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fix typo in 3.14
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kkinugasa committed Sep 26, 2024
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4 changes: 2 additions & 2 deletions src/content/3.14/lawvere-theories.tex
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Expand Up @@ -63,7 +63,7 @@ \section{Universal Algebra}
Lawvere theory $\cat{L}$: an object in the category $\cat{Law}$.
\item
Model $M$ of a Lawvere category: an object in the category\\
$\cat{Mod}(\cat{Law}, \Set)$.
$\cat{Mod}(\cat{L}, \Set)$.
\end{enumerate}

\begin{figure}[H]
Expand Down Expand Up @@ -352,7 +352,7 @@ \section{Lawvere Theories and Monads}
\[\cat{L}(n, -) \Colon \cat{L} \to \Set\]
To show that it's indeed free, all we have to do is to prove that it's a
left adjoint to the forgetful functor:
\[\cat{Mod}(\cat{L}(n, -), M) \cong \Set(n, U(M))\]
\[\cat{Mod}(\cat{L}, \Set)(\cat{L}(n, -), M) \cong \Set(n, U(M))\]
Let's simplify the right hand side:
\[\Set(n, U(M)) \cong \Set(n, M 1) \cong (M 1)^n \cong M n\]
(I used the fact that a set of morphisms is isomorphic to the
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