# The conceptual difference in notations of Cat

There have been some places in which a conceptual difference (that of criteria of identification, etc) is avoided in the categorical notation, namely

Let $Cat$ be at the same time the 2-category of small categories and its subjacent 1-category i.e. the category of small categories (...)

Why one can do this? The 2-category CAT and the 1-category Cat are not remarkably different? What is Cat (generally) in the literature..?

• It's even worse: I usually use Cat to denote the (2,1)-category of small categories (so a third possible meaning)! I imagine this happens because people routinely use only one of the meanings, and so never find useful to mention the other two possibilities... – Denis Nardin Oct 6 '17 at 14:59
• Like many other notations in mathematics, the meaning of "Cat" is context-dependent. If I'm writing a paper in which I need to discuss both meanings, I sometimes use different fonts, e.g. $\mathbf{Cat}$ for the 1-category and $\mathcal{C}\mathit{at}$ for the 2-category. – Mike Shulman Oct 6 '17 at 16:45
• To add more context to Mike Shulman's comment: Johnstone practices the same in 'A Topos Theory Compendium' (see 'Index of Notation'): $\mathbf{Cat}$ is for the 1-category, and $\mathfrak{Cat}$ is for (Johnstone's version of) the relevant 2-category. – Peter Heinig Oct 12 '17 at 10:46

As long as 2-category means strict 2-category this usage is exactly the same as using $M$ to name both a manifold and its set of points.
If $Cat$ is meant to be some weak 2-category then you need to specify what enrichment you want--or as you put it what criterion of identification. There is not only one possibility. In that case the usage only makes sense if one specified enrichment is being assumed in the context.