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Stream: learning: questions

Topic: Cylinder object model 2-category

Nico Beck (Jun 28 2023 at 12:30):

When $K$ is a model category with a compatible $\operatorname{Cat}$-enrichment , $K$ has tensors and $I$ is the walking isomorphism, will $I*A$ be a cylinder object of $A$? I can factor the fold map as $A\sqcup A \to I*A\to A$, but I don't know how to check the other conditions.

Reid Barton (Jun 28 2023 at 12:39):

Yes provided that $A$ is cofibrant and "compatible Cat-enrichment" is interpreted suitably (as in https://ncatlab.org/nlab/show/enriched+model+category, for a suitable model category structure on Cat).

Reid Barton (Jun 28 2023 at 12:44):

To check the conditions to be a cylinder object, apply the pushout product axiom to: (a) the cofibration $\varnothing \to A$ of $K$ and (b1) the acyclic cofibration $\{0\} \to I$ of Cat, (b2) the cofibration $\{0, 1\} \to I$ of Cat.