Model Category Structure Research Articles

Function complexes for diagrams of simplicial sets

Let S be the category of simplicial sets, let D be a small category and let S D denote the category of D-diagrams of simplicial sets. Then S D admits a closed simplicial model category structure and the aim of this note is to show that, for every cofibrant diagram X ϵ S D and every fibrant diagram Y ϵS D, the homotopy type of the function complex hom( X, Y) can be computed as a homotopy inverse limit involving function complexes in S between the simplicial sets that appear in X and Y.

Remarks on the closed model category structure of simplicial spectra

Remarks on the closed model category structure of simplicial spectra

Global matrix factorizations

We study matrix factorization and curved module categories for Landau-Ginzburg models (X,W) with X a smooth variety, extending parts of the work of Dyckerhoff. Following Positselski, we equip these categories with model category structures. Using results of Rouquier and Orlov, we identify compact generators. Via To\en's derived Morita theory, we identify Hochschild cohomology with derived endomorphisms of the diagonal curved module; we compute the latter and get the expected result. Finally, we show that our categories are smooth, proper when the singular locus of W is proper, and Calabi-Yau when the total space X is Calabi-Yau.

The homotopy theory of coalgebras over simplicial comonads

We apply the Acyclicity Theorem of Hess, Kerdziorek, Riehl, and Shipley (recently corrected by Garner, Kedziorek, and Riehl) to establishing the existence of model category structure on categories of coalgebras over comonads arising from simplicial adjunctions, under mild conditions on the adjunction and the associated comonad. We study three concrete examples of such adjunctions where the left adjoint is comonadic and show that in each case the component of the derived counit of the comparison adjunction at any fibrant object is an isomorphism, while the component of the derived unit at any 1-connected object is a weak equivalence. To prove this last result, we explain how to construct explicit fibrant replacements for 1-connected coalgebras in the image of the canonical comparison functor from the Postnikov decompositions of their underlying simplicial sets. We also show in one case that the derived unit is precisely the Bousfield-Kan completion map.

Local and stable homological algebra in Grothendieck abelian categories

We define model category structures on the category of chain complexes over a Grothendieck abelian category depending on the choice of a generating family, and we study their behaviour with respect to tensor products and stabilization. This gives convenient tools to construct and understand triangulated categories of motives.