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Universal Homotopy Theories

This entry is about the article

• Daniel Dugger, Universal homotopy theories, Advances in Mathematics 164, 144–176 (2001) (arXiv:math/0007070)

  Abstract Given a small category $C$, we show that there is a universal way of expanding $C$ into a model category, essentially by formally adjoining homotopy colimits. The technique of localization becomes a method for imposing “relations” into these universal gadgets. The paper develops this formalism and discusses various applications, for instance to the study of homotopy colimits, the Dwyer-Kan theory of framings, and to the homotopy theory of schemes.

The article discusses the projective global and local model structure on simplicial presheaves.

The general discussion is in parts is based on the unfinished but useful notes

• Dan Dugger, Sheaves and homotopy theory (web, dvi, pdf)

In particular this

• shows that and how the projective model structure on simplicial presheaves on $C$ is something like the free completion of $C$ under homotopy colimits. This is the model-category theoretic analog of the fact that the (∞,1)-category of (∞,1)-presheaves on $C$ is the
  free (∞,1)-cocompletion of $C$;

• announces the statement of Dugger’s theorem which states that every combinatorial model category arises as the left Bousfield localization of the projective global model structure on simplicial presheaves. The full proof of this is in the companion article

  • Daniel Dugger, Combinatorial model categories have presentations

• discusses cofibrant replacement in the projective model structures (necessarily both for the global as well as for the local structure)

• discusses geometric realization of $\mathrm{\infty }$-stacks on Diff essentially by using the geometric homotopy $\mathrm{\infty }$-groupoid functor as disscussed at homotopy groups in an (∞,1)-topos.

Free $(\mathrm{\infty },1)$-cocompletion

The main theorem of the article is the following. For more details see (∞,1)-category of (∞,1)-presheaves.

Definition

Let $A$ and $B$ be model categories, $D$ a plain category and

$$\begin{array}{ccc}D& \stackrel{r}{\to }& A\\ & {\searrow }_{\gamma }\\ & & B\end{array}$$\array{ D &\stackrel{r}{\to}& A \\ & \searrow_\gamma \\ && B }

two plain functors. Say that a model-category theoretic factorization of $\gamma$ through $A$ is

1. a Quillen adjunction $(L⊣R):A\stackrel{\stackrel{L}{\to }}{\underset{R}{\leftarrow }}B$

2. a natural weak equivalence $\eta :L\circ r\to \gamma$

   $$\begin{array}{ccccc}D& & \stackrel{r}{\to }& & A\\ & {\searrow }_{\gamma }& {}^{\eta }⇙& {\swarrow }_L\\ & & B\end{array}.$$\array{ D &&\stackrel{r}{\to}&& A \\ & \searrow_\gamma &{}^\eta\swArrow& \swarrow_{L} \\ && B } \,.

Let the category of such factorizations have morphisms $((L⊣R),\eta )\to ((L\prime ⊣R\prime ),\eta \prime )$ given by natural transformations $L\to L\prime$ such that for all all objects $d\in D$ the diagrams

$$\begin{array}{ccccc}L\circ r(d)& & \to & & L\prime \circ r(d)\\ & {}_{{\eta }_d}\searrow & & {\swarrow }_{\eta {\prime }_d}\\ & & \gamma ()\end{array}$$\array{ L\circ r(d) &&\to&& L'\circ r(d) \\ & {}_{\eta_{d}}\searrow && \swarrow_{\eta'_{d}} \\ && \gamma() }

commutes.

Notice that the (∞,1)-category presented by a model category – at least by a combinatorial model category – has all (∞,1)-categorical colimits, and that the Quillen left adjoint functor $L$ presents, via its derived functor, a left adjoint (∞,1)-functor that preserves $(\mathrm{\infty },1)$-categorical colimits. So the notion of factorization as above is really about factorizations through colimit-preserving $(\mathrm{\infty },1)$-functors into $(\mathrm{\infty },1)$-categories that have all colimits.

Theorem

(model category presentation of free $(\mathrm{\infty },1)$-cocompletion)

For $C$ a small category, the projective global model structure on simplicial presheaves $[C^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj}}$ on $C$ is universal with respect to such factorizations of functors out of $C$:

every functor $C\to B$ to any model category $B$ has a factorization through $[C^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj}}$ as above, and the category of such factorizations is contractible.

Proof

This is theorem 1.1 in the article.

category: reference