Timeline for Pull-backs which are push-outs
Current License: CC BY-SA 2.5
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 8, 2019 at 16:24 | comment | added | Oren Ben-Bassat | perfect, thanks! | |
| Aug 8, 2019 at 9:16 | comment | added | Mike Shulman | @OrenBen-Bassat If you mean the property that a pushout of a mono is also a pullback, I think this follows from the generalized Blakers-Massey theorem arxiv.org/abs/1703.09050. In particular, the "little Blakers-Massey theorem" (Corollary 4.1.4) says that the pushout of $f:A\to B$ and $g:A\to C$ is a pullback if $\Delta f \Box_A \Delta g$ is invertible. Since the diagonal $\Delta f$ of a mono is invertible (essentially by definition of "mono"), and any pushout product $\Box$ of an invertible map is invertible, the result follows. | |
| Aug 7, 2019 at 14:46 | comment | added | Todd Trimble | I'm copying @MikeShulman to alert him to your question (Oren), as he would have a much better chance of saying something meaningful. | |
| Aug 7, 2019 at 14:45 | comment | added | Todd Trimble | @OrenBen-Bassat Sadly, my feeling for higher topos theory is not all that it could or should be. However, ... | |
| Aug 7, 2019 at 14:29 | comment | added | Oren Ben-Bassat | Is there a property like this in Higher Topos Theory, using homotopy monomorphisms? | |
| Aug 16, 2012 at 2:38 | comment | added | Todd Trimble | @Colin: Yes, I see that point. | |
| Aug 15, 2012 at 16:58 | comment | added | Colin McLarty | @Todd: His point on the cat list is that there nothing new in the common theory of pretoposes and ab cats. As to elementary properties, the common ground of any two first order theories is just their disjunction. As to Horn properties or AE properties every product of pretoposes and abelian categories is a model. Conversely the interesting logical fact is that nothing else happens in this theory. Merely multiplying by 0 separates any model into one ab cat and one pretopos. this is the same trick as multiplying by idempotents in commutative algebra or in topos theory. | |
| Aug 15, 2012 at 12:22 | comment | added | Todd Trimble | @Colin: Interesting. Where does he say that? | |
| Aug 14, 2012 at 16:08 | comment | added | Colin McLarty | Actually Freyd intended his argument to show that AT categories are not interesting, and are a special case of a much more general uninteresting situation. | |
| Mar 10, 2011 at 22:22 | comment | added | Steve Lack | Fernando, pretoposes do not have the first property. In fact even the category Set does not have this property: consider the case where $C=0$. | |
| Mar 10, 2011 at 15:02 | comment | added | Fernando Muro | Hi Todd, thanks for your answer. I've had a look at your links and I've found that pretoposes have the second property I mentioned. Do they also have the first one? | |
| Mar 9, 2011 at 16:16 | comment | added | Todd Trimble | I've often wondered about that too, Mike. I don't have an answer to that. | |
| Mar 9, 2011 at 15:58 | comment | added | Mike Shulman | One might argue that the idea of AT category is so neat that it destroys itself as an object to be studied: since any AT category splits into an abelian category and a pretopos, why study AT categories rather than just abelian categories and pretopoi separately? | |
| Mar 9, 2011 at 15:09 | comment | added | Todd Trimble | Heh, thanks Peter! I was delighted to have this question fall into my lap, as it were. The idea of AT category is very cool, but I'm not aware that much has been done with it since those discussions between Freyd and Pratt. | |
| Mar 9, 2011 at 14:23 | vote | accept | Fernando Muro | ||
| Mar 9, 2011 at 14:04 | comment | added | Peter LeFanu Lumsdaine | The question could almost have been written as a deliberate feedline to give this wonderful answer a home on MO! | |
| Mar 9, 2011 at 13:56 | history | answered | Todd Trimble | CC BY-SA 2.5 |