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 𝖢 of morphisms equipped with

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

such that for any 𝑓,𝑔,ℎ ∈𝖢

1. 𝑓𝑔 exists iff dom⁡𝑓 =cod⁡𝑔
2. dom⁡(𝑓𝑔) =dom⁡𝑔
3. cod⁡(𝑓𝑔) =cod⁡𝑓
4. cod⁡(dom⁡𝑓) =dom⁡𝑓
5. dom⁡(cod⁡𝑓) =cod⁡𝑓
6. (cod⁡𝑓)𝑓 =𝑓 =𝑓(dom⁡𝑓)
7. (𝑓𝑔)ℎ =𝑓(𝑔ℎ) 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 Ob⁡𝖢. This unifies certain parts of the theory, as outlined below.

(Multi)functors

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

𝐹(𝑔𝑓)=(𝐹𝑔)(𝐹𝑓)dom⁡(𝐹𝑓)=𝐹(dom⁡𝑓)cod⁡(𝐹𝑓)=𝐹(cod⁡𝑓)

The concept of a multifunctor, e.g. 𝐹 :𝖢 ×𝖣 →𝖤, is also unified slightly. In particular, it is clear how one can produce a functor 𝐹( −,𝐴) :𝖢 →𝖤 by fixing an object 𝐴, since in this conception 𝐴 =id𝐴.

---

develop | en | SemBr