Define a category $\textbf{All}$ as follows:
- The objects of $\textbf{All}$ are the pairs $(X, C)$, where $C$ is a category and $X$ is an object of $C$.
- A morphism from $(X, C)$ to $(Y, D)$ in $\textbf{All}$ is an ordered pair $(F:C\to D, f:F(X)\to Y)$, where $F$ is a functor and $f$ is a morphism in $D$.
- Composition in $\textbf{All}$ is given by: $(F, f)\circ (G, g)=(GF, gG(f))$.
This category seems to capture the idea of taking the disjoint union of all categories, and then "bridging them together" with functors. The idea seems so natural that I feel it must have been defined before, but I can't find what it is called. Does anyone have a reference?
Best Answer
I don't know of a reference for the specific category $ \mathbf { All } $, but it's the result of applying the Grothendieck construction to the identity functor on $ \mathbf { Cat } $ (the category of small categories). That is, $ \mathbf { All } = \int \mathrm { id } _ { \mathbf { Cat } } = \int _ { C \in \mathbf { Cat } } C $.
The Grothendieck construction can be applied to any functor $ F \colon A \to \mathbf { Cat } $ from any category $ A $ to produce a category $ \int F = \int _ { a \in A } F ( a ) $. The notation $ \int $ for the Grothendieck construction is meant to suggest a more high-powered version of $ \sum $, which can be used for the disjoint union; any function $ f \colon A \to \mathbf { Set } $ gives a set $ \sum f = \sum _ { a \in A } f ( a ) $ whose elements are pairs $ ( a , x ) $ where $ a \in A $ and $ x \in f ( a ) $. So up one level, the objects of $ \int _ { a \in A } F ( a ) $ are pairs $ ( a , x ) $ where $ a \in A $ (meaning as an object) and $ x \in F ( a ) $, but there are also morphisms and composition (which I won't write out in full but it's like your construction of $ \mathbf { All } $).
As Peter hinted in a comment, the size of $ \int F $ is based on the size of $ A $ as well as the size of the elements of $ \mathbf { Cat } $, so $ \int F $ is small if $ A $ is small (since I said that $ \mathbf { Cat } $ consists of small categories), but $ \int F $ is large if $ A $ is large (even though I said that $ \mathbf { Cat } $ consists of small categories). In particular, $ \mathbf { All } $ is large even if it's made only out of small categories; and it has to be extra-large if you make it out of large categories. (This is also an issue one level down; $ \sum \mathrm { id } _ { \mathbf { Set } } $, the disjoint union of all sets, must be a proper class.)
Anyway, you can read about the Grothendieck construction at Wikipedia, the nLab, and the references on those two pages; but if you already know about that and want references for $ \int \mathrm { id } _ { \mathbf { Cat } } $ specifically, then I can't help you.