Category

Objects as identities

There is an equivalent formulation of the notion of category where an object is viewed as a special kind of morphism. Namely, one defines a category as a Class of morphisms equipped with

1. a unary operation (domain)
2. a unary operaton (dodomain)
3. a partial binary operation , also written as juxtaposition (composition)

such that for any

1. exists iff
7. iff either side exists

Thus an object is just a morphism that is the domain or codomain of another morphism, and these are precisely identities. The class of all such morphisms is denoted as . This unifies certain parts of the theory, as outlined below.

(Multi)functors

One advantage of this approach is that a functor becomes a single map on morphisms, preserving composition and (co)domains

The concept of a multifunctor, e.g. , is also unified slightly. In particular, it is clear how one can produce a functor by fixing an object , since in this conception .

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