158 research outputs found
On homotopy varieties
Given an algebraic theory \ct, a homotopy \ct-algebra is a simplicial set
where all equations from \ct hold up to homotopy. All homotopy \ct-algebras
form a homotopy variety. We give a characterization of homotopy varieties
analogous to the characterization of varieties. We will also study homotopy
models of limit theories which leads to homotopy locally presentable
categories. These were recently considered by Simpson, Lurie, To\"{e}n and
Vezzosi.Comment: Proposition 4.5 is not valid; see Remark 4.5(e) in the new version.
All other results are correct but there are gaps in proofs. They are fixed by
reducing simplicial categories to fibrant ones and replacing homotopy
colimits by fibrant ones, as wel
Are all cofibrantly generated model categories combinatorial?
G. Raptis has recently proved that, assuming Vop\v{e}nka's principle, every
cofibrantly generated model category is Quillen equivalent to a combinatorial
one. His result remains true for a slightly more general concept of a
cofibrantly generated model category. We show that Vop\v{e}nka's principle is
equivalent to this claim. The set-theoretical status of the original Raptis'
result is open
Homotopy locally presentable enriched categories
We develop a homotopy theory of categories enriched in a monoidal model
category V. In particular, we deal with homotopy weighted limits and colimits,
and homotopy local presentability. The main result, which was known for
simplicially-enriched categories, links homotopy locally presentable
V-categories with combinatorial model V-categories, in the case where has all
objects of V are cofibrant.Comment: 48 pages. Significant changes in v2, especially in the last sectio
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