# nLab (∞,1)-limit

Contents

### Context

#### $(\infty,1)$-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 $C_{/F}$:

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

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

A colimit in a quasi-category is accordingly a 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 $C_{/F}$ denotes the definition of over quasi-category induced from $\star$, while the notation $C^{/F}$ 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

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

## Properties

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

###### Proposition

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

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

is an acyclic Kan fibration.

###### Proposition

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

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

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

###### Proof

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

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

induces a span of slice $\infty$-categories

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

The right functor is an equivalence by prop. . 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 $(\infty,1)$-categories by quasi-categories. But more intrinsically in $(\infty,1)$-category theory, it should be true that there is an adjunction characterization of $(\infty,1)$-limits : limit and colimit, should be (pointwise or global) right and left adjoint (infinity,1)-functor of the constant diagram $(\infinity,1)$-functor, $const : K \to Func(K,C)$.

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

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

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

is a weak equivalence in $sSet_{Quillen}$.

But first consider the following pointwise characterization.

###### Proposition

Let $C$ be a quasi-category, $K$ a simplicial set. A co-cone diagram $\bar p : K \star \Delta[0] \to C$ 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

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

induced by the morpism $p \to const X$ that is encoded by $\bar p$ 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 $Hom_{Func(K,C)}(p, const Y)$ is given (up to equivalence) by the pullback $C^{p/} \times_C \{Y\}$ in sSet.

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

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

\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, $\phi_Y$ is the morphism

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

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

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

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

The strategy for the completion of the proof is: realize that the first condition of the proposition is equivalent to $\phi''$ being an acyclic Kan fibration, and the second statement equivalent to $\phi''_Y$ 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 $\kappa$ 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 $(\infty,1)$-categorical limits. In particular, under sending the Kan-complex enriched categories $C$ to quasi-categories $N(C)$ 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 $C$ and $J$ be Kan complex-enriched categories and let $F : J \to C$ be an sSet-enriched functor.

Then a cocone $\{\eta_i : F(i) \to c\}_{i \in J}$ 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

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

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

Here $N$ is the homotopy coherent nerve, $N(J)^{\triangleright}$ the join of quasi-categories with the point, $N(F)$ the image of the simplicial functor $F$ under the homotopy coherent nerve and $\bar{N(F)}$ 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 $\tau(S)$ the corresponding $sSet$-category under the left adjoint of the homotopy coherent nerve, we have an equivalence of quasi-categories

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

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 $a \in [C,sSet]$ a homotopy equivalence

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

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

### Commutativity of limits

The following proposition says that if for an $(\infty,1)$-functor $F : X \times Y \to C$ 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 $p : X^{\triangleleft} \times Y^{\triangleleft} \to C$ be a diagram. If

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

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

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

###### Proof

This is HTT, lemma 5.5.2.3

In other words, suppose that $\lim_x F(x,y)$ exists for all $y$ and $\lim_y F(x,y)$ exists for all $x$ and also that $\lim_y (\lim_x F(x,y))$ exists, then this object is also $\lim_x \lim_y F(x,y)$.

## Examples

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

#### Pullback / Pushout

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

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

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

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

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

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

and

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

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

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

is the pushout of the diagram $F$.

##### Pasting law of pushouts

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

###### Proposition

A pasting diagram of two squares is a morphism

$\Delta[2] \times \Delta[1] \to C \,.$

Schematically this looks like

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

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 $const_T : S \to Set$ the diagram parameterized by $S$ that is constant on $T$, we have

$\lim_{\leftarrow} const_T \simeq T^S \,.$

Accordingly the cotensoring

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

is defined by

$c^S := \lim_{\leftarrow} (S \stackrel{const_c}{\to} C) = \prod_{S} c \,.$

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

$Hom_C(d,c^S) = Hom_C(d, {\lim_{\leftarrow}}_S c) \simeq {\lim_{\leftarrow}}_S Hom_C(d,c) \simeq Set(S,Hom_C(d,C)) \,.$

Correspondingly if $C$ has colimits, then the tensoring

$(-) \otimes (-) : Set \times C \to C$

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

$Hom_C(S \otimes c, d) = Hom_C({\lim_{\to}}_S c,d) \simeq {\lim_{\leftarrow}}_S Hom_C(c,d) \simeq Set(S,Hom_C(c,d)) \,.$

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

##### Definition

We expect for $S, T \in$ ∞Grpd and for $const_T : S \to \infty Grpd$ the constant diagram, that

$\lim_{\leftarrow} const_T \simeq T^S \,,$

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

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

$(-)\otimes (-) : \infty Grpd \times C \to C$
$S \otimes c := {\lim_{\to}}_S c \,,$

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

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

$Hom_C(S \otimes c, d) = Hom_C({\lim_{\to}}_S c, d) \simeq {\lim_{\leftarrow}}_S Hom_C(c,d) \simeq \infty Grpd(S,Hom_C(c,d)) \,.$

The following proposition should assert that this is all true

###### Proposition

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

$Hom_C({\lim_{\to}}_S c, d) \simeq \infty Grpd(S, Hom_C(c,d))$

for all $d \in C$.

This is essentially HTT, corollary 4.4.4.9.

###### Corollary

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

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

This justifies the following definition

###### Definition

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

$(-) \otimes (-) : \infty Grpd \times C \to C$

given by

$S \otimes c = \lim_{\to} (const_c : S \to C) \,.$
##### Models

We discuss models for $(\infty,1)$-(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 $(\infty,1)$-tensoring and $(\infty,1)$-cotensoring of $C$ over ∞Grpd is modeled by the ordinary tensoring and powering of $A$ over sSet:

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

$c \otimes S \simeq \hat C \cdot S \,.$
###### Proof

The powering in $A$ satisfies the natural isomorphism

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

in sSet.

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

$Hom_C(c \cdot S, d) \simeq \infty Grpd(S, Hom_C(c,d)) \,.$

The claim then follows from the above proposition.

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

###### Proposition

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

###### Proof

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

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

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

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

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

\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 $Hom_F([r]\star D,C)$ is shorthand for the hom in the (ordinary) under category $sSet^{D/}$ from the canonical inclusion $D \to [r] \star D$ 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 $sSet$. 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

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

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

### Limits and colimits with values in $\infty Grpd$

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

Then let $X$ be any ∞-groupoid and

$F : X \to \infty Grpd$

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

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

Let the assumptions be as above. Then:

• The $\infty$-colimit of $F$ is equivalent to the (∞,1)-Grothendieck construction $E_F$:

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

$\underset{\longleftarrow}{\lim} \simeq \Gamma_X(E_F) \,.$

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

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

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

in the homotopy category of the presentation of $(\infty,1)$-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 $Func(D,C)$ 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 $K$ 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 $Func(D,C)$ has all $K$-indexed colimits;

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

###### Proof

This is HTT, corollary 5.1.2.3

• 2-limit

• $(\infty,1)$-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

A discussion of weighted $(\infty,1)$-limits is in

• Martina Rovelli, Weighted limits in an (∞,1)-category, 2019, arxiv:1902.00805

### In homotopy type theory

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