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Exercise. [frct-001T]

Prove that the total category \(\widetilde {\underline {B}}\) of the fundamental self-indexing is the arrow category \(B^{\to }\), and the projection is the codomain functor.

Context

The total category and its projection [frct-000A]

2022

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

An object of \(\widetilde {E}\) is given by a pair \((x,\bar {x})\) where \(x\in B\) and \(\bar {x}\in E_{x}\).
A morphism \({(x,\bar {x})}\xrightarrow {{}}{(y,\bar {y})}\) in \(\widetilde {E}\) is given by a pair \((f,\bar {f})\) where \({x}\xrightarrow {{f}}{y}\) and \({\bar {x}}\xrightarrow [f]{\bar {f}}{\bar {y}}:\bar {x}\).

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

Exercise. [frct-001T]

Prove that the total category \(\widetilde {\underline {B}}\) of the fundamental self-indexing is the arrow category \(B^{\to }\), and the projection is the codomain functor.

Related

The total category and its projection [frct-000A]

2022

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

An object of \(\widetilde {E}\) is given by a pair \((x,\bar {x})\) where \(x\in B\) and \(\bar {x}\in E_{x}\).
A morphism \({(x,\bar {x})}\xrightarrow {{}}{(y,\bar {y})}\) in \(\widetilde {E}\) is given by a pair \((f,\bar {f})\) where \({x}\xrightarrow {{f}}{y}\) and \({\bar {x}}\xrightarrow [f]{\bar {f}}{\bar {y}}:\bar {x}\).

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

Exercise. [frct-001T]

Prove that the total category \(\widetilde {\underline {B}}\) of the fundamental self-indexing is the arrow category \(B^{\to }\), and the projection is the codomain functor.

The fundamental self-indexing [frct-0003]

Construction. The fundamental self-indexing [frct-001X]

Let \(B\) be an ordinary category; there is an important displayed category \(\underline {B}\) over \(B\) given fiberwise by the slices of \(B\).

For \(x\in B\), we define \(\underline {B}_{x}\) to be the collection \({B}_{/x}\) of pairs \((\bar {x}\in B,{\bar {x}}\xrightarrow {{p_{x}}}{x})\).
For \({x}\xrightarrow {{f}}{y}\in B\), we define \(\underline {B}_{f}\) to be the collection of commuting squares in the following configuration:

Exercise. Pullbacks and cartesian maps [frct-001Y]

Prove that the fundamental self-indexing \(\underline {B}\) is a cartesian fibration if and only if \(B\) has pullbacks.