Timeline for Are inclusions "canonical" injections?
Current License: CC BY-SA 4.0
22 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 11, 2018 at 14:17 | vote | accept | Kevin Buzzard | ||
| Dec 10, 2018 at 22:03 | answer | added | user44191 | timeline score: 3 | |
| Dec 10, 2018 at 20:34 | comment | added | user44191 | Also: if your set-universe is the constructible universe, you can put a total order on all sets (constructing it through transfinite induction, for example). Then the set of all order-respecting injections $f: A \rightarrow Y, A \in \text{Card}$ is "good". | |
| Dec 10, 2018 at 20:01 | comment | added | user44191 | Is there any specific reason why, in condition 3), you assume $g$ is good? If $h$ is injective, then $f$ is injective. | |
| Dec 10, 2018 at 18:20 | answer | added | Mike Shulman | timeline score: 7 | |
| Dec 10, 2018 at 15:55 | comment | added | Asaf Karagila♦ | @Mike: Ah, serves me right for just skimming through the comments... :) | |
| Dec 10, 2018 at 15:49 | comment | added | Mike Shulman | @AsafKaragila Which is similar to what Peter suggested as a way to nontrivialize the question: "... which are not equivalent, as M-categories, to the standard one'. | |
| Dec 10, 2018 at 14:36 | comment | added | Asaf Karagila♦ | @Kevin: But that raises the obvious follow up. Given a class of "good morphisms", is there an endofunctor of $\bf Set$ which maps these good morphisms to inclusions? | |
| Dec 10, 2018 at 12:15 | answer | added | Simon L Rydin Myerson | timeline score: 4 | |
| Dec 10, 2018 at 11:39 | comment | added | Kevin Buzzard | Oh how dismal. So this trivially answers the question. | |
| Dec 9, 2018 at 22:44 | comment | added | Mike Shulman | Choose two sets $X$ and $Y$ and an isomorphism $e:X\cong Y$ that is not the identity. Define $F:{\rm Set}\to{\rm Set}$ by $F(X)=Y$, $F(Y)=X$, and $F(A)=A$ for all other sets $A$, and define the action of $F$ on arrows by composing with the chosen isomorphism $e$ whenever needed. Note that $F$ is an "automorphism" in the strict sense, i.e. a self-isomorphism (not equivalence), so any system of inclusions can be transported across it. But a subset inclusion $i:X'\hookrightarrow X$ is sent to an injection $e i : X' \to Y$ that is not an inclusion. | |
| Dec 9, 2018 at 22:12 | history | edited | Kevin Buzzard | CC BY-SA 4.0 |
clarifications after comments
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| Dec 9, 2018 at 22:11 | comment | added | Kevin Buzzard | Just to be clear @PeterLeFanuLumsdaine I am asking for more details of the assertion "Since one can certainly produce systems that are not equal to the usual one...". I need more hints! | |
| Dec 9, 2018 at 22:10 | comment | added | Kevin Buzzard | OK, dealing with comments. Todd -- yes, sorry, I've added the condition that all good functions are injections. Qfwfq sure you can ask the question for more general categories. I don't know the answer for sets though. Asaf -- sorry for being unclear. I've rewritten. I just mean that in contrast to the standard way of setting up functions in ZFC, a function here knows its codomain. Peter -- I can believe you can rephrase the question. Are you saying that it's currently trivial? I don't know any automorphisms of Set which move inclusions to something else. Can you answer the question as stated? | |
| Dec 9, 2018 at 15:01 | comment | added | Peter LeFanu Lumsdaine | and to make it a bit more non-trivial, one could add something like “…which are not equivalent, as M-categories, to the standard one”? (Since one can certainly produce systems that are not equal to the usual one, by transporting the usual one under an automorphism of Set that doesn’t respect it.) | |
| Dec 9, 2018 at 14:57 | comment | added | Peter LeFanu Lumsdaine | Mike Shulman’s answer to your earlier question is very relevant here, pointing to the notions of M-categories and (directed) (structural) systems of inclusions in the literature. The latter are a specific axiomatisation the properties of inclusions, which I don’t recall completely off the top of my head but had at least a similar flavour to your axioms here. The former are a more general abstraction of the overall structure formed. So your question could be written as something like “Does Set carry other systems of inclusions?”, [cont’d] | |
| Dec 9, 2018 at 12:58 | comment | added | Andreas Blass | @AsafKaragila I"m pretty sure "technically" is intended to mean "if you define a function as simply a suitable set of ordered pairs, without specifying a codomain". So all empty functions are technically equal. The OP wants a function to have a specified codomain, so $\varnothing$ has different functions into different sets. | |
| Dec 9, 2018 at 12:44 | comment | added | Asaf Karagila♦ | Wait, the empty function from $\varnothing$ into $\varnothing$ and into $\{\varnothing\}$ are two different functions in the sense that their codomains are different, but they are "technically equal". No? I'm confused by your clarification. | |
| Dec 9, 2018 at 12:39 | comment | added | Qfwfq | If you give a definition for a general category with "mono" in place of "injective", does something go wrong? | |
| Dec 9, 2018 at 12:36 | comment | added | Qfwfq | This may (or may not..) be tangentially relevant mathoverflow.net/questions/262490/… | |
| Dec 9, 2018 at 12:18 | comment | added | Todd Trimble | You want all good morphisms to be injections? | |
| Dec 9, 2018 at 11:56 | history | asked | Kevin Buzzard | CC BY-SA 4.0 |