# Has this "everything category" been defined before?

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?

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.
• @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.) Commented Feb 27 at 23:58