Straightening for 1-Categories (Grothendieck Construction)
Table of Contents
The Grothendieck construction is a special case of a very general question. In any category (or context) it is natural to ask which morphisms
\begin{equation*} \xymatrix{ E \ar[d]^p\\ B } \end{equation*}arise as a pullback along a classifying morphism \(c_p \colon B \rightarrow U \) to some universal object \(U \) of some universal morphism \(V \rightarrow U \). The Grothendieck construction describes this in the context of \(\Cat \). A functor \(p \colon \mathcal E \rightarrow \mathcal B \) is a Grothendieck fibration if and only if it is a pullback along \(\int_p \colon \mathcal B ^{op} \rightarrow \Cat \) of
1. QUESTION What is the universal morphism here?
Let \( p \colon \mathcal E \rightarrow \mathcal B \) be a functor of \( 1 \)-categories. \( p \) is a (Grothendieck) fibration (fibered category) if for every object \( y \in \mathcal E \) every morphism to \( p(y) \) lifts to a cartesion morphism in \( \mathcal E \). Explicitly for every object \(y \in \mathcal E \) every morphism \(g \colon z \rightarrow p(y) \) there is \(f \colon x \rightarrow y \) with \(p(f) = g \) such that every diagram of the below form has a unique lift:
\begin{equation*} \xymatrix{ \mathcal E \ar@{->}[ddd]_{p} & x' \ar@{->}[rd]^{h} \ar@{-->}[d]_{\exists! u} & \\ & x \ar@{->}[r]^{f} & y \\ & p(x') \ar@{->}[d]_{v} \ar@{->}[rd] & \\ \mathcal B & p(x) \ar@{->}[r]_{p(f) = g} & p(y) } \end{equation*}Let \(p \colon \mathcal E \rightarrow \mathcal B \) be a \(1 \)-functor. Then \(p \) is a fibration if and only if for every object \(y \in \mathcal E \) the canonical section \(s \colon \mathcal B _{/p(y)} \rightarrow \mathcal E _{/y} \) is an almost functor.
Proof.
Let \(F \colon \mathcal B ^{op} \rightarrow \Cat \) be a pseudofunctor of
discrete (op)fibration
discrete (op)fibrations are equivalent to (co)presheaves. Representable presheaves correspond to slices
categories fibered in groupoids