3.1. The total category and its projection [000A]

Note that any displayed category $E$ over $B$ can be viewed as an undisplayed category $\TotCat{E}$ equipped with a projection functor $p\Sub{E}: \TotCat{E}\to B$; in this case $\TotCat{E}$ is called the total category of $E$.

  1. An object of $\TotCat{E}$ is given by a pair $(x,\bar{x})$ where $x\in B$ and $\bar{x}\in E\Sub{x}$.
  2. A morphism $(x,\bar{x})\to (y,\bar{y})$ in $\TotCat{E}$ is given by a pair $(f,\bar{f})$ where $f:x\to y$ and $\bar{f}:\bar{x}\to\Sub{f}\bar{y}$.

The construction of the total category of a displayed category is called the Grothendieck construction.

[001T] Exercise 3.1·a.

Prove that the total category $\TotCat{\SelfIx{B}}$ of the canonical self-indexing (Section 2.1 [0003]) is the arrow category $B^{\to}$, and the projection is the codomain functor.