[Paper Review] A Giraud-type characterization of the simplicial categories associated to closed model categories as $\infty$-pretopoi

Carlos Simpson|ArXiv.org|Mar 29, 1999

Homotopy and Cohomology in Algebraic TopologyMathematics20 references18 citations

TL;DR

This paper establishes a Giraud-type characterization theorem for simplicial categories arising from cofibrantly generated closed model categories, showing they are precisely those admitting small homotopy colimits and generated by a small set of objects under these colimits. The key contribution is identifying such categories as ∞-pretopoi, with ∞-topoi defined as those satisfying additional exactness conditions, thereby generalizing Giraud’s theorem to the homotopical setting.

ABSTRACT

Theorem (after Giraud, SGA 4): Suppose $A$ is a simplicial category. The following conditions are equivalent: (i) There is a cofibrantly generated closed model category $M$ such that $A$ is equivalent to the Dwyer-Kan simplicial localization $L(M)$; (ii) $A$ admits all small homotopy colimits, and there is a small subset of objects of $A$ which are $A$-small, and which generate $A$ by homotopy colimits; (iii) There exists a small 1-category $C$ and a morphism $g:C o A$ sending objects of $C$ to $A$-small objects, which induces a fully faithful inclusion $i:A o \hat{C}$, such that $i$ admits a left homotopy-adjoint $ψ$. We call a Segal category $A$ which satisfies these equivalent conditions, an $\infty$-pretopos. Note that (i) implies that $A$ admits all small homotopy limits too. If furthermore there exists $C o A$ as in (iii) such that the adjoint $ψ$ preserves finite homotopy limits, then we say that $A$ is an ``$\infty$-topos''.

Motivation & Objective

• To generalize Giraud’s theorem from 1-categories to the homotopical setting of simplicial categories.
• To characterize which simplicial categories arise as Dwyer-Kan localizations of cofibrantly generated closed model categories.
• To introduce and formalize the notion of an ∞-pretopos as a homotopical analogue of a category of sheaves.
• To distinguish ∞-pretopoi from ∞-topoi by adding exactness conditions.
• To provide a foundational framework for ∞-topos theory using model category theory and homotopy coherent category theory.

Proposed method

• Use of Dwyer-Kan simplicial localization L(M) to associate a simplicial category to any closed model category M.
• Adoption of homotopy colimits as the primary homotopical analogue of colimits in classical Giraud’s theorem.
• Introduction of the notion of A-small objects and generation by homotopy colimits as internal conditions on the simplicial category A.
• Application of the small object argument and model category axioms (via [16] Lemma 2.5) to construct a new model structure M from a given category C and a functor ψ.
• Construction of a model category M with the same underlying category and cofibrations as N (Heller’s model of simplicial presheaves), but with weak equivalences defined via a functor ψ to A.
• Verification that the localization of L(M) recovers A, thereby proving equivalence between A and L(M) for a cofibrantly generated M.

Experimental results

Research questions

• RQ1 Which simplicial categories arise as Dwyer-Kan localizations of cofibrantly generated closed model categories?
• RQ2 Can the conditions for a simplicial category to be equivalent to L(M) be characterized by intrinsic, internal properties?
• RQ3 What is the homotopical analogue of Giraud’s theorem for categories of sheaves in the context of ∞-categories?
• RQ4 How do homotopy colimits and small generation relate to the structure of ∞-pretopoi?
• RQ5 What additional exactness conditions are needed to define ∞-topoi within the class of ∞-pretopoi?

Key findings

• A simplicial category A is equivalent to L(M) for some cofibrantly generated closed model category M if and only if A admits all small homotopy colimits and is generated by a small set of A-small objects under these colimits.
• The existence of small homotopy colimits and small generation implies the existence of small homotopy limits, though not full exactness in the classical sense.
• The class of such categories is termed ∞-pretopoi, with ∞-topoi defined as ∞-pretopoi satisfying additional exactness conditions.
• The proof constructs a model category M on the category of simplicial presheaves on a site C, with weak equivalences defined via a functor ψ to A, such that L(M) ≅ A.
• The model structure on M is cofibrantly generated and satisfies all axioms of a closed model category, relying on the small object argument and closure under retracts for weak equivalences.
• The localization of L(N) at the M-weak equivalences recovers A, confirming that A ≅ L(M) and completing the characterization.