A category is a **skeleton** if, roughly speaking, no two distinct objects within the category are isomorphic. To every category is associated a skeleton, and two categories are categorically "equivalent" if and only if their skeletons are isomorphic. A fuller definition can be found here.

Consider the subcategory of $\bf{Cat}$ which takes as objects those categories which are skeletons, and morphisms the functors between them; I will call this $\bf{{Cat}_{Skel}}$. Note that this is not the skeleton of $\bf{Cat}$ itself, but a subcategory of $\bf{Cat}$ in which the *objects* are skeletal categories.

Within this subcategory of skeletal categories, there are a number of objects which are, themselves, isomorphic. So we can take the skeleton of this category, hence obtaining a new category, which I will call $\bf{Skel({Cat}_{Skel})}$. (I don't care which skeleton you take; pick one.)

This category is noteworthy in that it contains one object for each *equivalence* class of categories in $\textbf{Cat}$, making it perhaps more useful than looking at $\bf{Skel({Cat})}$ itself, which only contains one object for each *isomorphism* class of categories. My questions are:

- Does $\bf{Skel({Cat}_{Skel})}$ have a name?
- Has this category been studied in any detail, and if so, can someone please reference me towards any research that's been done on its structure?
- Is there an essentially equivalent construction which might be defined more simply than the way I've laid it out here?
- Are there any useful areas of study in which this category naturally arises?

Lastly, I've glossed over the usual foundational issues which arise when considering $\bf{Cat}$, mostly because I don't care whether you use Grothendieck universes, or a class-set theory, or only look at small categories, or some other way of solving the problem. Feel free to use any foundational approach that you want which makes $\bf{Skel({Cat}_{Skel})}$ to be consistent.

lotsof nontrivial isomorphisms. They just all must be automorphisms. A category withnonontrivial isomoprhisms is "gaunt" ncatlab.org/nlab/show/gaunt+category. $\endgroup$ – Theo Johnson-Freyd Feb 9 '13 at 4:099more comments