### Finite Products Theories

#### Posted by John Baez

Here’s a classic theorem about finite products theories, also known as algebraic theories or Lawvere theories. You can find it in *Toposes, Triples and Theories*, but it must go way back to Lawvere’s thesis. In my work with Nina Otter on phylogenetic trees, we need a slight generalization of it… and if true, this generalization must already be known. So I really just want a suitable reference!

**Theorem.** Suppose $C$ is a category with finite products that is **single-sorted**: every object is a finite product of copies of a single object $x \in C$. Let $Mod(C)$ be the category of **models** of $C$: that is, product-preserving functors

$\phi : C \to Set$

and natural transformations between these. Let

$U : Mod(C) \to Set$

be the functor sending any model to its underlying set:

$U(\phi) = \phi(x)$

Then $U$ has a left adjoint

$F : Set \to Mod(C)$

and $Mod(C)$ is equivalent to the category of algebras of the monad

$U F : Set \to Set$

As a result, any model $\phi$ can be written as a coequalizer

$F U F U(\phi) \stackrel{\longrightarrow}{\longrightarrow} F U(\phi) \longrightarrow \phi$

where the arrows are built from the counit of the adjunction

$\epsilon : F U \to 1_{Mod(C)}$

in the obvious ways: $F U (\epsilon_{F U (\phi)})$, $\epsilon_{F U F U(\phi)}$ and $\epsilon_\phi$.

The generalization we need is quite mild, I hope. First, we need to consider multi-typed theories. Second, we need to consider models in $Top$ rather than $Set$. So, this is what we want:

**Conjecture.** Suppose $C$ is a category with finite products that is **$\Lambda$-sorted**: every object is a finite product of copies of certain objects $x_\lambda$, where $\lambda$ ranges over some index set $\Lambda$. Let $Mod(C)$ be the category of **topological models** of $C$: that is, product-preserving functors

$\phi : C \to Top$

and natural transformations between these. Let

$U : Mod(C) \to Top^\Lambda$

be the functor sending any model to its underlying spaces:

$U(\phi) = (\phi(x_\lambda))_{\lambda \in \Lambda}$

Then $U$ has a left adjoint

$F : Top^\Lambda \to Mod(C)$

and $Mod(C)$ is equivalent to the category of algebras of the monad

$U F : Top^\Lambda \to Top^\Lambda$

As a result, any model $\phi$ can be written as a coequalizer

$F U F U(\phi) \stackrel{\longrightarrow}{\longrightarrow} F U(\phi) \longrightarrow \phi$

where the arrows are built from the counit of the adjunction in the obvious ways.

#### Comments

1) There shouldn’t be anything terribly special about $Top$ here: any sufficiently nice category should work… but all I need is $Top$. What counts as ‘sufficiently nice’? If we need to replace $Top$ ‘convenient category of topological spaces’, to make it cartesian closed, that’s fine with me—just let me know!

2) I’m sure this conjecture, if true, follows from some super-duper-generalization. I don’t mind hearing about such generalizations, and I suspect some of you will be unable to resist mentioning them… so okay, go ahead, impress me…

… but remember: all I need is this puny little result!

In fact, all I really need is a puny special case of this puny result. If it works, it will be a small, cute application of algebraic theories to biology.

## Re: Finite Products Theories

I am certainly not an expert on this stuff, and I look forward to hearing what the experts have to say.

After having a bit of a non-expert nose about, I suspect it’s possible to extract what you need from Chapter 6 of Kelly’s

Basic Concepts of Enriched Category Theory.Obviously that goes much more general than you need, and specialising it to your situation would be a non-trivial amount of work. I hope someone will find a more suitable reference.