Dan Oneata wrote:

>Keith wrote:

>> I believe Puzzle 162 can be answered the same way using the initial objects and terminal objects of \$$\mathbf{Cat}\$$ and \$$\mathbf{Set}\$$.

>I'm not sure how to use this idea, so my attempt takes the long and tedious route.

What you did above I think is correct.

However, note that for all sets, there's a function from the empty set into any other set, therefore we must have,

\$\begin{matrix} \mathbf{Cat}[ & Disc(\varnothing) & \rightarrow & \mathcal{C} & ] \\\\ & & & & \\\\ & \Uparrow & & \Downarrow & \\\\ & & & & \\\\ \mathbf{Set}[ & \varnothing & \rightarrow & Ob(\mathcal{C}) & ] \end{matrix} \$

which must hold for every category. However, the discrete category on an empty set is the empty category, which does have a functor into every other category.

This gives an interesting special case,

\$\begin{matrix} \mathbf{Cat}[ & Disc(\varnothing) & \rightarrow & Disc(\varnothing) & ] \\\\ & & & & \\\\ & \Uparrow & & \Downarrow & \\\\ & & & & \\\\ \mathbf{Set}[ & \varnothing & \rightarrow & Ob(Disc(\varnothing)) & ] \end{matrix} \$

Likewise, for any one object, one morphism category there exists a unique functor into from any other category, therefore we must have,

\$\begin{matrix} \mathbf{Cat}[ & Disc(S) & \rightarrow & \mathbf{1} & ] \\\\ & & & & \\\\ & \Uparrow & & \Downarrow & \\\\ & & & & \\\\ \mathbf{Set}[ & S & \rightarrow & Ob(\mathbf{1}) & ] \end{matrix} \$

which must hold for every set. However, any object set on a one object category is just a one element set, for which every other set maps uniquely into.

This gives an interesting special case,

\$\begin{matrix} \mathbf{Cat}[ & Disc(Ob(\mathbf{1})) & \rightarrow & \mathbf{1} & ] \\\\ & & & & \\\\ & \Uparrow & & \Downarrow & \\\\ & & & & \\\\ \mathbf{Set}[ & Ob(\mathbf{1}) & \rightarrow & Ob(\mathbf{1}) & ] \end{matrix} \$