nLab cellular category

Idea

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.

Definition

Definition

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

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

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

such that

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

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

Examples

Example

(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).

Related concepts

• cell complex, relative cell complex

• cofibrantly generated model category

• model structure on functors

• transferred model structure

• the notions of cellular categories and of cellular model categories are vaguely related, but not as systematically as the (independently chosen) terminology might suggest

References

The notion is due to:

• M. Makkai, J. Rosický, Cellular categories, J. Pure Appl. Alg. 218 (2014) 1652-1664 [arXiv:1304.7572, doi:10.1016/j.jpaa.2014.01.005]