Timeline for Products of representables are regular on a regular skeletal Reedy category?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 10, 2018 at 2:28 | vote | accept | Harry Gindi | ||
| Oct 9, 2018 at 7:55 | answer | added | Harry Gindi | timeline score: 1 | |
| Jul 23, 2018 at 22:44 | history | edited | Harry Gindi | CC BY-SA 4.0 |
added 112 characters in body
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| Jul 23, 2018 at 22:22 | comment | added | Tim Campion | Good catch. That simplifies things (and makes your question nontrivial!)-- it just says that all the morphisms in $\mathcal A_+$ are monomorphisms. So all told, we have a Reedy category $\mathcal A$ where the morphisms in $\mathcal A_+$ are monomorphisms, the morphisms in $\mathcal A_-$ are split epimorphisms, and a morphism in $\mathcal A_-$ is determined by its sections. | |
| Jul 23, 2018 at 22:00 | comment | added | Harry Gindi | Also, for the category to be (pre)regular, that regularity property only needs to hold for representables, not all presheaves. | |
| Jul 23, 2018 at 21:58 | comment | added | Tim Campion | Drat! There's a typo in my first comment. In the second-to-last sentence, it should say "normale" rather than "reguliere". | |
| Jul 23, 2018 at 21:54 | comment | added | Tim Campion | Note that a categorie skeletique normale is the same thing as a Cisinski-generalized Reedy category which happens to be an ordinary Reedy category. So all told, a regular skeletal Reedy category is a Reedy category which satisfies the above condition (every nondegenerate map $A \to X$ in $\hat{\mathcal A}$ with $A$ representable is a monomorphism), as well as the condition from Definition 2.4 from the nlab page linked to above (to ensure that it is a categorie skeletique). | |
| Jul 23, 2018 at 21:51 | comment | added | Tim Campion | Let's see. "Regular skeletal Reedy category" is Cisinski's categorie skeletique reguliere. A categorie skeletique reguliere is a categorie skeletique normale $\mathcal A$ such that every nondegenerate map $A \to X$ in $\hat{\mathcal A}$ with $A$ representable is a monomorphism. A categorie skeletique reguliere in turn is a categorie skeletique with no nonidentity isomorphisms. And a categorie skeletique is what the nlab calls a Cisinski-generalized Reedy category. | |
| Jul 23, 2018 at 21:19 | history | asked | Harry Gindi | CC BY-SA 4.0 |