Limits and colimits
limits and colimits
limit and colimit
limits and colimits by example
commutativity of limits and colimits
connected limit, wide pullback
preserved limit, reflected limit, created limit
product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum
end and coend
The notion of disjoint coproduct is a generalization to arbitrary categories of that of disjoint union of sets.
One says that a coproduct of two objects in a category is disjoint if the intersection of with in is empty. In this case one writes for the coproduct and speaks of the disjoint union of with .
In a category
A coproduct in a category is disjoint if
the coprojections and are monic, and
their intersection is an initial object.
Equivalently, this means we have pullback squares
An arbitrary coproduct is disjoint if each coprojection is monic and the intersection of any two is initial. Note that every 0-ary coproduct (that, is initial object) is disjoint.
In a bicategory
Characterization of extensivity and of sheaf toposes
A category having all finitary disjoint coproducts is half of the condition for a category to be extensive.
Having all small disjoint coproducts is one of the conditions in Giraud's theorem characterizing sheaf toposes.
In coherent categories
This apears as (Johnstone, cor. A1.4.4).
(Johnstone, p. 34)
In a positive coherent category, every morphism into a coproduct factors through the coproduct coprojections:
Let be a postive coherent category, def. 1, and let be a morphism. Then the two subobjects and of , being the pullbacks in
are disjoint in and is their disjoint coproduct
This appears in (Johnstone, p. 34).
For instance page 34 in section A1.4.4 in