this post was submitted on 19 May 2025
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There is only one model structure that can be put on the category of small categories for which the weak equivalences coincide with honest equivalences of categories. It's called the Joyal-Tierney model structure. You can define the suspension of an object in any model category as the homotopy pushout to two terminals, then define an abstract notion of a sphere in any model category by setting the 0-sphere as the coproduct of two terminals and the (n+1)-sphere as the suspension of the n-sphere.
A small category is a CW-complex if and only if it is a groupoid.
I have absolutely no idea what you said. But I have a really awesome friend named Tierney.