equivalences in/of $(\infty,1)$-categories
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.
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 $C_{/F}$:
(well defined up to a ontractible space of choices).
A colimit in a quasi-category is accordingly an limit in the opposite quasi-category.
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
See HTT, prop 4.2.1.5.
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
is an acyclic Kan fibration.
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
between the slice (∞,1)-categories over $F$ (the $(\infty,1)$-category of $\infty$-cones over $F$) and over just $\underset{\leftarrow}{\lim}F$.
Let $\tilde F \colon \Delta^0 \star \mathcal{K} \to \mathcal{C}$ be the limiting cone. The canonical cospan of $\infty$-functors
induces a span of slice $\infty$-categories
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).
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, $const : K \to Func(K,C)$.
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
is a weak equivalence in $sSet_{Quillen}$.
But first consider the following pointwise characterization.
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
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).
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
Under this identification, $\phi_Y$ is the morphism
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.
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:
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.
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.
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
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 $\eta$.
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
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
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.
The quasi-category $N(A^\circ)$ presented by a combinatorial simplicial model category $A$ has all small quasi-categorical limits and colimits.
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.
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.
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
for every object $x \in X^{\triangleleft}$ (including the cone point) the restricted diagram $p_x : Y^{\triangleleft} \to C$ is a limit diagram;
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.
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 $\lim_x \lim_y F(x,y)$.
…
See also (∞,1)-pullback.
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
A sqare in a quasi-category $C$ is an image of this in $C$, i.e. a morphism
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:
and
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
is the pushout of the diagram $F$.
We have the following $(\infty,1)$-categorical analog of the familiar pasting law of pushouts in ordinary category theory:
A pasting diagram of two squares is a morphism
Schematically this looks like
If the left square is a pushout diagram in $C$, then the right square is precisely if the outer square is.
A proof appears as HTT, lemma 4.4.2.1
…
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
Accordingly the cotensoring
is defined by
And by continuity of the hom-functor this implies the required natural isomorphisms
Correspondingly if $C$ has colimits, then the tensoring
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
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.
We expect for $S, T \in$ ∞Grpd and for $const_T : S \to \infty Grpd$ the constant diagram, that
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
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
The following proposition should assert that this is all true
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
for all $d \in C$.
This is essentially HTT, corollary 4.4.4.9.
Every ∞-groupoid $S$ is the $(\infty,1)$-colimit in ∞Grpd of the constant diagram on the point over itself:
This justifies the following definition
For $C$ an $(\infty,1)$-category with colimits, the tensoring of $C$ over $\infty Grpd$ is the $(\infty,1)$-functor
given by
See HTT, section 4.4.4.
We discuss models for $(\infty,1)$-(co)limits in terms of ordinary category theory and homotopy theory.
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
The powering in $A$ satisfies the natural isomorphism
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
The claim then follows from the above 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 $C$.
Modelling $C$ as a quasi-category we have that $C/X$ is given by the simplicial set
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
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
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
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 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
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:
Let the assumptions be as above. Then:
The $\infty$-colimit of $F$ is equivalent to the (∞,1)-Grothendieck construction $E_F$:
The $\infty$-limit of $F$ is equivalent to the ∞-groupoid of sections of $E_F \to X$
The statement for the colimit is corollary 3.3.4.6 in HTT. The statement for the limit is corollary 3.3.3.4.
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 $Core(\inftyGrpd_{small})$ its small object classifier. See at internal (co-)limit – Groupoidal homotopy (co-)limits for more on this.
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
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.
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.
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.
This is HTT, corollary 5.1.2.3
$(\infty,1)$-limit
The definition of limit in quasi-categories is due to
A brief survey is on page 159 of
A detailed account is in definition 1.2.13.4, p. 48 in
A formalization of some aspects of $(\infty,1)$-limits in terms of homotopy type theory is Coq-coded in
See also