From 168b7d6b991275e2b3784dc458a30dc9c5043989 Mon Sep 17 00:00:00 2001 From: Ross Kirsling Date: Mon, 5 Nov 2018 09:55:26 -0800 Subject: [PATCH 1/2] Fix typesetting issues in Chapter 25. --- src/content/3.9/Algebras for Monads.tex | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/src/content/3.9/Algebras for Monads.tex b/src/content/3.9/Algebras for Monads.tex index de21c0bd..cf509978 100644 --- a/src/content/3.9/Algebras for Monads.tex +++ b/src/content/3.9/Algebras for Monads.tex @@ -119,7 +119,7 @@ \section{T-algebras} free functor. The left adjoint to $U^T$ is called $F^T$. It maps an object -$A$ in $\cat{C}$ to a free algebra in $\cat{C}^T$. The carrier +$a$ in $\cat{C}$ to a free algebra in $\cat{C}^T$. The carrier of this free algebra is $T\ a$. Its evaluator is a morphism from $T\ (T\ a)$ back to $T\ a$. Since $T$ is a monad, we can use the monadic $\mu_a$ (\code{join} in Haskell) as the @@ -247,8 +247,8 @@ \section{The Kleisli Category} We've seen the Kleisli category before. It's a category constructed from another category $\cat{C}$ and a monad $T$. We'll call this -category $\cat{C}^T$. The objects in the Kleisli category -$\cat{C}^T$ are the objects of $\cat{C}$, but the morphisms +category $\cat{C}_T$. The objects in the Kleisli category +$\cat{C}_T$ are the objects of $\cat{C}$, but the morphisms are different. A morphism $f_{\cat{K}}$ from $a$ to $b$ in the Kleisli category corresponds to a morphism $f$ from $a$ to $T\ b$ in the original category. We call this @@ -292,7 +292,7 @@ \section{The Kleisli Category} back to $\cat{C}$. It takes an object $a$ from the Kleisli category and maps it to an object $T\ a$ in $\cat{C}$. Its action on a morphism $f_{\cat{K}}$ corresponding to a Kleisli arrow: -\[f \Colon a -> T\ b\] +\[f \Colon a \to T\ b\] is a morphism in $\cat{C}$: \[T\ a \to T\ b\] given by first lifting $f$ and then applying $\mu$: From 3d398e0b7d06a9a91cd0c86c8827d137bcffba59 Mon Sep 17 00:00:00 2001 From: Ross Kirsling Date: Tue, 6 Nov 2018 09:30:47 -0800 Subject: [PATCH 2/2] Fix typesetting issue in Chapter 26. --- src/content/3.10/Ends and Coends.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/content/3.10/Ends and Coends.tex b/src/content/3.10/Ends and Coends.tex index f708d7d5..d964410b 100644 --- a/src/content/3.10/Ends and Coends.tex +++ b/src/content/3.10/Ends and Coends.tex @@ -117,7 +117,7 @@ \section{Ends} functions mapping the apex to the sets in the base. You may think of this family as one polymorphic function --- a function that's polymorphic in its return type: -\[\alpha \Colon \forall a\ .\ apex -> p\ a\ a\] +\[\alpha \Colon \forall a\ .\ apex \to p\ a\ a\] Unlike in cones, within a wedge we don't have any functions that would connect vertices of the base. However, as we've seen earlier, given any morphism $f \Colon a \to b$ in $\cat{C}$, we can connect both