Timeline for What should be required from a model category so that the category of algebraic objects in it has the natural model structure?
Current License: CC BY-SA 4.0
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| Dec 17, 2023 at 14:49 | history | edited | David White | CC BY-SA 4.0 |
Fixed minor typo, grammar.
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| Dec 17, 2023 at 14:21 | vote | accept | Arshak Aivazian | ||
| Dec 17, 2023 at 12:45 | answer | added | David White | timeline score: 3 | |
| Feb 7, 2023 at 18:57 | comment | added | Arshak Aivazian | @ZhenLin Of course, I mean the adjunction between $T[M]$ and $M^S$, where $T$ is an algebraic theory with a set of sorts $S$. The forgetting functor returns a tuple of all carriers. I really hoped that adjunction and monadicity were the same here, thanks, glad to hear that! | |
| Feb 6, 2023 at 22:59 | comment | added | Zhen Lin | The obstruction to monadicity is, almost always in practice, the existence of the left adjoint. Aside from that it is not hard to ensure that the conditions of the strict monadicity theorem are satisfied: so, indeed, the category of algebras of a single sorted Lawvere theory in abelian category will be monadic over the abelian category iff the left adjoint exists. The issue with multi sorted Lawvere theories is that there is no canonical candidate for a forgetful functor, so your question does not quite make sense in that context. | |
| Feb 6, 2023 at 19:44 | comment | added | Arshak Aivazian | Although the Cartesian-closure condition is somewhat restrictive, since excludes Abelian categories. Does the category of chain complexes in a good abelian category satisfy the condition under discussion? | |
| Feb 6, 2023 at 18:43 | comment | added | Arshak Aivazian | Thank you very much, this is very helpful! Then "bicomplete cartesian-closed category" is my answer to the first question (because it is natural sufficient condition for being a monoidally cocomplete category as defined by Todd Trimble). | |
| Feb 6, 2023 at 18:26 | comment | added | varkor | For your first question, see this discussion by Todd Trimble. | |
| Feb 6, 2023 at 17:46 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
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| Feb 6, 2023 at 17:39 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
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| Feb 6, 2023 at 17:22 | history | asked | Arshak Aivazian | CC BY-SA 4.0 |