# Twisted arrow categories

## Terminology

A twisted arrow category is an alternative name for a category of factorisations. That latter name is applied when discussing natural systems and Baues-Wirsching cohomology, whilst the name twisted arrow category is more often used in discussing Kan extensions and within the categorical literature.

## Definition

The twisted arrow category $\mathrm{Tw}\left(C\right)$ of $C$ a category is the category of elements of its hom-functor:

(1)$\mathrm{Tw}\left(C\right)=\mathrm{el}\left(\mathrm{hom}\right)=*/\mathrm{hom}$Tw(C) = el(hom) = * / hom

### Explicit description

Unpacking the well-known explicit construction of comma objects in $\mathrm{Cat}$ as comma categories, we get that $\mathrm{Tw}\left(C\right)$ has

• objects: $f$ an arrow in $C$, and

• morphisms: between $f$ and $g$ are pairs of arrows $\left(p,q\right)$ such that the following diagram commutes:

(2)$\begin{array}{ccc}A& \stackrel{p}{←}& C\\ f↓& & ↓g\\ B& \underset{q}{\to }& D\end{array}$\begin{matrix} A & \overset{p}{\leftarrow} & C \\ f \downarrow & & \downarrow g \\ B & \underset{q}{\to} & D \end{matrix}

you could view then morphisms from $f$ to $g$ as factorizations of $g$ through $f$.

### Origin of the name

From the description above, $\mathrm{Tw}\left(C\right)$ is the same as $\mathrm{Arr}\left(C\right)$ the arrow category of $C$, but with the direction of $p$ above in the def of morphism reversed, hence the twist.

## Properties

From its definition as a comma category, there’s a functor (a discrete opfibration, in fact)

(3)${\pi }_{C}:\mathrm{tw}\left(C\right)\to {C}^{\mathrm{op}}×C$\pi_C \colon tw(C) \to C^{op} \times C

which at the level of objects forgets the arrows:

(4)${\pi }_{C}\left(f:A\to B\right)=\left(A,B\right)$\pi_C(f \colon A \to B) = (A,B)

and keeps everything at the morphisms level.

### $\mathrm{tw}\left(C\right)$ and wedges

One could say that $\mathrm{tw}\left(C\right)$ classifies wedges?, in the sense that for any functor $F:{C}^{\mathrm{op}}×C\to B$,

are the same as

This can be used to give a proof of the reduction of ends to conical limits in the $\mathrm{Set}$-enriched case, and is used in the construction of ends in a derivator.

## References

The statement above is Ex. IX.6.3 in

• MacLane, Categories for the working mathematician - 2nd Edition

Revised on March 21, 2012 15:32:11 by Mike Shulman (71.136.234.110)