for model categories and in particular for simplicial model categories and in fact in all plain Kan complex-enriched categories – as described in more detail at homotopy Kan extension – is supposed to be a model for (,1)(\infty,1)-categorical limits. In particular, under sending the Kan-complex enriched categories CC to quasi-categories N(C)N(C) using the homotopy coherent nerve functor, homotopy limits should precisely corespond to quasi-categorical limits. That this is indeed the case is asserted by the following statements.

Proposition

Let CC and JJ be Kan complex-enriched categories and let F:JCF : J \to C be an sSet-enriched functor.

Then a cocone {η i:F(i)c} iJ\{\eta_i : F(i) \to c\}_{i \in J} under FF exhibits the object cCc \in C as a homotopy colimit (in the sense discussed in detail at homotopy Kan extension) precisely if the induced morphism of quasi-categories

N(F)¯:N(J) N(C) \bar {N(F)} : N(J)^{\triangleright} \to N(C)

is a quasi-categorical colimit diagram in N(C)N(C).

Here NN is the homotopy coherent nerve, N(J) N(J)^{\triangleright} the join of quasi-categories with the point, N(F)N(F) the image of the simplicial functor FF under the homotopy coherent nerve and N(F)¯\bar{N(F)} its extension to the join determined by the cocone maps η\eta.

Proof

This is HTT, theorem 4.2.4.1

A central ingredient in the proof is the fact, discused at (∞,1)-category of (∞,1)-functors and at model structure on functors, that sSet-enriched functors do model (∞,1)-functors, in that for AA a combinatorial simplicial model category, SS a quasi-category and τ(S)\tau(S) the corresponding sSetsSet-category under the left adjoint of the homotopy coherent nerve, we have an equivalence of quasi-categories

N(([C,A] proj) )Func(S,N(A )) N(([C,A]_{proj})^\circ) \simeq Func(S, N(A^\circ))

and the same is trued for AA itself replaced by a chunk? UAU \subset A.

With this and the discussion at homotopy Kan extension, we find that the cocone components η\eta induce for each a[C,sSet]a \in [C,sSet] a homotopy equivalence

C(c,a)[J op,C](jF,consta) C(c,a) \stackrel{}{\to} [J^{op}, C](j F, const a)

which is hence equivalently an equivalence of the corresponding quasi-categorical hom-objects. The claim follows then from the above discussion of characterization of (co)limits in terms of \infty-hom adjunctions.

Corollary

The quasi-category N(A )N(A^\circ) presented by a combinatorial simplicial model category AA has all small quasi-categorical limits and colimits.

Proof

This is HTT, 4.2.4.8.

It follows from the fact that AA has (pretty much by definition of model category and combinatorial model category) all homotopy limits and homotopy colimits (in fact all homotopy Kan extensions) by the above proposition.

Since (,1)(\infty,1)-categories equivalent to those of the form N(A )N(A^\circ) for AA a combinatorial simplicial model category are precisely the locally presentable (∞,1)-categories, it follows from this in particular that every locally presentable (,1)(\infty,1)-category has all limits and colimits.

Commutativity of limits

The following proposition says that if for an (,1)(\infty,1)-functor F:X×YCF : X \times Y \to C limits (colimits) over each of the two variables exist separately, then they commute.

Proposition

Let XX and YY be simplicial sets and CC a quasi-category. Let p:X ×Y Cp : X^{\triangleleft} \times Y^{\triangleleft} \to C be a diagram. If

  1. for every object xX x \in X^{\triangleleft} (including the cone point) the restricted diagram p x:Y Cp_x : Y^{\triangleleft} \to C is a limit diagram;

  2. for every object yYy \in Y (not including the cone point) the restricted diagram p y:X Cp_y : X^{\triangleleft} \to C is a limit diagram;

then, with cc denoting the cone point of Y Y^{\triangleleft}, the restricted diagram, p c:X Cp_c : X^{\triangleleft} \to C is also a limit diagram.

Proof

This is HTT, lemma 5.5.2.3

In other words, suppose that lim xF(x,y)\lim_x F(x,y) exists for all yy and lim yF(x,y)\lim_y F(x,y) exists for all xx and also that lim y(lim xF(x,y))\lim_y (\lim_x F(x,y)) exists, then this object is also lim xlim yF(x,y)\lim_x \lim_y F(x,y).

Examples

\infty-Limits of special shape

Coproduct

Pullback / Pushout

See also (∞,1)-pullback.

The non-degenerate cells of the simplicial set Δ[1]×Δ[1]\Delta[1] \times \Delta[1] obtained as the cartesian product of the simplicial 1-simplex with itself look like

(0,0) (1,0) (0,1) (1,1) \array{ (0,0) &\to& (1,0) \\ \downarrow &\searrow& \downarrow \\ (0,1) &\to& (1,1) }

A sqare in a quasi-category CC is an image of this in CC, i.e. a morphism

s:Δ[1]×Δ[1]C. s : \Delta[1] \times \Delta[1] \to C \,.

The simplicial square Δ[1] ×2\Delta[1]^{\times 2} is isomorphic, as a simplicial set, to the join of simplicial sets of a 2-horn with the point:

Δ[1]×Δ[1]{v}Λ[2] 2=(v 1 0 2) \Delta[1] \times \Delta[1] \simeq \{v\} \star \Lambda[2]_2 = \left( \array{ v &\to& 1 \\ \downarrow &\searrow& \downarrow \\ 0 &\to& 2 } \right)

and

Δ[1]×Δ[1]Λ[2] 0{v}=(0 1 2 v). \Delta[1] \times \Delta[1] \simeq \Lambda[2]_0 \star \{v\} = \left( \array{ 0&\to& 1 \\ \downarrow &\searrow& \downarrow \\ 2 &\to& v } \right) \,.

If a square Δ[1]×Δ[1]Λ[2] 0{v}C\Delta[1] \times \Delta[1] \simeq \Lambda[2]_0 \star \{v\} \to C exhibits {v}C\{v\} \to C as a colimit over F:Λ[2] 0CF : \Lambda[2]_0 \to C, we say the colimit

v:=lim F:=F(1) F(0)F(2) v := \lim_\to F := F(1) \coprod_{F(0)} F(2)

is the pushout of the diagram FF.

Pasting law of pushouts

We have the following (,1)(\infty,1)-categorical analog of the familiar pasting law of pushouts in ordinary category theory:

Proposition

A pasting diagram of two squares is a morphism

Δ[2]×Δ[1]C. \Delta[2] \times \Delta[1] \to C \,.

Schematically this looks like

x y z x y z. \array{ x &\to& y &\to& z \\ \downarrow && \downarrow && \downarrow \\ x' &\to& y' &\to& z' } \,.

If the left square is a pushout diagram in CC, then the right square is precisely if the outer square is.

Proof

A proof appears as HTT, lemma 4.4.2.1

Coequalizer

Quotients

Tensoring and cotensoring with an \infty-groupoid

Recap of the 1-categorical situation

An ordinary category with limits is canonically cotensored over Set:

For S,TS, T \in Set and const T:SSetconst_T : S \to Set the diagram parameterized by SS that is constant on TT, we have

lim const TT S. \lim_{\leftarrow} const_T \simeq T^S \,.

Accordingly the cotensoring

() ():Set op×CC (-)^{(-)} : Set^{op} \times C \to C

is defined by

c S:=lim (Sconst cC)= Sc. c^S := \lim_{\leftarrow} (S \stackrel{const_c}{\to} C) = \prod_{S} c \,.

And by continuity of the hom-functor this implies the required natural isomorphisms

Hom C(d,c S)=Hom C(d,lim Sc)lim SHom C(d,c)Set(S,Hom C(d,C)). Hom_C(d,c^S) = Hom_C(d, {\lim_{\leftarrow}}_S c) \simeq {\lim_{\leftarrow}}_S Hom_C(d,c) \simeq Set(S,Hom_C(d,C)) \,.

Correspondingly if CC has colimits, then the tensoring

()():Set×CC (-) \otimes (-) : Set \times C \to C

is given by forming colimits over constant diagrams: Sc:=lim ScS \otimes c := {\lim_{\to}}_S c, and again by continuity of the hom-functor we have the required natural isomorphism

Hom C(Sc,d)=Hom C(lim Sc,d)lim SHom C(c,d)Set(S,Hom C(c,d)). Hom_C(S \otimes c, d) = Hom_C({\lim_{\to}}_S c,d) \simeq {\lim_{\leftarrow}}_S Hom_C(c,d) \simeq Set(S,Hom_C(c,d)) \,.

Of course all the colimits appearing here are just coproducts and all limits just products, but for the generalization to (,1)(\infty,1)-categories this is a misleading simplification, it is really the notion of limit and colimit that matters here.

Definition

We expect for S,TS, T \in ∞Grpd and for const T:SGrpdconst_T : S \to \infty Grpd the constant diagram, that

lim const TT S, \lim_{\leftarrow} const_T \simeq T^S \,,

where on the right we have the internal hom of \infty-groupoids, which is modeled in the model structure on simplicial sets sSet QuillensSet_{Quillen} by the fact that this is a closed monoidal category.

Correspondingly, for CC an (,1)(\infty,1)-category with colimits, it is tensored over ∞Grpd by setting

()():Grpd×CC (-)\otimes (-) : \infty Grpd \times C \to C
Sc:=lim Sc, S \otimes c := {\lim_{\to}}_S c \,,

where now on the right we have the (,1)(\infty,1)-categorical colimit over the constant diagram const:SCconst : S \to C of shape SS on cc.

Then by the (,1)(\infty,1)-continuity of the hom, and using the above characterization of the internal hom in Grpd\infty Grpd we have the required natural equivalence

Hom C(Sc,d)=Hom C(lim Sc,d)lim SHom C(c,d)Grpd(S,Hom C(c,d)). Hom_C(S \otimes c, d) = Hom_C({\lim_{\to}}_S c, d) \simeq {\lim_{\leftarrow}}_S Hom_C(c,d) \simeq \infty Grpd(S,Hom_C(c,d)) \,.

The following proposition should assert that this is all true

Proposition

The (,1)(\infty,1)-categorical colimit lim c{\lim_{\to}} c over the diagram of shape SGrpdS \in \infty Grpd constant on cCc \in C is characterized by the fact that it induces natural equivalences

Hom C(lim Sc,d)Grpd(S,Hom C(c,d)) Hom_C({\lim_{\to}}_S c, d) \simeq \infty Grpd(S, Hom_C(c,d))

for all dCd \in C.

This is essentially HTT, corollary 4.4.4.9.

Corollary

Every ∞-groupoid SS is the (,1)(\infty,1)-colimit in ∞Grpd of the constant diagram on the point over itself:

Slim Sconst *. S \simeq {\lim_{\to}}_S const_* \,.

This justifies the following definition

Definition

For CC an (,1)(\infty,1)-category with colimits, the tensoring of CC over Grpd\infty Grpd is the (,1)(\infty,1)-functor

()():Grpd×CC (-) \otimes (-) : \infty Grpd \times C \to C

given by

Sc=lim (const c:SC). S \otimes c = \lim_{\to} (const_c : S \to C) \,.

See HTT, section 4.4.4.

Models

We discuss models for (,1)(\infty,1)-(co)limits in terms of ordinary category theory and homotopy theory.

Observation

If CC is presented by a simplicial model category AA, in that CA C \simeq A^\circ, then the (,1)(\infty,1)-tensoring and (,1)(\infty,1)-cotensoring of CC over ∞Grpd is modeled by the ordinary tensoring and powering of AA over sSet:

For c^A\hat c \in A cofibant and representing an object cCc \in C and for SsSetS \in sSet any simplicial set, we have an equivalence

cSC^S. c \otimes S \simeq \hat C \cdot S \,.
Proof

The powering in AA satisfies the natural isomorphism

sSet(S,A(c^,d^))A(c^S,d^) sSet(S,A(\hat c,\hat d)) \simeq A(\hat c \cdot S, \hat d)

in sSet.

For c^\hat c a cofibrant and d^\hat d a fibrant representative, we have that both sides here are Kan complexes that are equivalent to the corresponding derived hom spaces in the corresponding (,1)(\infty,1)-category CC, so that this translates into an equivalence

Hom C(cS,d)Grpd(S,Hom C(c,d)). Hom_C(c \cdot S, d) \simeq \infty Grpd(S, Hom_C(c,d)) \,.

The claim then follows from the above proposition.

Limits in over-(,1)(\infty,1)-categories

Proposition

For CC an (,1)(\infty,1)-category, X:DCX : D \to C a diagram, C/XC/X the over-(∞,1)-category and F:KC/XF : K \to C/X a diagram in the over-(,1)(\infty,1)-category, then the (∞,1)-limit lim F\lim_{\leftarrow} F in C/XC/X coincides with the (,1)(\infty,1)-limit lim F/X\lim_{\leftarrow} F/X in CC.

Proof

Modelling CC as a quasi-category we have that C/XC/X is given by the simplicial set

C/X:[n]Hom X([n]D,C), C/X : [n] \mapsto Hom_X([n] \star D, C) \,,

where \star denotes the join of simplicial sets. The limit lim F\lim_{\leftarrow} F is the initial object in (C/X)/F(C/X)/F, which is the quasi-category given by the simplicial set

(C/X)/F:[n]Hom F([n]K,C/X). (C/X)/F : [n] \mapsto Hom_{F}( [n] \star K, C/X) \,.

Since the join preserves colimits of simplicial sets in both arguments, we can apply the co-Yoneda lemma to decompose [n]K=lim [r][n]K[r][n] \star K = {\lim_{\underset{{[r] \to [n]\star K}}{\to}}} [r], use that the hom-functor sends colimits in the first argument to limits and obtain

Hom([n]K,C/X) Hom(lim r[r],C/X) lim rHom([r],C/X) lim rHom F([r]D,C) Hom F(lim r([r]D),C) Hom F((lim r[r])D,C) Hom F(([n]K)D,C) Hom F([n](KD),C). \begin{aligned} Hom([n] \star K, C/X) &\simeq Hom( {\lim_{\to}}_r [r], C/X) \\ & \simeq {\lim_{\leftarrow}}_r Hom([r], C/X) \\ & \simeq {\lim_{\leftarrow}}_r Hom_F( [r] \star D, C ) \\ & \simeq Hom_F({\lim_{\to}}_r ([r] \star D), C ) \\ & \simeq Hom_F( ({\lim_{\to}}_r[r]) \star D, C ) \\ & \simeq Hom_F(([n] \star K) \star D, C) \\ & \simeq Hom_F([n] \star (K \star D), C) \end{aligned} \,.

Here Hom F([r]D,C)Hom_F([r]\star D,C) is shorthand for the hom in the (ordinary) under category sSet D/sSet^{D/} from the canonical inclusion D[r]DD \to [r] \star D to X:DCX : D \to C. Notice that we use the 1-categorical analog of the statement that we are proving here when computing the colimit in this under-category as just the colimit in sSetsSet. We also use that the join of simplicial sets, being given by Day convolution is an associative tensor product.

In conclusion we have an isomorphism of simplicial sets

(C/X)/FC/(X/F) (C/X)/F \simeq C/(X/F)

and therefore the initial objects of these quasi-categories coincide on both sides. This shows that lim F\lim_{\leftarrow} F is computed as an initial object in C/(X/F)C/(X/F).

Limits and colimits with values in Grpd\infty Grpd

Limits and colimits over a (∞,1)-functor with values in the (∞,1)-category ∞-Grpd of ∞-groupoids may be reformulated in terms of the universal fibration of (∞,1)-categories, hence in terms of the (∞,1)-Grothendieck construction.

Let ∞Grpd be the (∞,1)-category of ∞-groupoids. Let the (∞,1)-functor Z| GrpdGrpd opZ|_{Grpd} \to \infty Grpd^{op} be the universal ∞-groupoid fibration whose fiber over the object denoting some \infty-groupoid is that very \infty-groupoid.

Then let XX be any ∞-groupoid and

F:XGrpd F : X \to \infty Grpd

an (∞,1)-functor. Recall that the coCartesian fibration E FXE_F \to X classified by FF is the pullback of the universal fibration of ∞-groupoids Z| GrpdZ|_{Grpd} along F:

E F Z| Grpd X F Grpd \array{ E_F &\to& Z|_{Grpd} \\ \downarrow && \downarrow \\ X &\stackrel{F}{\to}& \infty Grpd }
Proposition

Let the assumptions be as above. Then:

  • The \infty-colimit of FF is equivalent to the (∞,1)-Grothendieck construction E FE_F:

    limFE F \underset{\longrightarrow}{\lim} F \simeq E_F
  • The \infty-limit of FF is equivalent to the ∞-groupoid of sections of E FXE_F \to X

    limΓ X(E F). \underset{\longleftarrow}{\lim} \simeq \Gamma_X(E_F) \,.

The statement for the colimit is corollary 3.3.4.6 in HTT. The statement for the limit is corollary 3.3.3.4.

Remark

The form of the statement in prop. 10 is the special case of the general form of internal (co-)limits, here internal to the (∞,1)-topos ∞Grpd with Core(inftyGrpd small)Core(\inftyGrpd_{small}) its small object classifier. See at internal (co-)limit – Groupoidal homotopy (co-)limits for more on this.

Limits and colimits with values in (,1)(\infty,1)Cat

Proposition

For F:DF : D \to (∞,1)Cat an (∞,1)-functor, its \infty-colimit is given by forming the (∞,1)-Grothendieck construction F\int F of FF and then inverting the Cartesian morphisms.

Formally this means, with respect to the model structure for Cartesian fibrations that there is a natural isomorphism

lim F(F) \lim_\to F \simeq (\int F)^\sharp

in the homotopy category of the presentation of (,1)(\infty,1)-category by marked simplicial sets.

This is HTT, corollary 3.3.4.3.

For the special case that FF takes values in ordinary categories see also at 2-limit the section 2-limits in Cat.

Limits in \infty-functor categories

For CC an ordinary category that admits small limits and colimits, and for KK a small category, the functor category Func(D,C)Func(D,C) has all small limits and colimits, and these are computed objectwise. See limits and colimits by example. The analogous statement is true for an (∞,1)-category of (∞,1)-functors.

Proposition

Let KK and CC be quasi-categories, such that CC has all colimits indexed by KK.

Let DD be a small quasi-category. Then

  • The (∞,1)-category of (∞,1)-functors Func(D,C)Func(D,C) has all KK-indexed colimits;

  • A morphism K Func(D,C)K^\triangleright \to Func(D,C) is a colimiting cocone precisely if for each object dDd \in D the induced morphism K CK^\triangleright \to C is a colimiting cocone.

Proof

This is HTT, corollary 5.1.2.3

References

General

The definition of limit in quasi-categories is due to

  • André Joyal, Quasi-categories and Kan complexes Journal of Pure and Applied Algebra 175 (2002), 207-222.

A brief survey is on page 159 of

A detailed account is in definition 1.2.13.4, p. 48 in

In homotopy type theory

A formalization of some aspects of (,1)(\infty,1)-limits in terms of homotopy type theory is Coq-coded in

See also

Revised on November 4, 2014 11:52:14 by Urs Schreiber (185.26.182.38)