- k_{j}:X_{j}\rightarrow Y} are two
coproducts of the
family { X j } {\displaystyle \lbrace X_{j}\rbrace } , then (by the
definition of
coproducts)
there exists a unique...
- morphism.
Pushouts are
equivalent to
coproducts and
coequalizers (if
there is an
initial object) in the
sense that:
Coproducts are a
pushout from the initial...
- and only if it has
coequalizers and all (small)
coproducts, or, equivalently,
pushouts and
coproducts.
Finite completeness can be
characterized in several...
- (also
called the
direct sum, free union, free sum,
topological sum, or
coproduct) of a
family of
topological spaces is a
space formed by
equipping the...
-
properties of the
category of modules. In such a category,
finite products and
coproducts agree and the
direct sum is
either of them, cf. biproduct.
General case:...
- an
initial object. Both
being limits, they are not
finite products nor
coproducts. Thus an
additive category is
equivalently described as a preadditive...
-
colimit generalizes constructions such as
disjoint unions,
direct sums,
coproducts,
pushouts and
direct limits.
Limits and colimits, like the
strongly related...
- (equivalently, all
finite coproducts); note that
because C is also preadditive,
finite products are the same as
finite coproducts,
making them biproducts;...
-
using the
order relation as the morphisms. In this case the
products and
coproducts correspond to
greatest lower bounds (meets) and
least upper bounds (joins)...
- in a
preadditive category must also be a
coproduct, and conversely. In fact,
finite products and
coproducts in
preadditive categories can be characterised...
