# nLab limit in a quasi-category

### Context

#### $\left(\infty ,1\right)$-Category theory

(∞,1)-category theory

## Models

#### Limits and colimits

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

$\underset{←}{\mathrm{lim}}F≔\mathrm{TerminalObj}\left({C}_{/F}\right)$\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 $♢$ – of join, which yield results that differ as simplicial sets, though are equivalent as quasi-categories.

The notation ${C}_{/F}$ denotes the definition of over quasi-category induced from $*$, while the notation ${C}^{/F}$ denotes that induced from $♢$. 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

$\underset{←}{\mathrm{lim}}F≔\mathrm{TerminalObj}\left({C}^{/F}\right)\phantom{\rule{thinmathspace}{0ex}}.$\underset{\leftarrow}{\lim} F \coloneqq TerminalObj(C^{/F}) \,.

## Properties

### In terms of slice $\infty$-categories

###### Proposition

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

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

is an acyclic Kan fibration.

###### Proposition

For $F:𝒦\to 𝒞$ a diagram in an $\left(\infty ,1\right)$-category and $\underset{←}{\mathrm{lim}}F$ its limit, there is a natural equivalence of (∞,1)-categories

${𝒞}_{/F}\simeq {𝒞}_{/\underset{←}{\mathrm{lim}}F}$\mathcal{C}_{/F} \simeq \mathcal{C}_{/\underset{\leftarrow}{\lim} F}

between the slice (∞,1)-categories over $F$ (the $\left(\infty ,1\right)$-category of $\infty$-cones over $F$) and over just $\underset{←}{\mathrm{lim}}F$.

###### Proof

Let $\stackrel{˜}{F}:{\Delta }^{0}\star 𝒦\to 𝒞$ be the limiting cone. The canonical cospan of $\infty$-functors

$*\to {\Delta }^{0}\star 𝒦←𝒦$\ast \to \Delta^0 \star \mathcal{K} \leftarrow \mathcal{K}

induces a span of slice $\infty$-categories

${𝒞}_{/\underset{←}{\mathrm{lim}}F}←{𝒞}_{/\stackrel{˜}{F}}\to {𝒞}_{/F}\phantom{\rule{thinmathspace}{0ex}}.$\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 $\left(\infty ,1\right)$-categories by quasi-categories. But more intrinsically in $\left(\infty ,1\right)$-category theory, it should be true that there is an adjunction characterization of $\left(\infty ,1\right)$-limits : limit and colimit, should be (pointwise or global) right and left adjoint (infinity,1)-functor of the constant diagram $\left(\infty ,1\right)$-functor, $\mathrm{const}:K\to \mathrm{Func}\left(K,C\right)$.

$\left(\mathrm{colim}⊣\mathrm{const}⊣\mathrm{lim}\right):\mathrm{Func}\left(K,C\right)\stackrel{\stackrel{\mathrm{lim}}{\to }}{\stackrel{\stackrel{\mathrm{const}}{←}}{\underset{\mathrm{colim}}{\to }}}\mathrm{Func}\left(*,C\right)\simeq C\phantom{\rule{thinmathspace}{0ex}}.$(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 $\eta :{\mathrm{Id}}_{\mathrm{Func}\left(K,C\right)}\to \mathrm{const}\mathrm{colim}$ in $\mathrm{Func}\left(\mathrm{Func}\left(K,C\right),\mathrm{Func}\left(K,C\right)\right)$ such that for every $f\in \mathrm{Func}\left(K,C\right)$ and $Y\in C$ the induced morphism

${\mathrm{Hom}}_{C}\left(\mathrm{colim}\left(f\right),Y\right)\to {\mathrm{Hom}}_{\mathrm{Func}\left(K,C\right)}\left(\mathrm{const}\mathrm{colim}\left(f\right),\mathrm{const}Y\right)\stackrel{\mathrm{Hom}\left(\eta ,\mathrm{const}Y\right)}{\to }{\mathrm{Hom}}_{\mathrm{Func}\left(K,Y\right)}\left(f,\mathrm{const}Y\right)$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 ${\mathrm{sSet}}_{\mathrm{Quillen}}$.

But first consider the following pointwise characterization.

###### Proposition

Let $C$ be a quasi-category, $K$ a simplicial set. A co-cone diagram $\overline{p}:K\star \Delta \left[0\right]\to C$ with cone point $X\in C$ is a colimiting diagram (an initial object in ${C}_{p/}$) precisely if for every object $Y\in C$ the morphism

${\varphi }_{Y}:{\mathrm{Hom}}_{C}\left(X,Y\right)\to {\mathrm{Hom}}_{\mathrm{Func}\left(K,C\right)}\left(p,\mathrm{const}Y\right)$\phi_Y : Hom_C(X,Y) \to Hom_{Func(K,C)}(p, const Y)

induced by the morpism $p\to \mathrm{const}X$ that is encoded by $\overline{p}$ 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 ${\mathrm{Hom}}_{\mathrm{Func}\left(K,C\right)}\left(p,\mathrm{const}Y\right)$ is given (up to equivalence) by the pullback ${C}^{p/}{×}_{C}\left\{Y\right\}$ in sSet.

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

We have that ${\mathrm{Hom}}_{\mathrm{Func}\left(K,C\right)}\left(p,\mathrm{const}Y\right)$ is given by $\left({C}^{K}{\right)}^{p/}{×}_{{C}^{K}}\mathrm{const}Y$. We compute

$\begin{array}{rl}\left(\left({C}^{K}{\right)}^{p/}{×}_{{C}^{K}}\mathrm{const}Y{\right)}_{n}& ={\mathrm{Hom}}_{\Delta \left[0\right]/\mathrm{sSet}}\left(\Delta \left[0\right]♢\Delta \left[n\right],{C}^{K}\right){×}_{\left({C}^{K}{\right)}_{n}}\left\{\mathrm{const}Y\right\}\\ & ={\mathrm{Hom}}_{\Delta \left[0\right]/\mathrm{sSet}}\left(\Delta \left[0\right]\coprod _{\Delta \left[n\right]}\Delta \left[n\right]×\Delta \left[1\right],{C}^{K}\right){×}_{\left({C}^{K}{\right)}_{n}}\left\{\mathrm{const}Y\right\}& =\left\{p\right\}{×}_{\mathrm{Hom}\left(\Delta \left[0\right],{C}^{K}\right)}\mathrm{Hom}\left(\Delta \left[0\right],{C}^{K}\right){×}_{\mathrm{Hom}\left(\Delta \left[n\right],{C}^{K}\right)}\mathrm{Hom}\left(\Delta \left[n\right]×\Delta \left[1\right],{C}^{K}\text{Unknown character}\right){×}_{\mathrm{Hom}\left(\Delta \left[n\right],{C}^{K}\right)}\left\{\mathrm{const}Y\right\}\\ & =\left\{p\right\}{×}_{\mathrm{Hom}\left(K,C\right)}\mathrm{Hom}\left(K,C\right){×}_{\mathrm{Hom}\left(\Delta \left[n\right]×K,C\right)}\mathrm{Hom}\left(\Delta \left[n\right]×K×\Delta \left[1\right],C\right){×}_{\mathrm{Hom}\left(\Delta \left[n\right]×K,C\right)}\mathrm{Hom}\left(\Delta \left[n\right],C\right){×}_{\Delta \left[n\right],C}\left\{Y\right\}\\ & =\left\{p\right\}{×}_{\mathrm{Hom}\left(K,C\right)}\mathrm{Hom}\left(K♢\Delta \left[n\right],C\right){×}_{\mathrm{Hom}\left(\Delta \left[n\right],C\right)}\left\{Y\right\}\\ & =\left({C}^{p/}{×}_{C}\left\{Y\right\}{\right)}_{n}\end{array}$\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, ${\varphi }_{Y}$ is the morphism

$\left({C}^{X/}\stackrel{\varphi \prime }{\to }{C}^{\overline{p}/}\stackrel{\varphi ″}{\to }{C}^{p/}\right){×}_{C}\left\{Y\right\}\phantom{\rule{thinmathspace}{0ex}},$\left( C^{X/} \stackrel{\phi'}{\to} C^{\bar p/} \stackrel{\phi''}{\to} C^{p/} \right) \times_C \{Y\} \,,

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

The morphism $\varphi ″$ is a left fibration (using HTT, prop. 4.2.1.6)

One finds that the morphism $\varphi ″$ is a left fibration.

The strategy for the completion of the proof is: realize that the first condition of the proposition is equivalent to $\varphi ″$ being an acyclic Kan fibration, and the second statement equivalent to $\varphi {″}_{Y}$ 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 $\kappa$ be a regular cardinal. An (∞,1)-category $C$ has all $\kappa$-small limits precisely if it has equalizers and $\kappa$-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 $\left(\infty ,1\right)$-categorical limits. In particular, under sending the Kan-complex enriched categories $C$ to quasi-categories $N\left(C\right)$ 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 $C$ and $J$ be Kan complex-enriched categories and let $F:J\to C$ be an sSet-enriched functor.

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

$\overline{N\left(F\right)}:N\left(J{\right)}^{▹}\to N\left(C\right)$\bar {N(F)} : N(J)^{\triangleright} \to N(C)

is a quasi-categorical colimit diagram in $N\left(C\right)$.

Here $N$ is the homotopy coherent nerve, $N\left(J{\right)}^{▹}$ the join of quasi-categories with the point, $N\left(F\right)$ the image of the simplicial functor $F$ under the homotopy coherent nerve and $\overline{N\left(F\right)}$ its extension to the join determined by the cocone maps $\eta$.

###### 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 $A$ a combinatorial simplicial model category, $S$ a quasi-category and $\tau \left(S\right)$ the corresponding $\mathrm{sSet}$-category under the left adjoint of the homotopy coherent nerve, we have an equivalence of quasi-categories

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

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

With this and the discussion at homotopy Kan extension, we find that the cocone components $\eta$ induce for each $a\in \left[C,\mathrm{sSet}\right]$ a homotopy equivalence

$C\left(c,a\right)\stackrel{}{\to }\left[{J}^{\mathrm{op}},C\right]\left(jF,\mathrm{const}a\right)$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 $N\left({A}^{\circ }\right)$ presented by a combinatorial simplicial model category $A$ has all small quasi-categorical limits and colimits.

###### Proof

This is HTT, 4.2.4.8.

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

### Commutativity of limits

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

###### Proposition

Let $X$ and $Y$ be simplicial sets and $C$ a quasi-category. Let $p:{X}^{◃}×{Y}^{◃}\to C$ be a diagram. If

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

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

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

###### Proof

This is HTT, lemma 5.5.2.3

In other words, suppose that ${\mathrm{lim}}_{x}F\left(x,y\right)$ exists for all $y$ and ${\mathrm{lim}}_{y}F\left(x,y\right)$ exists for all $x$ and also that ${\mathrm{lim}}_{y}\left({\mathrm{lim}}_{x}F\left(x,y\right)\right)$ exists, then this object is also ${\mathrm{lim}}_{x}{\mathrm{lim}}_{y}F\left(x,y\right)$.

## Examples

### $\infty$-Limits of special shape

#### Pullback / Pushout

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

$\begin{array}{ccc}\left(0,0\right)& \to & \left(1,0\right)\\ ↓& ↘& ↓\\ \left(0,1\right)& \to & \left(1,1\right)\end{array}$\array{ (0,0) &\to& (1,0) \\ \downarrow &\searrow& \downarrow \\ (0,1) &\to& (1,1) }

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

$s:\Delta \left[1\right]×\Delta \left[1\right]\to C\phantom{\rule{thinmathspace}{0ex}}.$s : \Delta[1] \times \Delta[1] \to C \,.

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

$\Delta \left[1\right]×\Delta \left[1\right]\simeq \left\{v\right\}\star \Lambda \left[2{\right]}_{2}=\left(\begin{array}{ccc}v& \to & 1\\ ↓& ↘& ↓\\ 0& \to & 2\end{array}\right)$\Delta[1] \times \Delta[1] \simeq \{v\} \star \Lambda[2]_2 = \left( \array{ v &\to& 1 \\ \downarrow &\searrow& \downarrow \\ 0 &\to& 2 } \right)

and

$\Delta \left[1\right]×\Delta \left[1\right]\simeq \Lambda \left[2{\right]}_{0}\star \left\{v\right\}=\left(\begin{array}{ccc}0& \to & 1\\ ↓& ↘& ↓\\ 2& \to & v\end{array}\right)\phantom{\rule{thinmathspace}{0ex}}.$\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 $\Delta \left[1\right]×\Delta \left[1\right]\simeq \Lambda \left[2{\right]}_{0}\star \left\{v\right\}\to C$ exhibits $\left\{v\right\}\to C$ as a colimit over $F:\Lambda \left[2{\right]}_{0}\to C$, we say the colimit

$v:=\underset{\to }{\mathrm{lim}}F:=F\left(1\right)\coprod _{F\left(0\right)}F\left(2\right)$v := \lim_\to F := F(1) \coprod_{F(0)} F(2)

is the pushout of the diagram $F$.

##### Pasting law of pushouts

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

###### Proposition

A pasting diagram of two squares is a morphism

$\Delta \left[2\right]×\Delta \left[1\right]\to C\phantom{\rule{thinmathspace}{0ex}}.$\Delta[2] \times \Delta[1] \to C \,.

Schematically this looks like

$\begin{array}{ccccc}x& \to & y& \to & z\\ ↓& & ↓& & ↓\\ x\prime & \to & y\prime & \to & z\prime \end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ x &\to& y &\to& z \\ \downarrow && \downarrow && \downarrow \\ x' &\to& y' &\to& z' } \,.

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

###### Proof

A proof appears as HTT, lemma 4.4.2.1

#### Tensoring and cotensoring with an $\infty$-groupoid

##### Recap of the 1-categorical situation

An ordinary category with limits is canonically cotensored over Set:

For $S,T\in$ Set and ${\mathrm{const}}_{T}:S\to \mathrm{Set}$ the diagram parameterized by $S$ that is constant on $T$, we have

$\underset{←}{\mathrm{lim}}{\mathrm{const}}_{T}\simeq {T}^{S}\phantom{\rule{thinmathspace}{0ex}}.$\lim_{\leftarrow} const_T \simeq T^S \,.

Accordingly the cotensoring

$\left(-{\right)}^{\left(-\right)}:{\mathrm{Set}}^{\mathrm{op}}×C\to C$(-)^{(-)} : Set^{op} \times C \to C

is defined by

${c}^{S}:=\underset{←}{\mathrm{lim}}\left(S\stackrel{{\mathrm{const}}_{c}}{\to }C\right)=\prod _{S}c\phantom{\rule{thinmathspace}{0ex}}.$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

${\mathrm{Hom}}_{C}\left(d,{c}^{S}\right)={\mathrm{Hom}}_{C}\left(d,{\underset{←}{\mathrm{lim}}}_{S}c\right)\simeq {\underset{←}{\mathrm{lim}}}_{S}{\mathrm{Hom}}_{C}\left(d,c\right)\simeq \mathrm{Set}\left(S,{\mathrm{Hom}}_{C}\left(d,C\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$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 $C$ has colimits, then the tensoring

$\left(-\right)\otimes \left(-\right):\mathrm{Set}×C\to C$(-) \otimes (-) : Set \times C \to C

is given by forming colimits over constant diagrams: $S\otimes c:={{\mathrm{lim}}_{\to }}_{S}c$, and again by continuity of the hom-functor we have the required natural isomorphism

${\mathrm{Hom}}_{C}\left(S\otimes c,d\right)={\mathrm{Hom}}_{C}\left({\underset{\to }{\mathrm{lim}}}_{S}c,d\right)\simeq {\underset{←}{\mathrm{lim}}}_{S}{\mathrm{Hom}}_{C}\left(c,d\right)\simeq \mathrm{Set}\left(S,{\mathrm{Hom}}_{C}\left(c,d\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$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 $\left(\infty ,1\right)$-categories this is a misleading simplification, it is really the notion of limit and colimit that matters here.

##### Definition

We expect for $S,T\in$ ∞Grpd and for ${\mathrm{const}}_{T}:S\to \infty \mathrm{Grpd}$ the constant diagram, that

$\underset{←}{\mathrm{lim}}{\mathrm{const}}_{T}\simeq {T}^{S}\phantom{\rule{thinmathspace}{0ex}},$\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 ${\mathrm{sSet}}_{\mathrm{Quillen}}$ by the fact that this is a closed monoidal category.

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

$\left(-\right)\otimes \left(-\right):\infty \mathrm{Grpd}×C\to C$(-)\otimes (-) : \infty Grpd \times C \to C
$S\otimes c:={\underset{\to }{\mathrm{lim}}}_{S}c\phantom{\rule{thinmathspace}{0ex}},$S \otimes c := {\lim_{\to}}_S c \,,

where now on the right we have the $\left(\infty ,1\right)$-categorical colimit over the constant diagram $\mathrm{const}:S\to C$ of shape $S$ on $c$.

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

${\mathrm{Hom}}_{C}\left(S\otimes c,d\right)={\mathrm{Hom}}_{C}\left({\underset{\to }{\mathrm{lim}}}_{S}c,d\right)\simeq {\underset{←}{\mathrm{lim}}}_{S}{\mathrm{Hom}}_{C}\left(c,d\right)\simeq \infty \mathrm{Grpd}\left(S,{\mathrm{Hom}}_{C}\left(c,d\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$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 $\left(\infty ,1\right)$-categorical colimit ${\mathrm{lim}}_{\to }c$ over the diagram of shape $S\in \infty \mathrm{Grpd}$ constant on $c\in C$ is characterized by the fact that it induces natural equivalences

${\mathrm{Hom}}_{C}\left({\underset{\to }{\mathrm{lim}}}_{S}c,d\right)\simeq \infty \mathrm{Grpd}\left(S,{\mathrm{Hom}}_{C}\left(c,d\right)\right)$Hom_C({\lim_{\to}}_S c, d) \simeq \infty Grpd(S, Hom_C(c,d))

for all $d\in C$.

This is essentially HTT, corollary 4.4.4.9.

###### Corollary

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

$S\simeq {\underset{\to }{\mathrm{lim}}}_{S}{\mathrm{const}}_{*}\phantom{\rule{thinmathspace}{0ex}}.$S \simeq {\lim_{\to}}_S const_* \,.

This justifies the following definition

###### Definition

For $C$ an $\left(\infty ,1\right)$-category with colimits, the tensoring of $C$ over $\infty \mathrm{Grpd}$ is the $\left(\infty ,1\right)$-functor

$\left(-\right)\otimes \left(-\right):\infty \mathrm{Grpd}×C\to C$(-) \otimes (-) : \infty Grpd \times C \to C

given by

$S\otimes c=\underset{\to }{\mathrm{lim}}\left({\mathrm{const}}_{c}:S\to C\right)\phantom{\rule{thinmathspace}{0ex}}.$S \otimes c = \lim_{\to} (const_c : S \to C) \,.
##### Models

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

###### Observation

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

For $\stackrel{^}{c}\in A$ cofibant and representing an object $c\in C$ and for $S\in \mathrm{sSet}$ any simplicial set, we have an equivalence

$c\otimes S\simeq \stackrel{^}{C}\cdot S\phantom{\rule{thinmathspace}{0ex}}.$c \otimes S \simeq \hat C \cdot S \,.
###### Proof

The powering in $A$ satisfies the natural isomorphism

$\mathrm{sSet}\left(S,A\left(\stackrel{^}{c},\stackrel{^}{d}\right)\right)\simeq A\left(\stackrel{^}{c}\cdot S,\stackrel{^}{d}\right)$sSet(S,A(\hat c,\hat d)) \simeq A(\hat c \cdot S, \hat d)

in sSet.

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

${\mathrm{Hom}}_{C}\left(c\cdot S,d\right)\simeq \infty \mathrm{Grpd}\left(S,{\mathrm{Hom}}_{C}\left(c,d\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$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-$\left(\infty ,1\right)$-categories

###### Proposition

For $C$ an $\left(\infty ,1\right)$-category, $X:D\to C$ a diagram, $C/X$ the over-(∞,1)-category and $F:K\to C/X$ a diagram in the over-$\left(\infty ,1\right)$-category, then the (∞,1)-limit ${\mathrm{lim}}_{←}F$ in $C/X$ coincides with the $\left(\infty ,1\right)$-limit ${\mathrm{lim}}_{←}F/X$ in $C$.

###### Proof

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

$C/X:\left[n\right]↦{\mathrm{Hom}}_{X}\left(\left[n\right]\star D,C\right)\phantom{\rule{thinmathspace}{0ex}},$C/X : [n] \mapsto Hom_X([n] \star D, C) \,,

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

$\left(C/X\right)/F:\left[n\right]↦{\mathrm{Hom}}_{F}\left(\left[n\right]\star K,C/X\right)\phantom{\rule{thinmathspace}{0ex}}.$(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 $\left[n\right]\star K={\mathrm{lim}}_{\underset{\left[r\right]\to \left[n\right]\star K}{\to }}\left[r\right]$, use that the hom-functor sends colimits in the first argument to limits and obtain

$\begin{array}{rl}\mathrm{Hom}\left(\left[n\right]\star K,C/X\right)& \simeq \mathrm{Hom}\left({\underset{\to }{\mathrm{lim}}}_{r}\left[r\right],C/X\right)\\ & \simeq {\underset{←}{\mathrm{lim}}}_{r}\mathrm{Hom}\left(\left[r\right],C/X\right)\\ & \simeq {\underset{←}{\mathrm{lim}}}_{r}{\mathrm{Hom}}_{F}\left(\left[r\right]\star D,C\right)\\ & \simeq {\mathrm{Hom}}_{F}\left({\underset{\to }{\mathrm{lim}}}_{r}\left(\left[r\right]\star D\right),C\right)\\ & \simeq {\mathrm{Hom}}_{F}\left(\left({\underset{\to }{\mathrm{lim}}}_{r}\left[r\right]\right)\star D,C\right)\\ & \simeq {\mathrm{Hom}}_{F}\left(\left(\left[n\right]\star K\right)\star D,C\right)\\ & \simeq {\mathrm{Hom}}_{F}\left(\left[n\right]\star \left(K\star D\right),C\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 ${\mathrm{Hom}}_{F}\left(\left[r\right]\star D,C\right)$ is shorthand for the hom in the (ordinary) under category ${\mathrm{sSet}}^{D/}$ from the canonical inclusion $D\to \left[r\right]\star D$ to $X:D\to C$. 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 $\mathrm{sSet}$. 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

$\left(C/X\right)/F\simeq C/\left(X/F\right)$(C/X)/F \simeq C/(X/F)

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

### Limits and colimits with values in $\infty \mathrm{Grpd}$

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 $Z{\mid }_{\mathrm{Grpd}}\to \infty {\mathrm{Grpd}}^{\mathrm{op}}$ be the universal ∞-groupoid fibration whose fiber over the object denoting some $\infty$-groupoid is that very $\infty$-groupoid.

Then let $X$ be any ∞-groupoid and

$F:X\to \infty \mathrm{Grpd}$F : X \to \infty Grpd

an (∞,1)-functor. Recall that the coCartesian fibration ${E}_{F}\to X$ classified by $F$ is the pullback of the universal fibration of (∞,1)-categories $Z$ along F:

$\begin{array}{ccc}{E}_{F}& \to & Z{\mid }_{\mathrm{Grpd}}\\ ↓& & ↓\\ X& \stackrel{F}{\to }& \infty \mathrm{Grpd}\end{array}$\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 $F$ is equivalent to the (∞,1)-Grothendieck construction ${E}_{F}$:

$\underset{\to }{\mathrm{lim}}F\simeq {E}_{F}$\lim_\to F \simeq E_F
• The $\infty$-limit of $F$ is equivalent to the (∞,1)-groupoid of sections? of ${E}_{F}\to X$

${\Gamma }_{X}\left({E}_{F}\right)\simeq \mathrm{lim}F\phantom{\rule{thinmathspace}{0ex}}.$\Gamma_X(E_F) \simeq lim F \,.
###### Proof

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

### Limits and colimits with values in $\left(\infty ,1\right)$Cat

###### Proposition

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

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

$\underset{\to }{\mathrm{lim}}F\simeq \left(\int F{\right)}^{♯}$\lim_\to F \simeq (\int F)^\sharp

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

This is HTT, corollary 3.3.4.3.

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

### Limits in $\infty$-functor categories

For $C$ an ordinary category that admits small limits and colimits, and for $K$ a small category, the functor category $\mathrm{Func}\left(D,C\right)$ 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 $K$ and $C$ be quasi-categories, such that $C$ has all colimits indexed by $K$.

Let $D$ be a small quasi-category. Then

• The (∞,1)-category of (∞,1)-functors $\mathrm{Func}\left(D,C\right)$ has all $K$-indexed colimits;

• A morphism ${K}^{▹}\to \mathrm{Func}\left(D,C\right)$ is a colimiting cocone precisely if for each object $d\in D$ the induced morphism ${K}^{▹}\to C$ is a colimiting cocone.

###### Proof

This is HTT, corollary 5.1.2.3

• limit

• 2-limit

• $\left(\infty ,1\right)$-limit

## 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 $\left(\infty ,1\right)$-limits in terms of homotopy type theory is Coq-coded in