# Finitely presentable objects and the Kleisli category

There is a correspendence between Lawvere theories $L$ and finitary monads $\mathbb{T}_L$ (associated to $L$), due to Lawvere: the category $Mod(L)$ of models of $L$ (in $\mathbf{Set}$) is equivalent to the Eilenberg-Moore category $\mathbb{T}_L$-$\mathbf{Alg}$ of algebras over the monad $\mathbb{T}_L$.

Now, this category $\mathbb{C}=Mod(L)$ is a locally finitely presentable category (by definition). Write $\mathbb{C}_f$ for the (full) subcategory of finitely presentable objects of $\mathbb{C}$. Can $\mathbb{C}_f$ be constructed somehow from the Kleisli category $\mathbf{Kl}(\mathbb{T}_L)$ of the monad $\mathbb{T}_L$? It seems to me that there must be some relationship between the two (at first I thought $\mathbb{C}_f=\mathbf{Kl}(\mathbb{T}_L)^{\text{op}}$, but this is definitely not the case).

You can also distinguish the free algebras $A$ which are finitely presentable in $\mathbb{T}_L-\mathbf{Alg}$ from those which are not by the fact that $\mathbf{KL}(\mathbb{T}_L)(A,-)$ commutes with the filtered colimit $B \to 2\cdot B \to \dots \to \omega \cdot B \to \dots \to \kappa \cdot$ for every infinite $\kappa$, where $B$ is any object of $\mathbf{KL}(\mathbb{T}_L)$ and $\alpha \cdot B$ is the $\alpha$-fold coproduct of $B$ with itself.
Putting these together, you can pass from $\mathbf{KL}(\mathbb{T}_L)$ to its finite rank objects to their reflexive coequalizer completion. Of course, you could also just construct $\mathbb{T}_L-\mathbf{Alg}$ from $\mathbf{KL}(\mathbb{T}_L)$ (I think it's also a reflexive coequalizer completion) and then just take the finitely presentable objects in the resulting category.
• Thanks! Just a clarification, $L$ (the Lawvere theory) is actually $(\mathbf{Kl}(\mathbb{T}_L)_f)^{\text{op}}$ always, right? (the category opposite to the full subcategory of finitely presentable objects of the Kleisli category). I think I am getting confused over that too now.. – MmOmT Aug 1 '16 at 15:03
• Right. It's kind of confusing. Personally, I tend to think of $L^\mathrm{op} = \mathbf{Kl}(\mathbb{T}_L)$ as the "primary object". – tcamps Aug 1 '16 at 17:57
• $L^{\text{op}}$ is the whole of Kleisli, not its finitely presentable objects? – MmOmT Aug 2 '16 at 13:25
• Oh, that was silly of me, of course not. $L^\mathrm{op}$ is just the finitely presentable objects of the Kleisli category. – tcamps Aug 2 '16 at 13:35