Page 16 - Textos de Matemática Vol. 45

8 Chapter 1. Unital categories and intrinsic commutation Proof. Given a morphism g : A → D such that gk = 0, we have that gs makes
the triangle below commutative:
K[f] k //Aoo f //B
s
gs
g
D.
Indeed:
and since k and s are jointly (strongly) epimorphic, we have that gsf = g.
gsfs=gs and gsfk=0=gk, Moreover, given any h: B → D such that hf = g, we have that
h = hfs = gs.

Definition 1.3.1. The category C is said to be C′-unital when, for any object A ∈ C′ and any object B ∈ C, the morphisms ⟨1A,0⟩ and ⟨0,1B⟩ in the following diagram are jointly strongly epimorphic:
1.3 C′ -unital categories
Let C′ be a full subcategory of a pointed category C with finite products.
A
oo πA ⟨1A ,0⟩
πB //
// A×B oo B. (1.3.3)
⟨0,1B ⟩
In a C′-unital category we can still speak of cooperating pairs (f,g) of morphisms, provided that the domain X of f belongs to C′. More generally, X × Y being isomorphic to Y × X, we can speak of cooperating pair as soon as the domain of one of the two maps is in C′. Accordingly, we can still speak of commutative objects in C′.
Proposition 1.3.2. Suppose that C is C′-unital and that C′ is closed under finite products (and thus it contains the zero object 0). Then C′ is unital.
Proof. Straightforward. 1.4 The fibration of points
Let E be any category. We shall denote by PtE the category whose objects are the split epimorphisms in E and whose arrows are the commuting squares