Every category $C$ gives rise to an arrow category $Arr(C)$ such that the objects of $Arr(C)$ are the morphisms (or arrows, hence the name) of $C$.
For $C$ any category, its arrow category $Arr(C)$ is the category such that:
in $C$;
This is isomorphic to the functor category
for $I$ the interval category $\{0 \to 1\}$. $Arr(C)$ is also written $[\mathbf{2},C]$, $C^{\mathbf{2}}$, $[\Delta[1],C]$, or $C^{\Delta[1]}$, since $\mathbf{2}$ and $\Delta[1]$ (for the $1$-simplex) are common notations for the interval category.
The arrow category $Arr(C)$ is equivalently the comma category $(id/id)$ for the case that $id\colon C \to C$ is the identity functor.
$Arr(C)$ plays the role of a directed path object for categories in that functors
are the same as natural transformations between functors between $X$ and $Y$.
(arrow category is Grothendieck construction on slice categories)
For $\mathcal{S}$ any category, let
be the pseudofunctor which sends
an object $B \,\in\, \mathcal{S}$ to the slice category $\mathcal{S}_{/B}$,
a morphism $f \colon B \to B'$ to the left base change functor $f_! \,\colon\, \mathcal{C}_{B} \to \mathcal{C}_{/B'}$ given by post-composition in $\mathcal{C}$.
The Grothendieck construction on this functor is the arrow category $Arr(\mathcal{S})$ of $\mathcal{S}$:
This follows readily by unwinding the definitions. In the refinement to the Grothendieck construction for model categories (here: slice model categories and model structures on functors) this equivalence is also considered for instance in Harpaz & Prasma (2015), above Cor. 6.1.2.
The correponding Grothendieck fibration is also known as the codomain fibration.
Last revised on April 1, 2023 at 16:23:38. See the history of this page for a list of all contributions to it.