Page 14 - Textos de Matemática Vol. 45

6 Chapter 1. Unital categories and intrinsic commutation
Definition 1.2.1. Let C be a pointed category with finite products. Given two objects A and B in C, consider the following diagram
πB //
// A×B oo B. (1.2.1)
⟨0,1B ⟩
oo πA ⟨1A ,0⟩
A
The category C is said to be unital if, for every pair of objects A, B ∈ C, the
morphisms ⟨1A,0⟩ and ⟨0,1B⟩ are jointly strongly epimorphic.
When C is finitely complete, this is equivalent to the fact that the object A×B is the supremum of the two subobjects ⟨1A,0⟩ and ⟨0,1B⟩; namely, any monomorphism j : J A × B containing A and B, as in the following diagram
<< J cc
j
ooπA πB//
A // A × B oo B, ⟨1A ,0⟩ ⟨0,1B ⟩
is an isomorphism.
Example 1.2.2. The category Mon of monoids is unital.
Proof. It suffices to prove that, for any pair of monoids A and B, any submonoid
M → A×B containing all the elements (a,1) and (1,b) is equal to A×B. Unital categories are a setting where it is possible to express a categorical
notion of commutativity.
Definition 1.2.3 ([10]). Let C be a unital category. Two morphisms with the same codomain f: X → Z and g: Y → Z are said to cooperate (or to com- mute) if there exists a morphism φ : X × Y → Z such that both triangles in the following diagram commute:
X ⟨1X ,0⟩// X × Y oo⟨0,1Y ⟩ Y
φ
## {{ Z.
The morphism φ is necessarily unique, because ⟨1X , 0⟩ and ⟨0, 1Y ⟩ are jointly (strongly) epimorphic, and it is called the cooperator of f and g.
The uniqueness of the cooperator makes commutativity a property and not an additional structure in the category C.
f
g