Timeline for When is a fold monomorphic/epimorphic
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 10, 2019 at 14:11 | comment | added | Mike Shulman | Well of course you would have to put conditions on $X$ too. | |
| Mar 10, 2019 at 11:45 | comment | added | Valery Isaev | @MikeShulman I mean a sufficient condition on $F$ cannot be of the form "$F$ preserves something", or "$F$ is a polynomial functor", or anything like that since the identity functor satisfies these conditions. Of course, we can consider conditions such as "$F$ does not preserve the initial object", but it seems less natural to me. | |
| Mar 10, 2019 at 4:35 | comment | added | Mike Shulman | However I'm not sure what you mean by "It is hard to imagine some nice sufficient properties of $F$ which are not true for the identity functor." There are plenty of endofunctors for which we want to consider initial algebras that don't preserve the initial object; indeed as you note, if $F0=0$ then $A=0$, so any functor with an interesting initial algebra must not preserve the initial object. For instance, $F(X) = X+1$, whose initial algebra is the natural numbers. | |
| Mar 10, 2019 at 4:33 | comment | added | Mike Shulman | I think in place of local finite presentability, it also suffices to assume $C$ is a Grothendieck topos, or more generally an exhaustive category (ncatlab.org/nlab/show/exhaustive+category), since then $F^\alpha(0)\to X$ is a union of a chain of subobjects of $X$ and hence also a subobject. | |
| Mar 8, 2019 at 23:15 | history | answered | Valery Isaev | CC BY-SA 4.0 |