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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 CC, we show that there is a universal way of expanding CC 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

Free (,1)(\infty,1)-cocompletion

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

Definition

Let AA and BB be model categories, DD a plain category and

D r A γ B \array{ D &\stackrel{r}{\to}& A \\ & \searrow_\gamma \\ && B }

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

  1. a Quillen adjunction (LR):ARLB(L \dashv R) : A \stackrel{\overset{L}{\to}}{\underset{R}{\leftarrow}} B

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

    D r A γ η L B. \array{ D &&\stackrel{r}{\to}&& A \\ & \searrow_\gamma &{}^\eta\swArrow& \swarrow_{L} \\ && B } \,.

Let the category of such factorizations have morphisms ((LR),η)((LR),η)((L \dashv R), \eta ) \to ((L' \dashv R'), \eta' ) given by natural transformations LLL \to L' such that for all all objects dDd \in D the diagrams

Lr(d) Lr(d) η d η d γ() \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 LL presents, via its derived functor, a left adjoint (∞,1)-functor that preserves (,1)(\infty,1)-categorical colimits. So the notion of factorization as above is really about factorizations through colimit-preserving (,1)(\infty,1)-functors into (,1)(\infty,1)-categories that have all colimits.

Theorem

(model category presentation of free (,1)(\infty,1)-cocompletion)

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

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

Proof

This is theorem 1.1 in the article.

category: reference

Revised on April 15, 2010 08:44:49 by Urs Schreiber (87.212.203.135)