# Algebraic theories and forgetful functor

Let $\bf{N}$ be the category whose objects are all the nonegative integers while the morphisms $m\longrightarrow n$ are the mappings from $m$ to $n$, considered as finite sets (so $\bf{N}$ is the skeleton of the category of finite sets and mappings). In $\bf{N}$ we have an operation of coproduct or sum, $m+n$, represented by the disjoint union of sets. In particular, each $n\in Obj\bf{N}$ can be obtained as a sum of $n$ terms: $n=1+1+\ldots +1$.

By an algebraic theory one means a category $\mathscr{P}$ containing $\bf{N}$ as a subcategory, with the same objects and coproducts maps as $\bf{N}$. Thus $\mathscr{P}$ differs from $\bf{N}$ in having (possibly) more morphisms.

Now a $\mathscr{P}$-algebra is a controavariant functor $F$ from $\mathscr{P}$ to $Set$, which converts coproducts to products. Thus $Fn=A^n$, where $A^n$ is the product of $n$ copies of $A^1=A$, and the $\mathscr{P}$-morphisms $1\longrightarrow n$ define mappings $A^n\longrightarrow A$, i.e. $n$-ary operations on $A$.

Given an algebraic theory $\mathscr{P}$, we can form the category $Funct(\mathscr{P}^{op},Set)$ of all product preserving functors from $\mathscr{P}^{op}$ to $Set$.

QUESTION 1: how can I define a forgetful functor $U:Funct(\mathscr{P}^{op},Set)\longrightarrow Set$, associating to each $\mathscr{P}$-algebra its "underlying" set, in such a way that this functor $U$ has a left adjoint $F$ providing the free $\mathscr{P}$-algebra on a given set?

QUESTION 2: can you suggest me some references (books, articles, ...) where I can find a complete description of this topic?

• @berci thank u, where can I find the most commonly used definition of algebraic theory? – Danae Kissinger Jun 8 '15 at 10:18

Q1: You already sketched the structure. A functor $F:\mathscr P^{op}\to\mathcal{Set}$ plays the role of an algebra on set $A=F1$ with projections and possible operations $A^n\to A$.
This suggests that the forgetful functor is going to be $F\mapsto F1$.