Category-less Category Theory. Note Quote.



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.


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


Let us prove the uniqueness for ι (for ι′ the proof goes analogously). Let ι1 and ι2 be identities, and αι1 and αι2 exist. Then αι1 = α and (αι12 = αι2. From axiom (1) it follows that ι1ι2 is determined. But ι1ι2 exists if an only if ι1 = ι2. Indeed, let us assume that ι1ι2 exist then ι1 = ι1ι2 = ι2. And vice versa, assume that ι1 = ι2. Then from axiom (2) it follows that there exists an identity ι such that ιι1 exists, and hence is equal to ι (because ι1 is an identity). This, in turn, means that (ιι12 exists, because (ιι12 = ιι2 = ιι1 = ι. Therefore, ι1ι2 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.


If for any two identities ι1 and ι2 the class ⟨ι12⟩ = {α : d(α) = ι1, c(α) = ι2},

is a set then objectless category theory is called small.


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 C0 denote the class of all identities of C. If ι123 ∈ C0, we define the composition

mC0ι123 : ⟨ι1, ι2⟩ × ⟨ι2, ι3⟩ → ⟨ι1, ι3

by mC0 (α, β) = βα. Class C is called objectless category.


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


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 C0 of objects,
(II) for each A,B ∈ C0,
a collection ⟨A,B⟩ C0 of morphisms from A to B,

(III) for each A,B,C ∈ C0, if α ∈ ⟨A,B⟩ C0 and β ∈ ⟨B,C⟩ C0, the composition

mC0 : ⟨A,B⟩ C0 × ⟨B,C⟩ C0 → ⟨A,C⟩ C0

is defined by mC0A,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 ∈ C0 there exists a morphism ιB ∈ ⟨B,B⟩C0 such that

A∈C0α∈⟨A,B⟩C0 ιBα = α, ∀C∈C0β∈⟨B,C⟩C0 βιB = 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.


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