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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the twisted diagonal of a

category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...

(also called the twisted arrow category), which makes the morphisms of a category into the objects of a new category, whose morphisms are then pairs of morphisms connecting domain and codomain with the twist coming from them being in opposite directions. It can be constructed as the

category of elements In category theory, a branch of mathematics, the category of elements of a presheaf is a category associated to that presheaf whose objects are the elements of sets in the presheaf. It and its generalization are also known as the Grothendieck cons ...

of the

Hom functor In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between object (category theory), objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applicati ...

, which makes the twist come from the fact that it is contravariant in the first entry and covariant in the second entry. It can be generalized to the twisted diagonal of a simplicial set to which it corresponds under the

nerve A nerve is an enclosed, cable-like bundle of nerve fibers (called axons). Nerves have historically been considered the basic units of the peripheral nervous system. A nerve provides a common pathway for the Electrochemistry, electrochemical nerv ...

construction.

Definition

For a category

\mathcal

, its ''twisted diagonal''

\operatorname(\mathcal)

is a category, whose objects are its arrows: :

\operatorname(\operatorname(\mathcal))
=\operatorname(\mathcal)

and for which the morphisms between two such objects

f\colon A\rightarrow B

and

g\colon X\rightarrow Y

are the pairs

p\colon X\rightarrow A

and

q\colon B\rightarrow Y

of morphisms in

\mathcal

so that

g=q\circ f\circ p

. If

/math> denotes the category 0\rightarrow 1 with two objects and one non-trivial morphism (with the notation taken from the

simplex category In mathematics, the simplex category (or simplicial category or nonempty finite ordinal category) is the category of non-empty finite ordinals and order-preserving maps. It is used to define simplicial and cosimplicial objects. Formal definition ...

), then the twisted arrow category

\operatorname(\mathcal)

is ''not'' the functor category

\operatorname(\mathcal)

since the morphisms between the domains is reversed. An alternative definition is as the

of the

: :

\operatorname(\mathcal)
:=\operatorname(\operatorname_\mathcal).

There is a canonical functor: :

\operatorname(\mathcal)\rightarrow\mathcal^\mathrm\times\mathcal,(f\colon X\rightarrow Y)\mapsto(X,Y).

Properties

* Under the

, the twisted diagonal of categories corresponds to the twisted diagonal of simplicial sets. For a category

\mathcal

, one has:Kerodon
Proposition 8.1.1.10.
/ref> *:

N\operatorname(\mathcal)
=\operatorname(N\mathcal).

* Slice and coslice categories arise through pullbacks from the twisted arrow category. For a category

\mathcal

, one has:Kerodon
Remark 8.1.0.6.
/ref> *:

X\backslash\mathcal
\cong\\times_\operatorname(\mathcal),

\mathcal/X
\cong\operatorname(\mathcal)\times_\mathcal\{X\}.

References

External links

Twisted Arrows and Cospans
on Kerodon Category theory