nLab
twisted arrow category

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

(1)$$\mathrm{Tw}(C)=\mathrm{el}(\mathrm{hom})=*/\mathrm{hom}$$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 $\mathrm{Tw}(C)$ 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)$$\begin{array}{ccc}A& \stackrel{p}{\leftarrow }& C\\ f\downarrow & & \downarrow g\\ B& \underset{q}{\to }& D\end{array}$$\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, $\mathrm{Tw}(C)$ is the same as $\mathrm{Arr}(C)$ 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)$${\pi }_C:\mathrm{tw}(C)\to C^{\mathrm{op}}\times C$$\pi_C \colon tw(C) \to C^{op} \times C

which at the level of objects forgets the arrows:

(4)$${\pi }_C(f:A\to B)=(A,B)$$\pi_C(f \colon A \to B) = (A,B)

and keeps everything at the morphisms level.

$\mathrm{tw}(C)$ and wedges

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

• extranatural transformations $e\stackrel{•}{\to }F$

are the same as

• natural transformations $e\to F\circ {\pi }_C$

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