nLab
filtered homotopy colimit

Contents

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

(,1)(\infty,1)-Limits and colimits

Contents

Idea

homotopy colimits of filtered diagrams?

Characterization

Proposition

In a combinatorial model category, for every sufficiently large regular cardinal κ\kappa the following holds:

This appears as (Dugger 00, prop 7.3).

Proof

The point is to choose κ\kappa such that all domains and codomains of the generating cofibrations are κ\kappa-compact object. This is possible since by assumption that CC is a locally presentable category all its objects are small objects, hence each a λ\lambda-compact object for some cardinal λ\lambda. Take κ\kappa to be the maximum of these.

Let F,G:JCF, G : J \to C be κ\kappa-filtered diagrams in CC and FGF \to G a natural transformation that is degreewise a weak equivalence. Using the functorial factorization provided by the small object argument this may be factored as FHGF \to H \to G where the first transformation is objectwise an acyclic cofibration and the second objectwise an acyclic fibration, and by functoriality of the factorization this sits over a factorization

lim Flim Hlim G. \lim_\to F \stackrel{\simeq}{\hookrightarrow} \lim_\to H \stackrel{}{\to}\lim_\to G \,.

It remains to show that the second morphism is a weak equivalence. But by our factorization and by 2-out-of-3 applied to our componentwise weak equivalences, we have that all its components H(j)G(j)H(j) \to G(j) are acyclic fibrations.

At small object it is described in detail how κ\kappa-smallness of an object XX implies that morphisms from XX into a κ\kappa-filtered colimit lift to some component of the colimit

H(j1) H(j) H(j+1) f^ X f lim H. \array{ \cdots&\to&H(j-1) &\to& H(j) &\to& H(j+1) &\to& \cdots \\ &&&{}^{\mathllap{\exists \hat f}}\nearrow&\downarrow & \swarrow \\ &&X& \stackrel{\forall f}{\to} &\lim_\to H } \,.

So given a diagram

X lim H I Y lim G \array{ X &\to& \lim_\to H \\ \downarrow^{\mathrlap{\in I}} && \downarrow \\ Y &\to& \lim_\to G }

we are guaranteed, by the κ\kappa-smallness of XX and YY that we established above, a lift

X H(j) lim H I rlp(I) Y G(j) lim G \array{ X &\to& H(j) &\to& \lim_\to H \\ \downarrow^{\mathrlap{\in I}} && \downarrow^{\in \mathrlap{\in rlp(I)}} && \downarrow \\ Y &\to& G(j) &\to& \lim_\to G }

into some component at jJj \in J and hence a lift

X H(j) lim H I rlp(I) Y G(j) lim G. \array{ X &\to& H(j) &\to& \lim_\to H \\ \downarrow^{\mathrlap{\in I}} & \nearrow & \downarrow^{\in \mathrlap{\in rlp(I)}} && \downarrow \\ Y &\to& G(j) &\to& \lim_\to G } \,.

Thereby lim Hlim G\lim_\to H \to \lim_\to G is in rlp(I)Wrlp(I) \subset W.

Corollary

In the situation of prop , since finite homotopy limits are given, after fibrant resolution, by finite limits, it follows from the ordinary commutativity of filtered colimits with finite limits that also filtered homotopy colimits commute with finite homotopy limits.

References

Last revised on May 20, 2014 at 02:33:32. See the history of this page for a list of all contributions to it.