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James Deikun (Jan 18 2023 at 17:38):I've heard the description of [[relative cell complexes]] as "transfinite compositions of pushouts of coproducts of generating arrows" but I feel like that description is a little vague. Like, the coproducts must be happening in the arrow category, the transfinite compositions must be happening in the base category, but where do the pushouts happen? (Maybe an illustration of how the simplicial infinity-sphere arises as a cell complex would be illuminating?)
Tom Hirschowitz (Jan 18 2023 at 19:53):I think another way to phrase the definition is: relative cell complexes are the smallest class containing generating arrows, and closed under transfinite composition and cobase change (=pushouts) along arbitrary morphisms. (You could add closure under coproduct of arrows, but I think this is redundant.) Closure under cobase change means that, in any pushout square
if is a relative cell complex, then so is . Is this any clearer?
James Deikun (Jan 18 2023 at 22:30):Yes, much!
Reid Barton (Jan 18 2023 at 22:38):Though it is also true and convenient to know that you only have to allow the steps in that order: "transfinite compositions of pushouts of generating arrows".
Reid Barton (Jan 18 2023 at 22:39):(Just because a pushout of a transfinite composition is a transfinite composition of pushouts, and a pushout of a pushout is a pushout.)