Is any elementary topos a concretizable category?

Question

The following special cases are obvious:

A Grothendieck topos is concretizable ($F \mapsto \times_i F(i)$)
A well-pointed topos is concretizable ($X \mapsto \rm{Hom}(1, X)$)

I looked at some more different examples and they are all obviously concretizable (but I'm just starting to learn topos theory). Is there an elementary topos from which there is no faitfull functor to the category of sets?

Actually, a well-pointed topos may fail to be concretisable if it is not locally small. Or are you implicitly assuming size conditions here? — Zhen Lin, Nov 06 '22 at 22:45
Because regular subobjects form a set, every elementary topos is concretizable by the lazy Isbell criterion. Check Freyd's papers on concreteness. — Ivan Di Liberti, Nov 06 '22 at 23:05
@IvanDiLiberti Indeed, thank you very much! You can post it as an answer, I'll accept it. — Arshak Aivazian, Nov 06 '22 at 23:19
I wrote down the answer (as a community wiki). If you post yours, I'll accept it and delete mine. — Arshak Aivazian, Nov 06 '22 at 23:25
(Obviously, every non-locally small category is not concretizable, so the question only makes sense for locally small ones) — Arshak Aivazian, Nov 06 '22 at 23:32
Which categories are concretizable depends on the foundation you work in; it can be argued that all categories are concretizable under an appropriate ‘change of foundation’. — Alec Rhea, Nov 07 '22 at 02:26
@AivazianArshak, thanks but I do not need points. I am happy to have been of help. — Ivan Di Liberti, Nov 07 '22 at 06:13

score 13 · Accepted Answer · edited Nov 07 '22 at 00:48

13

As Ivan Di Liberti suggests in the comments, according to Lemma 1.2 (Freyd's paper, Concretness, JPAA 1973) every regular well-powered category with equalizers is concretizable. In particular, every locally small elementary topos is concretizable.

edited Nov 07 '22 at 00:48

David Roberts

33,851

answered Nov 06 '22 at 23:25

Arshak Aivazian

6,474

Is any elementary topos a concretizable category?

1 Answers1