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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?

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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.

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    $\begingroup$ Worth perhaps pointing up the size issue a little more prominently: Whatever collection of “all categories” you start with, the resulting category All will be too big to appear in it. E.g. if you start with all small categories as this answer suggests, then All will be large. This can be shown by various proofs, e.g. the Burali-Forti paradox. $\endgroup$ Commented Feb 27 at 10:16
  • $\begingroup$ @PeterLeFanuLumsdaine : Yes, that's a good point. $ \int _ { a \in A } F ( a ) $ lives in the smallest universe that contains $ A $ and all of the categories of the form $ F ( a ) $, so it must be large if $ A $ is large, even if every $ F ( a ) $ is small. (And if you don't have the Axiom of Replacement in your universe, then you might need an even larger one.) $\endgroup$ Commented Feb 27 at 23:58

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