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

a unary operation $d o m$ (domain)
a unary operaton $c o d$ (dodomain)
a partial binary operation $(\circ)$ , also written as juxtaposition (composition)

such that for any $𝑓, 𝑔, ℎ \in 𝖢$

$𝑓 𝑔$ exists iff $d o m 𝑓 = c o d 𝑔$
$d o m (𝑓 𝑔) = d o m 𝑔$
$c o d (𝑓 𝑔) = c o d 𝑓$
$c o d (d o m 𝑓) = d o m 𝑓$
$d o m (c o d 𝑓) = c o d 𝑓$
$(c o d 𝑓) 𝑓 = 𝑓 = 𝑓 (d o m 𝑓)$
$(𝑓 𝑔) ℎ = 𝑓 (𝑔 ℎ)$ 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 $O b 𝖢$ . This unifies certain parts of the theory, as outlined below.

(Multi)functors

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

𝐹 (𝑔 𝑓) = (𝐹 𝑔) (𝐹 𝑓) d o m (𝐹 𝑓) = 𝐹 (d o m 𝑓) c o d (𝐹 𝑓) = 𝐹 (c o d 𝑓)

The concept of a multifunctor, e.g. $𝐹 : 𝖢 \times 𝖣 \to 𝖤$ , is also unified slightly. In particular, it is clear how one can produce a functor $𝐹 (-, 𝐴) : 𝖢 \to 𝖤$ by fixing an object $𝐴$ , since in this conception $𝐴 = i d 𝐴$ .

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