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\section*{Lawvere&#39;s reconstruction theorem}

\hypertarget{lawveres_reconstruction_theorem}{}\subsection*{{Lawvere's Reconstruction Theorem}}\label{lawveres_reconstruction_theorem}

This is a famous result from William Lawvere's thesis.

A \textbf{finite products theory} or [[Lawvere theory]] $C$ is a category with finite [[products]] where all objects are finite products of copies of a given object $x$. A \textbf{model} is a functor $F: C \to Set$ preserving finite products, and a morphism of models is a natural transformation between such functors. This gives us a category $Mod(C)$ of models of $C$.

What is going on here? A model $F$ is really just a set $F(x)$ together with a bunch of $n$-ary operations coming from the morphisms in $C$, satisfying equational laws coming from the equations between morphisms in $C$. Any sort of algebraic gadget that's just a set with a bunch of $n$-ary operations satisfying equations can be described using a theory of this sort. For example: monoids, groups, abelian groups, rings\ldots{} and so on. We can describe any of these using a suitable algebraic theory, and in each case, the category $Mod(C)$ will be the category of these algebraic gadgets.

There is a functor

\begin{displaymath}
R: Mod(C) \to Set
\end{displaymath}
which carries each model $F$ to the set $F(x)$. We can think of this as a functor which forgets all the operations of our algebraic gadget and remembers only the underlying set. This should make you desire a [[adjoint functor|left adjoint]]

\begin{displaymath}
L: Set \to Mod(C)
\end{displaymath}
sending each set to the ``free'' algebraic gadget on this set. Indeed, such a left adjoint exists!

Given this pair of adjoint functors we we can talk about the category of ``finitely generated free models'' of our theory. The objects here are objects of $Mod(C)$ of the form $L(S)$ where $S$ is a finite set, and the morphisms are the usual morphisms in $Mod(C)$. Let us call this category $fg Free Mod(C)$.

Here is a way to reconstruct $C$ from its category of finitely generated free models:

\begin{utheorem}
If $C$ is a Lawvere theory, $C$ is equivalent to $fg Free Mod(C)^{op}$ (i.e., $fg Free Mod(C)$ is equivalent to the opposite of the category $C$).

\end{utheorem}
\begin{proof}
See William F. Lawvere's Ph.D. thesis, \href{http://www.tac.mta.ca/tac/reprints/articles/5/tr5abs.html}{Functorial Semantics of Algebraic Theories}.

\end{proof}
In other words, you can reconstruct a Lawvere theory from its category of finitely generated free algebras in the simplest manner imaginable: just reversing the direction of all the morphisms!

The above all continues to apply in suitable form even when we replace ``finite products'' throughout by any other structure given by limits; e.g., ``finite limits'', ``arbitrary (small) products'', or ``arbitrary (small) limits''. (Of course, for each case, we must also look at a correspondingly wider class of ``free'' models than merely those freely generated on finite sets; specifically, we recover the theory from its representable models, comprising one ``freely generated'' model for each object of $C$). In particular, in the ``arbitrary (small) products'' case, we can recover the theory from its free models on arbitrary sets. Indeed, in this case $L$ takes the particularly nice definition $L(k)(b) = Hom_C(x^k, b)$, and, in fact, the adjunction between $L$ and $R$ is [[monadic]]; that is, $Mod(C)$ is equivalent to the [[Eilenberg–Moore category]] of the monad $RL(-) = Hom_C(x^{(-)}, x)$, with the functors $R$ and $L$ corresponding under this equivalence to the forgetful and free functors of the Eilenberg--Moore construction. In fact, every monad can be seen to arise uniquely in this way, taking $C$ as the dual of its [[Kleisli category]] (automatically a category with small products generated by a single object, as $Set$ is a category with small coproducts generated by the single object $1$, and the Kleisli category is generated by a left adjoint applied to $Set$); thus, we have a correspondence between monads on $Set$ and categories with small products generated by a single object, both equally well representing single-sorted, infinitary algebraic theories, and, in the former setting, the above recoverability result is just the immediate fact that a monad can be recovered from the forgetful functor on its Eilenberg--Moore category (as, indeed, it can of course be recovered from any adjunction giving rise to it).

(This all also works just as well for inescapably multi-sorted theories (that is, even without the restriction that $C$ be generated by a single object); in this case, we can't help but recognize that there are multiple forgetful functors from $Mod(C)$ to $Set$, one for each object of $C$, each with a corresponding free model on $1$ (by the [[Yoneda lemma]]), and, again, the theory is recovered as the dual of the category of all these models (by the [[Yoneda embedding lemma]]))

[[!redirects Lawvere reconstruction theorem]] [[!redirects Lawvere's reconstruction theorem]]



\end{document}