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 literature¹.

Definition

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

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

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:

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

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

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 definition of morphism reversed, hence the twist.

As a lax colimit

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

Example

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

Properties

• From its definition as a comma category, there is a functor (a discrete opfibration, in fact)
  $$\pi_C \colon Tw(C) \to C^{op} \times C$$

  which at the level of objects forgets the arrows:

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

  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 $F \colon C^{op} \times C \to B$,

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

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 $\mathbf{Set}$-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 $(\infty,1)$-category theory?, see at twisted arrow (∞,1)-category.

Related entries

• category of factorizations

• twisted arrow (∞,1)-category

References

• M. Bunge, S. Niefield, Exponentiability and single universes, JPAA 148 (2000) pp.217-250.

• L. Errington, Twisted Systems, PhD thesis Imperial College London 1999. (doi)

• Peter Johnstone, A Note on Discrete Conduché Fibrations, TAC 5 no.1 (1999) pp.1-11. (pdf)

• Fred Linton, Autonomous categories and duality of functors, J. Algebra 2 no.3 (1965) pp.315-349.

• 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)

• Paul-André Melliès, Noam Zeilberger, Isbell Duality for Refinement Types, arXiv:1501.05115 (2015).

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