2
$\begingroup$

I have this MSE question with a two hundred bounty but even with the bounty this post got underviewed. So maybe here is a more suitable place to post it. The question follows:

(a) Define a sketch $s_{\mathbf{Grp}}$ and a equivalence functor $$E: \mathbf{Grp}\to \mathbf{Mod}(s_{\mathbf{Grp}},\mathbf{Set})$$ (b) Knowing that finite limits commute with filtered colimits in $\mathbf{Set}$, use the result in (a) to prove that they also commute in $\mathbf{Grp}$.

(c) Prove that $\mathbf{Ab} \backsimeq \mathbf{Mod}(s_{\mathbf{Grp}},\mathbf{Grp})$

I found a useful example at nlab's sketch article. Example 3.2 especifies the directed graph, diagrams, cones and cocones of a sketch which has unital magmas as models (sets with a binary operation which has a two sided unit).

So I thought taking this same sketch and "interpreting" the arrows $e$ as the identity of a group and $m$ as its multiplication, all this via the equivalence functor the exercise request us to build. But I don't even know how to finish the construction of $E$ and in fact I don't see why it will be an equivalence functor at all.

Could you help me? Also is there any result that I'm missing on (b)? Because I think this shouldn't be so difficult.

Improve this question
$\endgroup$
1
  • 1
    $\begingroup$ It looks as though the MSE bounty is expired. I could help (over the next few days), but I think I'd prefer to post over there. The problem may be that full details are somewhat tedious; this may account for lack of response. $\endgroup$
    – Todd Trimble
    Commented Feb 14, 2021 at 1:56

1 Answer 1

Reset to default
2
$\begingroup$

The case of semigroups is spelled out in all details in Sec 2 of Chap 7 of Category theory for Computer Science by Barr and Wells.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .