nLab filtered homotopy colimit

Idea

homotopy colimits of filtered diagrams

Characterization

Proposition

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

• $\kappa$-filtered colimits preserve weak equivalences;

• hence $\kappa$-filtered colimits are already homotopy colimits.

This appears as (Dugger 00, prop 7.3).

Proof

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

Let $F, G : J \to C$ be $\kappa$-filtered diagrams in $C$ and $F \to G$ a natural transformation that is degreewise a weak equivalence. Using the functorial factorization provided by the small object argument this may be factored as $F \to H \to G$ 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

$$\lim_\to F \stackrel{\simeq}{\hookrightarrow} \lim_\to H \stackrel{}{\to}\lim_\to G \,.$$

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 $H(j) \to G(j)$ are acyclic fibrations.

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

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

So given a diagram

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

we are guaranteed, by the $\kappa$-smallness of $X$ and $Y$ that we established above, a lift

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

into some component at $j \in J$ and hence a lift

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

Thereby $\lim_\to H \to \lim_\to G$ is in $rlp(I) \subset W$.

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.

Related concepts

• filtered (∞,1)-colimit

References

• Daniel Dugger, Combinatorial model categories have presentations (arXiv:math/0007068)