(co)limits of models of a Lawvere theory

Question

Somehow, I got the impression from this answer that, for a Lawvere theory $L,$ the category $\text{Mod}(L,\text{Set})$ of models of a Lawvere theory has all small limits and colimits. Assuming this is true, my question is, how do I find such limits and colimits ?

As I understand it, $\text{Mod}(L,\text{Set})$ is just the full subcategory of $[L,\text{Set}],$ on product preserving functors. Here $[L,\text{Set}]$ is the category of functors from $L$ to $\text{Set}.$ Given a functor $\mathcal{C} \overset{F}{\rightarrow} \text{Mod}(L,\text{Set}),$ can I find the (co)limit of $F$ just by calculating the (co)limit of $\mathcal{C} \overset{F}{\rightarrow} \text{Mod}(L,\text{Set}) \hookrightarrow [L,\text{Set}]$ (in other words, is it okay to pretend I'm working in the larger category $[L,\text{Set}],$ where I know how to find the (co)limits), or is there some other method I should use ?

Answer 1

Your impression is correct, because the categories of models of Lawvere theories are all locally presentable; so they are complete and cocomplete (cocomplete, as part of one among the many equivalent definitions of LPC, and complete, for example as a consequence of the adjoint functor theorem -a very slick proof is Theorem 6.1.4 of Makkai-Paré book, or of a more "by hand" argument: see Borceux's handbook, tome II, Corollary 5.2.8).

Computing limits is "easy", they are just limits in the underlying category, because the forgetful functor $U : Mod({\cal T}) \to Set$ preserves are reflects them (Borceux II, 3.4.1).

Computing colimits requires more care: they exist, but their shape may change drastically in the category of models; as an example take the category of monoids or groups: given $G,H$ their free group $G*H$ is the coproduct in $Grp$, but it's very far from being the coproduct of the underlying sets with a certain group operation.

So, the answer to

Given a functor $\mathcal{C} \overset{F}{\rightarrow} \text{Mod}(L,\text{Set}),$ can I find the (co)limit of $F$ just by calculating the (co)limit of $\mathcal{C} \overset{F}{\rightarrow} \text{Mod}(L,\text{Set}) \hookrightarrow [L,\text{Set}]$ (in other words, is it okay to pretend I'm working in the larger category $[L,\text{Set}],$

is a loud "no!" :-) some colimits change a lot.

This said filtered colimits are easy: the forgetful functor preserves and reflects them (Borceux II, 3.4.2).

I strongly advise you to study in depth all Chapter 3 in Borceux II, as your latest questions all seem to stem from the same ballpark!