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.
objects: an arrow in , and
morphisms: between and are pairs of arrows such that the following diagram commutes:
you could view then morphisms from to as factorizations of through : .
From the description above, is the same as the arrow category of , but with the direction of 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)
which at the level of objects forgets the arrows:
and keeps everything at the morphisms level.
One could say that classifies wedges, in the sense that for any functor ,
are the same as
The twisted arrow category is a special case of a category of judgments in the sense of (Melliès-Zeilberger 15).
The construction generalizes to -categories (cf. Lurie 11, sec.4.2).
L. Errington, Twisted Systems , PhD thesis Imperial College London 1999.(doi)
Jacob Lurie, Derived Algebraic Geometry X: Formal Moduli Problems , ms. (2011).
Saunders Mac Lane, Categories for the Working Mathematician , Springer Heidelberg 1998². (cf. exercise IX.6.3, p.227)