Limits and colimits
limits and colimits
limit and colimit
limits and colimits by example
commutativity of limits and colimits
connected limit, wide pullback
preserved limit, reflected limit, created limit
product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum
end and coend
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.
In terms of slice -categories
For a diagram in an -category and its limit, there is a natural equivalence of (∞,1)-categories
between the slice (∞,1)-categories over (the -category of -cones over ) and over just .
Let be the limiting cone. The canonical cospan of -functors
induces a span of slice -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. 22.214.171.124).
This appears for instance in (Lurie, proof of prop. 126.96.36.199).
In terms of -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 -categories by quasi-categories. But more intrinsically in -category theory, it should be true that there is an adjunction characterization of -limits : limit and colimit, should be (pointwise or global) right and left adjoint (infinity,1)-functor of the constant diagram -functor, .
By the discussion at adjoint (∞,1)-functor (HTT, prop. 188.8.131.52) this requires exhibiitng a morphism in such that for every and the induced morphism
is a weak equivalence in .
But first consider the following pointwise characterization.
Let be a quasi-category, a simplicial set. A co-cone diagram with cone point is a colimiting diagram (an initial object in ) precisely if for every object the morphism
induced by the morpism that is encoded by is an equivalence (i.e. a homotopy equivalence of Kan complexes).
This is HTT, lemma 184.108.40.206.
The key step is to realize that is given (up to equivalence) by the pullback in sSet.
Here is a detailed way to see this, using the discussion at hom-object in a quasi-category.
We have that is given by . We compute
Under this identification, is the morphism
in sSet where is a section of the map , (which one checks is an acyclic Kan fibration) obtained by choosing composites of the co-cone components with a given morphism .
The morphism is a left fibration (using HTT, prop. 220.127.116.11)
One finds that the morphism is a left fibration.
The strategy for the completion of the proof is: realize that the first condition of the proposition is equivalent to being an acyclic Kan fibration, and the second statement equivalent to 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:
This is HTT, prop. 18.104.22.168.
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 -categorical limits. In particular, under sending the Kan-complex enriched categories to quasi-categories 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.
Here is the homotopy coherent nerve, the join of quasi-categories with the point, the image of the simplicial functor under the homotopy coherent nerve and its extension to the join determined by the cocone maps .
This is HTT, theorem 22.214.171.124
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 combinatorial simplicial model category, a quasi-category and the corresponding -category under the left adjoint of the homotopy coherent nerve, we have an equivalence of quasi-categories
and the same is trued for itself replaced by a chunk? .
With this and the discussion at homotopy Kan extension, we find that the cocone components induce for each 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 -hom adjunctions.
Since -categories equivalent to those of the form for a combinatorial simplicial model category are precisely the locally presentable (∞,1)-categories, it follows from this in particular that every locally presentable -category has all limits and colimits.
Commutativity of limits
The following proposition says that if for an -functor limits (colimits) over each of the two variables exist separately, then they commute.
Let and be simplicial sets and a quasi-category. Let be a diagram. If
for every object (including the cone point) the restricted diagram is a limit diagram;
for every object (not including the cone point) the restricted diagram is a limit diagram;
then, with denoting the cone point of , the restricted diagram, is also a limit diagram.
In other words, suppose that exists for all and exists for all and also that exists, then this object is also .
-Limits of special shape
Pullback / Pushout
See also (∞,1)-pullback.
The non-degenerate cells of the simplicial set obtained as the cartesian product of the simplicial 1-simplex with itself look like
A sqare in a quasi-category is an image of this in , i.e. a morphism
The simplicial square is isomorphic, as a simplicial set, to the join of simplicial sets of a 2-horn with the point:
If a square exhibits as a colimit over , we say the colimit
is the pushout of the diagram .
Pasting law of pushouts
We have the following -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 , then the right square is precisely if the outer square is.
Tensoring and cotensoring with an -groupoid
Recap of the 1-categorical situation
An ordinary category with limits is canonically cotensored over Set:
For Set and the diagram parameterized by that is constant on , we have
Accordingly the cotensoring
is defined by
And by continuity of the hom-functor this implies the required natural isomorphisms
Correspondingly if has colimits, then the tensoring
is given by forming colimits over constant diagrams: , 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 -categories this is a misleading simplification, it is really the notion of limit and colimit that matters here.
We expect for ∞Grpd and for the constant diagram, that
where on the right we have the internal hom of -groupoids, which is modeled in the model structure on simplicial sets by the fact that this is a closed monoidal category.
Correspondingly, for an -category with colimits, it is tensored over ∞Grpd by setting
where now on the right we have the -categorical colimit over the constant diagram of shape on .
Then by the -continuity of the hom, and using the above characterization of the internal hom in we have the required natural equivalence
The following proposition should assert that this is all true
The -categorical colimit over the diagram of shape constant on is characterized by the fact that it induces natural equivalences
for all .
This is essentially HTT, corollary 126.96.36.199.
Every ∞-groupoid is the -colimit in ∞Grpd of the constant diagram on the point over itself:
This justifies the following definition
For an -category with colimits, the tensoring of over is the -functor
See HTT, section 4.4.4.
We discuss models for -(co)limits in terms of ordinary category theory and homotopy theory.
If is presented by a simplicial model category , in that , then the -tensoring and -cotensoring of over ∞Grpd is modeled by the ordinary tensoring and powering of over sSet:
For cofibant and representing an object and for any simplicial set, we have an equivalence
The powering in satisfies the natural isomorphism
For a cofibrant and a fibrant representative, we have that both sides here are Kan complexes that are equivalent to the corresponding derived hom spaces in the corresponding -category , so that this translates into an equivalence
The claim then follows from the above proposition.
Limits in over--categories
For an -category, a diagram, the over-(∞,1)-category and a diagram in the over--category, then the (∞,1)-limit in coincides with the -limit in .
Modelling as a quasi-category we have that is given by the simplicial set
where denotes the join of simplicial sets. The limit is the initial object in , 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 , use that the hom-functor sends colimits in the first argument to limits and obtain
Here is shorthand for the hom in the (ordinary) under category from the canonical inclusion to . 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 . 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 is computed as an initial object in .
Limits and colimits with values in
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 be the universal ∞-groupoid fibration whose fiber over the object denoting some -groupoid is that very -groupoid.
Then let be any ∞-groupoid and
an (∞,1)-functor. Recall that the coCartesian fibration classified by is the pullback of the universal fibration of ∞-groupoids along F:
Let the assumptions be as above. Then:
The -colimit of is equivalent to the (∞,1)-Grothendieck construction :
The -limit of is equivalent to the ∞-groupoid of sections of
The statement for the colimit is corollary 188.8.131.52 in HTT. The statement for the limit is corollary 184.108.40.206.
Limits and colimits with values in Cat
This is HTT, corollary 220.127.116.11.
For the special case that takes values in ordinary categories see also at 2-limit the section 2-limits in Cat.
Limits in -functor categories
For an ordinary category that admits small limits and colimits, and for a small category, the functor category 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 and be quasi-categories, such that has all colimits indexed by .
Let be a small quasi-category. Then
The (∞,1)-category of (∞,1)-functors has all -indexed colimits;
A morphism is a colimiting cocone precisely if for each object the induced morphism is a colimiting cocone.
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 18.104.22.168, p. 48 in
In homotopy type theory
A formalization of some aspects of -limits in terms of homotopy type theory is Coq-coded in