Timeline for Writing categories as slice categories
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Sep 23, 2016 at 22:35 | answer | added | fosco | timeline score: 2 | |
| Sep 23, 2016 at 22:21 | comment | added | Mathemologist | That's exactly what I was looking for, thank you for your time! (It's evident, but I want to note that I would upvote your comments if I had the reputation to do so.) | |
| Sep 23, 2016 at 22:18 | comment | added | Todd Trimble | Mathemologist: what base category? I suppose, given categories $D$ and $C$, one can ask about circumstances under which $D$ is equivalent to a slice of $C$, and these might most usefully take the form of suitable comonadic functors $G: D \to C$ (i.e., equivalent to a forgetful functor of the form $\Sigma_X: C/X \to C$, which is comonadic). For this one should consult (co)monadicity theorems, together with conditions which ensure that the relevant comonad is of the form $X \times -: C \to C$ (possibly something to do with creation/reflection of connected limits). | |
| Sep 23, 2016 at 18:55 | comment | added | Mathemologist | @JohnWiltshire-Gordon That's an interesting question to me as well, and probably captures the nontrivial cases I had in mind. Can we look for functors between the indexed categories and check somehow that they are induced by a morphism of underlying objects? I suppose there is some clever condition that classifies this property. | |
| Sep 23, 2016 at 18:53 | comment | added | Mathemologist | @ToddTrimble What about more interesting examples? Is there something that I can look for in a category which suggests that we can write it as a slice category over a nonterminal object in the base category? | |
| Sep 23, 2016 at 18:02 | comment | added | John Wiltshire-Gordon | Here is what I thought the question would be: "Given a family of categories indexed by the objects of some other category, under what circumstances can these be made the slice categories for some functor?" | |
| Sep 23, 2016 at 18:00 | comment | added | Todd Trimble | But my first comment told you: a category is equivalent to a slice category if and only if it has a terminal object. | |
| Sep 23, 2016 at 17:35 | history | edited | Mathemologist | CC BY-SA 3.0 |
Edit for clarification.
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| Sep 23, 2016 at 17:28 | comment | added | Mathemologist | Yes, I suppose I need to weaken what I am looking for. In the case that $C$ has a terminal object, what other obstructions do we have? I think what I am trying to do is classify what categories are equivalent to slice categories. | |
| Sep 23, 2016 at 17:25 | comment | added | Todd Trimble | Of course, but then you're changing the category $C$ (to something inequivalent to $C$). Not sure what you're driving at... | |
| Sep 23, 2016 at 17:24 | comment | added | Mathemologist | Yes, I was thinking about that, but can't we freely give a category a terminal object if it does not have one? | |
| Sep 23, 2016 at 17:21 | comment | added | Todd Trimble | No, certainly not, since slice categories have a terminal object. But if $C$ does have a terminal object $1$, then trivially $C \simeq C/1$. | |
| Sep 23, 2016 at 16:55 | review | First posts | |||
| Sep 23, 2016 at 17:01 | |||||
| Sep 23, 2016 at 16:54 | history | asked | Mathemologist | CC BY-SA 3.0 |