Timeline for Category theory without axiom of choice
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| when toggle format | what | by | license | comment | |
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| Oct 28, 2017 at 0:16 | comment | added | David Roberts♦ | @OmerRosler Makkai's paper introducing anafunctors does not deal with universes or foundation that I recall, merely the lack of AC. Another way to view an anafunctor, and more helpful to my mind, is that it is a span of functors $C\leftarrow C' \to D$ where the left-pointing arrow is surjective on objects and fully faithful. Makkai gives this, but then reverts to his pointwise definition, which uses quite frankly clunky notation. | |
| Oct 27, 2017 at 17:47 | vote | accept | Omer Rosler | ||
| Oct 27, 2017 at 16:03 | answer | added | Mike Shulman | timeline score: 12 | |
| Oct 27, 2017 at 13:07 | comment | added | Simon Henry | ... The only place where you are getting into trouble is that without some very weak choice principles (discussed at the end of the paper. you end up with the fact that the category of anafunctors between two small category may fails to be small it self (see mathoverflow.net/questions/264585/… ) | |
| Oct 27, 2017 at 13:05 | comment | added | Simon Henry | @OmerRosler : roughly yes, more precesely an anafunctor is something to which object $c \in C$ attach a 'set' (even a class in some cases) of objects $F(c) \rightarrow D$ with canonical isomorphism between each of these objects (and the action on morphims as well). for example if you want to define the anafunctor $C \times C \rightarrow C$ which to each pairs of objects $C$ associate their products then you just associate the set of all possible products and their canonical isomorphisms between them instead of having to 'chose' one product for each pair. The theory works very nicely... | |
| Oct 27, 2017 at 12:28 | comment | added | Omer Rosler | @AsafKaragila I completely agree, I might have needed to state this more precisely: the point was there is no foundational issue with dropping AC, so the theory can be built, and what I was looking for is exactly that theory, which is Makkai's work. | |
| Oct 27, 2017 at 12:20 | comment | added | Asaf Karagila♦ | Yes, but I'm saying that the focus on universes is a bit irrelevant. It's just a small observation. There are other things which require the use of choice, in an actual and substantial way. Universes are entirely boring. Yes, you need to formulate exactly what you mean by an inaccessible... hurray... what else is new in the choiceless world? | |
| Oct 27, 2017 at 12:16 | comment | added | Omer Rosler | @AsafKaragila Exactly, and that is what I'm looking for, which is found in Makkai's work, where the universe/foundation issues are handled without choice. | |
| Oct 27, 2017 at 12:14 | comment | added | Omer Rosler | @SimonHenry I know that universes are not essential, but I specifically work with them so this was my starting point | |
| Oct 27, 2017 at 12:09 | comment | added | Asaf Karagila♦ | Talking about universes in ZF is the same as in ZFC. The difference is that now the term "inaccessible cardinal" can be interpreted in several ways, which yield non-equivalent definitions, so if you want to talk about universes it is important to talk about "the right type of inaccessible cardinals". But as @Simon says (pun not intended!), this is a red herring to the actual work. You want to ask yourself more about the structure of certain categories, or the existence of projective, injective, and otherwise categorical constructions. And there choice might actually be a culprit. | |
| Oct 27, 2017 at 12:09 | comment | added | Omer Rosler | @SimonHenry I was not aware of this, and this seems to be exactly what I'm looking for; if I understand this corrrectly, an anafunctor $F:C\rightarrow D$ basically is a mapping from "iso classes of $C$" to "iso classes of $D$" without specifying "the value of the functor at specific object" which constitutes choice. Is this correct? | |
| Oct 27, 2017 at 11:06 | comment | added | Simon Henry | Also, for a very large part of category theory you really don't need any inaccessible cardinal or Grothendieck univers. Something like Neuman-Bernays-Godel set theory where you just have a notion of class but basically do not require anything about it is often enough. This framework is equiconsistant with ZF and the absence of choice do not pose any problems. That is the framework that Makkai's is using. (and in my opinion, 'very large part' = all, but I know lots of people will argue against) | |
| Oct 27, 2017 at 10:35 | comment | added | Simon Henry | Are you familiar with Makkai's paper "Avoiding the axiom of choice in general category theory" ? It is a bit orthogonal to what you mention in your question (As far as I remember it does not discuss universe) but it develop very good notions which avoid all those anoying problems of having to chose limits and so one... | |
| Oct 27, 2017 at 10:35 | comment | added | Alec Rhea | Possibly related: mathoverflow.net/questions/279558/… mathoverflow.net/questions/273551/… | |
| Oct 27, 2017 at 10:25 | review | First posts | |||
| Oct 27, 2017 at 10:35 | |||||
| Oct 27, 2017 at 10:24 | history | asked | Omer Rosler | CC BY-SA 3.0 |