A category with one object is a monoid. A groupoid with one object is a group. A linear category with one object is a ring.
This may be a trivial question, but what is a model category with one object? Do such things occur "in nature"?
2026-04-06 23:18:35.1775517515
Model category with one object
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A model category has all limits and colimits (or at least all finite limits and colimits, depending on your definition). In particular, it has an initial object, so if there is only one object, it is initial and has only the identity map as an endomorphism. So a model category with one object is just the terminal category which can trivially be seen to admit a unique model structure (the identity must be a weak equivalence, fibration, and cofibration).