# 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 literature1.

## Definition

The twisted arrow category $Tw(C)$ of $C$ a category is the category of elements of its hom-functor:

(1)$Tw(C) = el(hom) = * / hom$

### Explicit description

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

• objects: $f$ an arrow in $C$, and

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

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

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

### Origin of the name

From the description above, $Tw(C)$ is the same as $Arr(C)$ 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)$\pi_C \colon tw(C) \to C^{op} \times C$

which at the level of objects forgets the arrows:

(4)$\pi_C(f \colon A \to B) = (A,B)$

and keeps everything at the morphisms level.

### $tw(C)$ and wedges

One could say that $tw(C)$ classifies wedges, in the sense that for any functor $F \colon C^{op} \times C \to B$,

are the same as

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

## Generalizations

• The twisted arrow category is a special case of a category of judgments in the sense of (Melliès-Zeilberger 15).

• The construction generalizes to $(\infty,1)$-categories (cf. Lurie 11, sec.4.2).

## References

1. Lawvere (‘Equality in hyperdoctrines’, 1970) uses the term ‘twisted morphism category’.

Revised on May 28, 2015 08:24:58 by Thomas Holder (89.204.139.157)