nLab cellular category



Category theory

Limits and colimits



A cellular category in the sense of Makkai & Rosický 2014 is a (cocomplete) category equipped with a minimum of extra information indicating that some of its morphisms behave like relative cell complexes.



(Makkai & Rosický 2014, Def. 2.1)
A cellular category is:

  1. a cocomplete category 𝒞\mathcal{C},

  2. a class of morphisms Cell(𝒞)Mor(𝒞)Cell(\mathcal{C}) \,\subset\, Mor(\mathcal{C}),

such that

  1. Cell(𝒞)Cell(\mathcal{C}) contains all isomorphisms,

  2. Cell(𝒞)Cell(\mathcal{C}) is closed under pushouts and transfinite composition.



(model categories with their cofibrations)
Given a model category (𝒞,(W,Fib,Cof))\big(\mathcal{C}, (\mathrm{W},\, Fib, Cof) \big), then – understood as equipped (just) with its cofibrations CofCof or (just) with its acyclic cofibrations CofWCof \cap \mathrm{W} – it is a cellular category in the sense of Def. .

More to the point, if such a model category is cofibrantly generated by generating sets ICofI \,\subset\, Cof (and JCofWJ \,\subset\, Cof \cap \mathrm{W}), then 𝒞\mathcal{C} equipped with the closure of II under pushout and transfinite composition is cellular (by construction) and in this case the class Cell()Cell(-) is that which one ordinarily addresses as relative cell complexes with attaching maps in II. (The full class of cofibrations is the re-obtained by further closing the class of relative cell complexes under forming retracts).


The notion is due to:

Created on August 17, 2022 at 14:10:12. See the history of this page for a list of all contributions to it.