In this post John Baez defines a categorification of a set $S$ as a map $$ p:\DeclareMathOperator{Decat}{Decat}\Decat(\mathscr C)\to S $$ where $\Decat(\mathscr C)$ is the set of isomorphism classes of a category $\mathscr C$. He mentions that there is also a notion of categorifying a map between sets. What precisely is this notion?

My guess is that a categorification of a map between sets $f:X\to Y$ is a commutative diagram $$ \begin{array}{ccc} \DeclareMathOperator{Ob}{Ob}\Ob(\mathscr C) & \xrightarrow{\Ob(F)} & \Ob(\mathscr D) \\ \scriptsize{u}\downarrow & &\downarrow \\ X & \xrightarrow{f} & Y \end{array} $$ where $u$ is a bijection, $F:\mathscr C\to\mathscr D$ is a functor, and $\Ob:\mathsf{Cat}\to\mathsf{Set}$ is the obvious functor. Is this the "correct" notion? Also, are there any introductory references on categorification?

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    $\begingroup$ A categorification of a map between sets should be a functor between categories. Categorification is a big topic so it really depends on what you're interested in. $\endgroup$ – Qiaochu Yuan Jul 21 '14 at 21:22
  • $\begingroup$ @QiaochuYuan For example, would the diagram $\begin{array}{ccc} \DeclareMathOperator{Ob}{Ob}\Ob(\mathsf{CW}) & \xrightarrow{\Ob(H_*)} & \Ob(C^b(\mathsf{Ab})) \\ \scriptsize{\DeclareMathOperator{id}{id}\id}\downarrow & &\downarrow \\ \Ob(\mathsf{CW}) & \xrightarrow{\chi} & \Bbb Z \end{array}$ justify the statement "homology categorifies euler characteristic"? $\endgroup$ – Brian Fitzpatrick Jul 21 '14 at 21:30
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    $\begingroup$ homology is of course more than a function on objects; it's also a functor. $\endgroup$ – Qiaochu Yuan Jul 21 '14 at 21:50

If $S$ has a categorification $\mathrm{Decat}(\mathcal{C}) \to S$ and $T$ has a categorification $\mathrm{Decat}(\mathcal{D}) \to T$, then a categorificiation of a map $S \to T$ is a functor $\mathcal{C} \to \mathcal{D}$ such that the induced map $\mathrm{Decat}(\mathcal{C}) \to \mathrm{Decat}(\mathcal{D})$ makes the diagram $$\begin{array}{cc} \mathrm{Decat}(\mathcal{C})& \rightarrow & \mathrm{Decat}(\mathcal{D}) \\ \downarrow && \downarrow \\ S & \rightarrow & T \end{array}$$ commutative.

Example. The map $\mathbb{N} \times \mathbb{N} \to \mathbb{N}$, $(x,y) \mapsto x+y$ categorifies to the functor $\mathsf{FinSet} \times \mathsf{FinSet} \to \mathsf{FinSet}$, $(X,Y) \mapsto X \coprod Y$.

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    $\begingroup$ Ahh of course. Thanks. $\endgroup$ – Brian Fitzpatrick Jul 21 '14 at 21:34

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