nLab arrow category




Every category CC gives rise to an arrow category Arr(C)Arr(C) such that the objects of Arr(C)Arr(C) are the morphisms (or arrows, hence the name) of CC.


For CC any category, its arrow category Arr(C)Arr(C) is the category such that:

  • an object aa of Arr(C)Arr(C) is a morphism a:a 0a 1a\colon a_0 \to a_1 of CC;
  • a morphism f:abf\colon a \to b of Arr(C)Arr(C) is a commutative square
    a 0 f 0 b 0 a b a 1 f 1 b 1 \array { a_0 & \overset{f_0}\to & b_0 \\ \llap{a}\downarrow & & \rlap{b}\downarrow \\ a_1 & \underset{f_1}\to & b_1 }

    in CC;

  • composition in Arr(C)Arr(C) is given simply by placing commutative squares side by side to get a commutative oblong.

This is isomorphic to the functor category

Arr(C):=Funct(I,C)=[I,C]=C I Arr(C) := Funct(I,C) = [I,C] = C^I

for II the interval category {01}\{0 \to 1\}. Arr(C)Arr(C) is also written [2,C][\mathbf{2},C], C 2C^{\mathbf{2}}, [Δ[1],C][\Delta[1],C], or C Δ[1]C^{\Delta[1]}, since 2\mathbf{2} and Δ[1]\Delta[1] (for the 11-simplex) are common notations for the interval category.



The arrow category Arr(C)Arr(C) is equivalently the comma category (id/id)(id/id) for the case that id:CCid\colon C \to C is the identity functor.


Arr(C)Arr(C) plays the role of a directed path object for categories in that functors

XArr(Y) X \to Arr(Y)

are the same as natural transformations between functors between XX and YY.


(arrow category is Grothendieck construction on slice categories)
For 𝒮\mathcal{S} any category, let

𝒮 ():𝒮Cat \mathcal{S}_{(-)} \,\colon\, \mathcal{S} \longrightarrow Cat

be the pseudofunctor which sends

The Grothendieck construction on this functor is the arrow category Arr(𝒮)Arr(\mathcal{S}) of 𝒮\mathcal{S}:

Arr(𝒮) B𝒮𝒮 /B. Arr(\mathcal{S}) \;\;\; \simeq \;\;\; \int_{B \in \mathcal{S}} \mathcal{S}_{/B} \mathrlap{\,.}

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.