The Clowder Project

Throughout this work, we will sometimes denote a function $f\colon X\to Y$ by

1. 1.

   For example, given a function

   \[ \Phi \colon \operatorname {\mathrm{Hom}}_{\mathsf{Sets}}(X,Y)\to K \]

   taking values on a set of functions such as $\operatorname {\mathrm{Hom}}_{\mathsf{Sets}}(X,Y)$, we will sometimes also write

   \[ \Phi (f)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\Phi ([\mspace {-3mu}[x\mapsto f(x)]\mspace {-3mu}]). \]
2. 2.

   This notational choice is based on the lambda notation

   \[ f\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}(\lambda x.\ f(x)), \]

   but uses a “$\mathord {\mapsto }$” symbol for better spacing and double brackets instead of either:

   1. (a)

      Square brackets $[x\mapsto f(x)]$;

   2. (b)

      Parentheses $(x\mapsto f(x))$;

   hoping to improve readability when dealing with e.g.:

   1. (a)

      Equivalence classes, cf.:

      1. (i)

         $[\mspace {-3mu}[[x]\mapsto f([x])]\mspace {-3mu}]$

      2. (ii)

         $[[x]\mapsto f([x])]$

      3. (iii)

         $(\lambda [x].\ f([x]))$

   2. (b)

      Function evaluations, cf.:

      1. (i)

         $\Phi ([\mspace {-3mu}[x\mapsto f(x)]\mspace {-3mu}])$

      2. (ii)

         $\Phi ((x\mapsto f(x)))$

      3. (iii)

         $\Phi ((\lambda x.\ f(x)))$

3. 3.

   We will also sometimes write $-$, $-_{1}$, $-_{2}$, etc. for the arguments of a function. Some examples include:

   1. (a)

      Writing $f(-_{1})$ for a function $f\colon A\to B$.

   2. (b)

      Writing $f(-_{1},-_{2})$ for a function $f\colon A\times B\to C$.