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:
a cocomplete category $\mathcal{C}$,
a class of morphisms $Cell(\mathcal{C}) \,\subset\, Mor(\mathcal{C})$,
such that
$Cell(\mathcal{C})$ contains all isomorphisms,
$Cell(\mathcal{C})$ is closed under pushouts and transfinite composition.
(model categories with their cofibrations)
Given a model category $\big(\mathcal{C}, (\mathrm{W},\, Fib, Cof) \big)$, then – understood as equipped (just) with its cofibrations $Cof$ or (just) with its acyclic cofibrations $Cof \cap \mathrm{W}$ – 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 $I \,\subset\, Cof$ (and $J \,\subset\, Cof \cap \mathrm{W}$), then $\mathcal{C}$ equipped with the closure of $I$ under pushout and transfinite composition is cellular (by construction) and in this case the class $Cell(-)$ is that which one ordinarily addresses as relative cell complexes with attaching maps in $I$. (The full class of cofibrations is the re-obtained by further closing the class of relative cell complexes under forming retracts).
the notions of cellular categories and of cellular model categories are vaguely related, but not as systematically as the (independently chosen) terminology might suggest
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.