Page 15 - Textos de Matemática Vol. 45

1.2. Unital categories 7 Definition 1.2.4. An object A of a unital category C is said to be commutative
if the identity 1A cooperates with itself.
The cooperator m: A × A → A of a commutative object A endows A with a canonical structure of internal commutative monoid in C, i.e. the binary operation m satisfies the axioms of an internal commutative monoid (see the Appendix).
Proposition 1.2.5. In Mon, two morphisms f and g, as in Definition 1.2.3, cooperate if and only if
f(x)g(y) = g(y)f(x) for all x ∈ X, y ∈ Y. (1.2.2) Proof. If Condition 1.2.2 is true, we can define the cooperator as
φ(x, y) = f (x)g(y).
It is easy to show that φ is a morphism and φ⟨1X,0⟩ = f, φ⟨0,1Y ⟩ = g.
Conversely, if a cooperator φ exists, then we have
φ(x, y) = φ((x, 1)(1, y)) = φ(x, 1)φ(1, y) = f (x)g(y),
and
foranyx∈X andy∈Y,andhencewegetCondition1.2.2.
Corollary 1.2.6. A monoid A is commutative in the categorical sense if and only if it is commutative in the classical sense, i.e. xy = yx for any x, y ∈ X.
Any functor U : C → D between unital categories which preserves products preserves the cooperating pairs as well.
Definition 1.2.7. A split epimorphism (A, B, f, s) in C is said to be a strongly split epimorphism if the pair (k, s) in the associated split sequence
f
Lemma 1.2.8. ([26], Proposition 2.6 and [12], Proposition 1.9) If (A, B, f, s) is a strongly split epimorphism, then f is the cokernel of k. In other words the split sequence
//k//oos //
0 K[f] A ////B 0
f
φ(x, y) = φ((1, y)(x, 1)) = φ(1, y)φ(x, 1) = g(y)f (x)
k // oo s K[f] A
// // B,
with fs = 1B and k = ker f, is jointly strongly epimorphic.
is exact.