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Let take a first order theory $T$ in a language $\mathcal L$. One can form the category $\mathrm{Mod}(T)$ whose

  • objects are the models of $T$ (those $\mathcal L$-structures $\mathfrak M$ such that $\mathfrak M \models T$),
  • arrows are the homomorphism of $\mathcal L$-structures. (One might also want to take only the embedding as arrows to have more elementary-like properties.)

Then, how much does the theory $T$ (or an axiomatisation of $T$) tell about the shape of the category $\mathrm{Mod}(T)$ ?

This question came to me when studying $\mathsf C$-valued (pre)sheafs on a topological space, when $\mathsf C$ was a category among : sets, groups, abelian groups, rings, etc. For all those categories $\mathsf C$, the underlying sets of a stalk of a (pre)sheaf is the stalk of the postcomposite (pre)sheaf with the forgetful functor to sets. This for the one and good reason that the forgetful functor $U \colon \mathsf C \to \mathsf{Set}$ commutes with filtered colimits. The question is :

Could such a property about the forgetful functor be determined from the properties of the theory $T$ ?

To continue with the example of $U$ commuting with filtered colimits, it seems that the theory of groups (respectively abelian groups, rings) admitting a universal axiomatization (in the correct language of course) has something to do with it… (I'm not entirely sure, this insight comes from the special case of a growing union.)

I principally look for references or keywords, as I have no clue about the name of such a field.

P.S. : along my search on the net, I came across the book Accessible Categories: The Foundations of Categorical Model Theory by Makkai and Paré, but it seems kind of hard to apprehend and I can not determine if it could answer my question.

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First-order theories are far too general. All the examples you listed are algebraic theories, a.k.a. varieties in the sense of universal algebra. – Zhen Lin Jan 31 '14 at 17:26
@ZhenLin My understanding is that universal algebra is model theory without relation symbols, and varieties are then (something of) model-complete theories. So I was talking in the larger context. But still, if first order theories are to general, my question still makes sense restricting to universal algebra, and I would be interested if you have knowledge of any work/book/notes about such a thing. – Pece Jan 31 '14 at 17:38
The point is this: if you have an algebraic theory, then the category of its models has all the familiar properties: it will be complete and cocomplete, with limits and filtered colimits computed as in $\mathbf{Set}$. This is well-known and straightforward. If one knows a bit more, one can even deduce a converse. – Zhen Lin Jan 31 '14 at 19:03
@ZhenLin Thanks. What is exactly "a bit more" ? – Pece Jan 31 '14 at 20:07

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