nLab
arrow category

Contents

Idea

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.

Definition

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.

Up to equivalence, this is the same as 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.

Properties

Revised on July 26, 2016 10:04:27 by Brian Schroeder? (24.188.96.3)