As a reminder, we say an $\infty$-category $\mathcal(C)$). In the literature on $\infty$-categories, a great deal of attention appears to be given to so-called presentable $\infty$-categories. Despite that, I have always been uncomfortable with their existence, although this discomfort is mostly rooted in a personal aesthetic ideal: I see technical digressions on cardinals, universes, and transfiniteness as a stain on otherwise clean mathematical theories. I have never cared much for set-theoretic size issues, for they seem not to cause much trouble in my day-to-day mathematical life. The notion of locally presentable category is, at least roughly, an analogue for categories of the notion of a finitely generated module.Ī locally presentable category 𝒞 \mathcal.Let me start out with a confession. The general idea is that a locally presentable category is a large category generated from small data: from small objects under small colimits. Presentation by combinatorial model categories.Locally presentable ( ∞, 1 ) (\infty,1)-categories.Localizations of ( ∞, 1 ) (\infty,1)-categories.Basic idea in ( ∞, 1 ) (\infty,1)-category theory.Model structure on simplicial presheaves.
#Locally presentable category free
Generation exhibited by epimorphism from a free object. Locally presentable category: generated from colimits over small objects. This page means to give an introduction to the notion of locally presentable category, and its related notions in higher category theory and survey some fundamental properties.įor the first section Basic idea in category theory the reader is assumed to be familiar with basic notions of category theory such as presheaves and colimits.įor the section Basic idea in model category theory the reader is assumed to be familiar with basic notions in model category theory such as cofibrantly generated model categories and homotopy colimits.įor the third section Basic idea in (∞,1)-category theory the reader is assumed to be familiar with basic concepts of (∞,1)-category theory such as (∞,1)-categories of (∞,1)-presheaves and (∞,1)-colimits. Model structure for (2,1)-sheaves/for stacksĮquivalences in/ of ( ∞, 1 ) (\infty,1)-categories Global model structure/ Cech model structure/ local model structure On dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrationsįor weak ∞-categories as weak complicial setsįor ( ∞, 1 ) (\infty,1)-sheaves / ∞ \infty-stacks Joyal model for quasi-categories (and its cubical version)įor stable ( ∞, 1 ) (\infty,1)-categories Related by the monoidal Dold-Kan correspondenceįor L-∞ algebras: on dg-Lie algebras, on dg-coalgebras, on simplicial Lie algebras On dg-algebras and on on simplicial rings/ on cosimplicial rings Model structure on differential graded-commutative superalgebras Model structure on differential-graded commutative algebras On modules over an algebra over an operad On simplicial T-algebras, on homotopy T-algebras Model structure on reduced simplicial sets Model structure on equivariant dgc-algebras Model structure on equivariant chain complexes 
Model structure on cosimplicial simplicial setsįine model structure on topological G-spacesĬoarse model structure on topological G-spacesįor rational equivariant ∞ \infty-groupoids On chain complexes/ model structure on cosimplicial abelian groups On simplicial sets, on semi-simplicial sets Model structure on presheaves over a test category Presentation of ( ∞, 1 ) (\infty,1)-categories Grothendieck construction for model categories ( relative category, homotopical category)Ĭartesian closed model category, locally cartesian closed model category
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