twisted arrow category

Twisted arrow categories


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.


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

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

Explicit description

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

  • objects: ff an arrow in CC, and

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

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

    you could view then morphisms from ff to gg as factorizations of gg through ff: g=qfpg=qfp.

Origin of the name

From the description above, Tw(C)Tw(C) is the same as Arr(C)Arr(C) the arrow category of CC, but with the direction of pp above in the def of morphism reversed, hence the twist.


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

(3)π C:tw(C)C op×C \pi_C \colon tw(C) \to C^{op} \times C

which at the level of objects forgets the arrows:

(4)π C(f:AB)=(A,B) \pi_C(f \colon A \to B) = (A,B)

and keeps everything at the morphisms level.

tw(C)tw(C) and wedges

One could say that tw(C)tw(C) classifies wedges, in the sense that for any functor F:C op×CBF \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 Set\mathbf{Set}-enriched case, and is used in the construction of ends in a derivator.


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

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


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

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