A decade ago, Gil Kalai asked the question What precisely Is "Categorification"?
After seeing some answers and some online pages, I think one of the meanings of categorification is the following:
It is not just that we want to associate a category for a set or a group or a manifold or a topological space. Along with assigning a category $F(A)$ with a set $A$, given a map of sets $A\rightarrow B$, we want to associate a functor $F(A)\rightarrow F(B)$. We want little more than this depending on the situation.
Consider a collection of objects that forms a category $\mathcal{C}$.
I think a categorification of (elements of objects of) $\mathcal{C}$ is a functor $F:\mathcal{C}\rightarrow \text{Cat}$, where $\text{Cat}$ is the category of all categories, such that the functor $F$ is an embedding.
Consider the collection of groups, the category formed by them is denoted by $\text{Grp}$. Then, a categorification is a functor $\text{Grp}\rightarrow \text{Cat}$ such that, this functor is an embedding of catgeories.
The objects of $\text{Grp}$ are not just sets but they come with two maps, namely $^{-1}:A\rightarrow A$ and $\cdot:A\times A\rightarrow A$. Then, categorification of $A$, should also come with (atleast) two functors $\cdot :F(A)\times F(A)\rightarrow F(A)$ and $^{-1}:F(A)\rightarrow F(A)$. Any such functor, I would like to call a categorification of the category of groups.
Question : Is this notion of categorification compatible with the already existing notion of categorification? Is my understanding correct? In this sense, what could be an example of a categorification of category of topological spaces? Should $\text{Man}\rightarrow \text{Cat}$ defined as $M\rightarrow (M\rightrightarrows M)$ considered as categorification of manifolds? Should $\text{LieGrp}\rightarrow \text{Cat}$ defined as $G\mapsto (G\rightrightarrows *)$ considered as a categorification of Lie groups? What are the other possible categorifications in these examples?
