Sheafifiable homotopy model categories

By Tibor Beke. Mathematical Proceedings of the Cambridge Philosophical Society, 2000

242 citations recorded by the source corpus.

If a Quillen model category can be specified using a certain logical syntax (intuitively, ‘is algebraic/combinatorial enough’), so that it can be defined in any category of sheaves, then the satisfaction of Quillen's axioms over any site is a purely formal consequence of their being satisfied over the category of sets. Such data give rise to a functor from the category of topoi and geometric morphisms to Quillen model categories and Quillen adjunctions.

Subfields: Geometry and Topology, Mathematical Physics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/43eb59fd11447a78ce7d33090ae3f3f4a906da28

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Citations in the corpus

On left and right model categories and left and right Bousfield localizations (2010, background): “Many of the results here have satisfactory proofs in print; the first section of [1] in particular is a very nice reference.”
Homotopy Theory of Higher Categories: THE MODEL STRUCTURE (2010, background): “Of course, these model structures have a long history going back to Bousfield-Kan and others, more recently considered by Hirschhorn [116], Smith (see Beke [27], Dugger [84]), Blander [41], Barwick [16].”
Kaplansky classes and derived categories (2007, background): “Proof G is locally κ-presentable for some regular cardinal κ by Proposition 3.10 of [ 3 ].”
T-motives (2016, background): “Finally, remark that we have a natural model category structure on the category of chain complexes over any Grothendieck abelian category where cofibrations are the monomorphisms and weak equivalences the quasi-isomorphisms (see [18] and [10]).”
Homotopy theory of comodules over a Hopf algebroid (2003, background): “Now, if A happens to be a Grothendieck abelian category, then it is automatically complete and cocomplete, and every object in A is -presented, and so a fortiori -small, for some . This latter statement is an immediate corollary of the fact that Grothendieck abelian categories are locally presentabl”
Weak complicial sets I. Basic homotopy theory (2008, methodology): “Combining this with the lifting property established above, it follows that p also has the RLP with respect to the thin horn inclusion Λ0[2]′ ⊆r Δ0[2]′′.”
Mapping spaces in Quasi-categories (2009, background): “The interested reader may consult Beke [1] for full details.”
Left-determined model categories and universal homotopy theories (2003, methodology): “(a) there are vertices b1 ∈ U ∗B, b2 ∈ U ∗B − (U ∗f)(U ∗A), an edge e in B from”
Hochschild cohomology of abelian categories and ringed spaces (2004, methodology): “A very efficient proof for the existence of the latter has been given in [6] by Beke.”
Approximations and adjoints in homotopy categories (2010, background): “In particular, the category DpAq has small Hom-sets.”
Cotorsion pairs, model category structures, and representation theory (2002, methodology): “Let ChA denote the category of unbounded chain complexes on A. The injective model structure on ChA was first constructed by Joyal [Joy84], and written down by Beke [ Bek00 ].”
Dendroidal sets and simplicial operads (2011, methodology): “Moreover, the canonical interval Δ[1] is naturally endowed with a structure of cocommutative counital comonoid (the comultiplication is the diagonal Δ[1] −→ Δ[1] × Δ[1], while the counit is the unique map Δ[1] −→ Δ[0]), so that the existence of this model category structure on the category sOper C f”
Model structures on commutative monoids in general model categories (2014, background): “It is clear from this description of CAlg(R) that ifM is combinatorial thenCAlg(R) is combinatorial (see [3], 2.”