MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

In the article of M. Barr "Representation of Categories" J. Pure Appl. Algebra, 41 (1986), 113–137 link: ftp://ftp.math.mcgill.ca/pub/barr/pdffiles/represen.pdf

M. Barr shows a classical (but hard) theorem :

"every regular category had a full, regular embedding into a category of set-valued functors".

Anyway I think that in that article there are some mistakes, or he takes for granted things that I do not know or I haven't seen in the literature.

Then I had some difficulties in reassembling the demonstration of the fundamental theorem and other questions that I asked myself while reading.

M. Barr starts from a small category $C$, and lets $FL(C)$ be the full sub category of copreshaves category $C^<:=Cat[C , Set]$ formed by the functors that preserve finite limits.
Then we have the Yoneda immersion $h^-: C^{op} \to FL(C)$ and the dual is a immersion $Y: C \to \widetilde{C} $ by $\widetilde{C}:=FL(C)^{op} $.

Then Barr proves that if $C$ is regular then also $\widetilde{C} $ is regular (in the proof one needs the observation that the Yoneda functor $h^-$ sends a (reg.Epi, Mono) factorization to an (Epi, reg.Mono) one, easy to prove). And this is ok for me.

Then Barr gives a proof (not clear I think) that if $C$ is a pretopos then also $\widetilde{C} $ is pretopos.

For what I know (Sketches of an Elephant, Peter T. Johnstone, Vol 1, part 1.4.7):

A pretopos is a coherent effective categories by disjoint finite coproducts.

And a coherent category is a regular one, by unions that are preservated by pullback's. Effective means that any equivalence realtion is a ker-pair of some morphism (I guess).

Among other difficulties, there is that the completeness of $FL(C)$ follows from being a reflective subcategory of $C^<$ (a hard theorem) and then the immersion $FL(C) \subset C^<$ dosen't preserve colimits (it creates limits of course).

I ask simply: If $C$ is coherent (resp. effective, a pretopos), how I can prove that $\widetilde{C} $ is coherent (resp. effective, a pretopos)?

share|cite|improve this question
    
I've made some (I hope) purely typographical changes. If I have changed the meaning of anything you wrote, I sincerely apologize, and you should feel free to roll back the edits. (I was unsure how to fix "A pretopos is a coherent effective categories by disjoint finite coproducts.", which doesn't parse for me, and I felt that any best guess would too likely change the content of (or introduce errors) to the post.) – Theo Johnson-Freyd Dec 20 '10 at 3:22
    
I fixed a small TeX-typo that was throwing out the markdown – David Roberts Dec 20 '10 at 6:27
    
And another, after Buschi's latest edit. I'm doing this because it was cutting out half of a sentence, making the idea hard to grasp. – David Roberts Dec 20 '10 at 8:21
    
I'm in way of solution. No so hard as I guessed in first time, (fog is vanishing). Of course dont seem honest reply to a mine question. But if someone is intesting I can post (or Emailing) a "more detalied" proofs and explanation about BArr article. The result of Barr article is a important goal in category theory tools, and generally the proofs are very long and hard. Thank you for your interset and patience, I'm sorry for my poor English . PS. someone understand Italian too? – Buschi Sergio Dec 21 '10 at 14:31

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.