Multicoequalizer diagrams in complete lattices #
We introduce the notion of bicartesian square (Lattice.BicartSq
) in a lattice T
.
This consists of elements x₁
, x₂
, x₃
and x₄
such that x₂ ⊔ x₃ = x₄
and
x₂ ⊓ x₃ = x₁
.
It shall be shown (TODO) that if T := Set X
, then the image of the
associated commutative square in the category Type _
is a bicartesian square
in a categorical sense (both pushout and pullback).
More generally, if T
is a complete lattice, x : T
, u : ι → T
, v : ι → ι → T
,
we introduce a property MulticoequalizerDiagram x u v
which says that x
is
the supremum of u
, and that for all i
and j
, v i j
is the minimum of u i
and u j
.
Again, when T := Set X
, we shall show (TOOD) that we obtain a multicoequalizer diagram
in the category of types.
The commutative square associated to a bicartesian square in a lattice.
A multicoequalizer diagram in a complete lattice T
consists of families of elements
u : ι → T
, v : ι → ι → T
, and an element x : T
such that x
is the supremum of u
,
and for any i
and j
, v i j
is the minimum of u i
and u j
.
Instances For
The multispan index in the category associated to the complete lattice T
given by the objects u i
and the minima v i j = u i ⊓ u j
,
when d : MulticoequalizerDiagram x u v
.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The multicofork in the category associated to the complete lattice T
associated to d : MulticoequalizerDiagram x u v
with x : T
.
(In the case T := Set X
, this multicofork becomes colimit after the application
of the obvious functor Set X ⥤ Type _
.)
Equations
- d.multicofork = CategoryTheory.Limits.Multicofork.ofπ d.multispanIndex x (fun (i : (CategoryTheory.Limits.MultispanShape.prod ι).R) => CategoryTheory.homOfLE ⋯) ⋯