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Any category $\mathcal C$ with finite limits and finite colimits can be seen as a model category where the weak equivalences are the isomorphisms, and the fibrations and cofibrations are all maps.

The nlab called it the trivial model structure on $\mathcal C$. But trivial is already overloaded, especially in model categories (with trivial cofibrations, trivial fibrations). Another name that could fit is discrete as in such a category homotopy is equality: the spaces deduced from the hom-sets of the model category are discrete. But again, discrete already denotes a set seen as a category; in particular discrete model category is ambiguous.

Is there any other established name for this model structure?

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  • $\begingroup$ I've only ever seen it called the trivial model structure. I'm not sure it matters so much since the model structure tells you nothing you didn't already know. $\endgroup$ – Randall Aug 15 '17 at 13:29
  • $\begingroup$ @Randall My situation is quite unusual : I have model categories that I want to consider sometimes with the trivial model structure for a bit before returning to their plain model structure. So I need a name to be precise about which model structure I'm considering. $\endgroup$ – Pece Aug 15 '17 at 13:33
  • $\begingroup$ Understood. I just looked in my own papers (not that I should be trusted) and I've used the term trivial. If I did that, it's probably because Hovey or Hirschhorn did, and they are actual trustworthy authorities. $\endgroup$ – Randall Aug 15 '17 at 13:47
  • $\begingroup$ @Randall Thanks. I guess I'll use trivial then, and be careful to use acyclic (co)fibrations to keep things clear. (Because of course, I also have Grothendieck fibrations, so I let you imagine the mess if I say "trivial fibration" without much context.) $\endgroup$ – Pece Aug 15 '17 at 13:50

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