The Factorisation Category of a Category #
Factorisation f is the category containing as objects all factorisations of a morphism
We show that
Factorisation f always has an initial and a terminal object.
TODO: Show that
Factorisation f is isomorphic to a comma category in two ways.
MonoFactorisation f a special case of a
- mid : C
The midpoint of the factorisation.
- ι : X ⟶ s.mid
The morphism into the factorisation midpoint.
- π : s.mid ⟶ Y
The morphism out of the factorisation midpoint.
The factorisation condition.
Factorisations of a morphism
f as a structure, containing, one object, two morphisms,
and the condition that their composition equals
- h : d.mid ⟶ e.mid
The morphism between the midpoints of the factorizations.
The left commuting triangle of the factorization morphism.
The right commuting triangle of the factorization morphism.
Factorisation f consist of morphism between their midpoints and the obvious
Composition of morphisms in