Definition:

Let us axiomatically define a theory which we shall call an objectless or object free category theory. In this theory, the only primitive concepts (besides the usual logical concepts and the equality concept) are:

(I) α is a morphism,

(II) the composition αβ is defined and is equal to γ, The following axioms are assumed:

1. Associativity of compositions: Let α, β, γ be morphisms. If the compositions βα and γβ exist, then

• the compositions γ(βα) and (γβ)α exist and are equal;

• if γ(βα) exists, then γβ exists, and if (γβ)α exists then βα exists.

2. Existence of identities: For every morphism α there exist morphisms ι and ι′, called identities, such that

• βι = β whenever βι is defined (and analogously for ι′),

• ιγ = γ whenever ιγ is defined (and analogously for ι′).

• αι and ι′α are defined.

Lemma:

Identities ι and ι′ of axiom (2) are uniquely determined by the morphism α.

Proof:

Let us prove the uniqueness for ι (for ι′ the proof goes analogously). Let ι_{1} and ι_{2} be identities, and αι_{1} and αι_{2} exist. Then αι_{1} = α and (αι_{1})ι_{2} = αι_{2}. From axiom (1) it follows that ι_{1}ι2 is determined. But ι_{1}ι_{2} exists if an only if ι_{1} = ι_{2}. Indeed, let us assume that ι_{1}ι_{2} exist then ι_{1} = ι_{1}ι_{2} = ι_{2}. And vice versa, assume that ι_{1} = ι_{2}. Then from axiom (2) it follows that there exists an identity ι such that ιι_{1} exists, and hence is equal to ι (because ι_{1} is an identity). This, in turn, means that (ιι_{1})ι_{2} exists, because (ιι_{1})ι_{2} = ιι_{2} = ιι_{1} = ι. Therefore, ι_{1}ι_{2} exists by Axiom 1.

Let us denote by d(α) and c(α) identities that are uniquely determined by a morphism α, i.e. such that the compositions αd(α) and c(α)α exist (letters d and c come from “domain” and “codomain”, respectively).

Lemma 2.2 The composition βα exists if and only if c(α) = d(β), and consequently,

d(βα) = d(α) and c(βα) = c(β).

Proof. Let c(α) = d(β) = ι, then βι and ια exist. From axiom (1) it follows that there exists the composition (βι)α = βα. Let us now assume that βα exists, and let us put ι = c(α). Then ια exists which implies that βα = β(ια) = (βι)α. Since βι exists then d(β) = ι.

Definition:

If for any two identities ι_{1} and ι_{2} the class ⟨ι_{1},ι_{2}⟩ = {α : d(α) = ι_{1}, c(α) = ι_{2}},

is a set then objectless category theory is called small.

Definition:

Let us choose a class C of morphisms of the objectless category theory (i.e. C is a model of the objectless category theory), and let C^{0} denote the class of all identities of C. If ι_{1},ι_{2},ι_{3} ∈ C^{0}, we define the composition

m^{C0}_{ι1,ι2,ι3} : ⟨ι_{1}, ι_{2}⟩ × ⟨ι_{2}, ι_{3}⟩ → ⟨ι_{1}, ι_{3}⟩

by m^{C0} (α, β) = βα. Class C is called objectless category.

Proposition:

The objectless category definition is equivalent to the standard definition of category.

Proof:

To prove the theorem it is enough to reformulate the standard category definition in the following way. A category C consists of

(I) a collection C^{0} of objects,

(II) for each A,B ∈ C^{0},

a collection ⟨A,B⟩ _{C0} of morphisms from A to B,

(III) for each A,B,C ∈ C^{0}, if α ∈ ⟨A,B⟩ _{C0} and β ∈ ⟨B,C⟩ _{C0}, the composition

m^{C0} : ⟨A,B⟩ _{C0} × ⟨B,C⟩ _{C0} → ⟨A,C⟩ _{C0}

is defined by m^{C0}_{A,B,C} (α, β). The following axioms are assumed

1. Associativity: If α ∈ ⟨A,B⟩_{C0}, β ∈ ⟨B,C⟩_{C0} , γ ∈ ⟨C,D⟩_{C0} then γ(βα) = (γβ)α.

2. Identities: For every B ∈ C^{0} there exists a morphism ι_{B} ∈ ⟨B,B⟩_{C0} such that

∀ _{A∈C0} ∀ _{α∈⟨A,B⟩C0} ι_{B}α = α, ∀_{C∈C0} ∀_{β∈⟨B,C⟩C0} βι_{B} = β.

To see the equivalence of the two definitions it is enough to suitably replace in the above definition objects by their corresponding identities.

This theorem creates three possibilities to look at the category theory: (1) the standard way, in terms of objects and morphisms, (2) the objectless way, in terms of morphisms only, (3) the hybrid way in which we take into account the existence of objects but, if necessary or useful, we regard them as identity morphisms.