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src/content/1.8/functoriality.tex
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@ -39,7 +39,9 @@ is indeed a category.
An easier way to think about bifunctors would be to consider them functors in
each argument separately. So instead of translating functorial laws ---
associativity and identity preservation --- from functors to bifunctors,
it would be enough to check them separately for each argument. However, in general, separate functoriality is not enough to prove joint functoriality. Categories in which joint functoriality fails are called \textit{premonoidal}.
it would be enough to check them separately for each argument. However, in general,
separate functoriality is not enough to prove joint functoriality. Categories in which
joint functoriality fails are called \newterm{premonoidal}.
Let's define a bifunctor in Haskell. In this case all three categories
are the same: the category of Haskell types. A bifunctor is a type
@ -61,7 +63,9 @@ at once. The result is a lifted function,
\code{(f a b -> f c d)}, operating on types
generated by the bifunctor's type constructor. There is a default
implementation of \code{bimap} in terms of \code{first} and
\code{second}. (As mentioned before, this doesn't always work, because the two maps may not commute, that is \code{first g . second h} may not be the same as \code{second h . first g}.)
\code{second}. (As mentioned before, this doesn't always work, because
the two maps may not commute, that is \code{first g . second h} may not
be the same as \code{second h . first g}.)
\noindent