twisted arrow category


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.

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 definition of morphism reversed, hence the twist.

As a lax colimit

The (opposite of the) category Tw(C)Tw(C) can be described as the lax colimit of the diagram CCat:cC/cC \to Cat\colon c\mapsto C/c.


If CC is a partially ordered set, then Tw(C)Tw(C) is isomorphic to the set of nonempty intervals [a,b]={cC|acb}[a,b]=\{c\in C| a\leq c\leq \b\} with aba\leq b in C ordered by inclusion (cf. Johnstone 1999).


  • 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 December 14, 2016 14:09:38 by Anonymous (