nLab
subquotient
Redirected from "subquotient objects".
Contents
Context
Limits and colimits
limits and colimits
1-Categorical
limit and colimit
limits and colimits by example
commutativity of limits and colimits
small limit
filtered colimit
sifted colimit
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
finite limit
Kan extension
weighted limit
end and coend
fibered limit
2-Categorical
(∞,1)-Categorical
Model-categorical
Category theory
Contents
Definition
A subquotient of an object X X in some category is
or
a subobject of a quotient object of X X .
Properties
The two definitions are equivalent whenever quotients are stable under pullback and subobjects are stable under pushout, such as in a topos .
Just as with subobjects and quotient objects, we have that X X is a subquotient of itself, and subquotients of subquotients of X X are themselves subquotients of X X in a natural way.
Just as subobjects of a set X X are in correspondence with predicate s on X X and quotients of X X are in correspondence with equivalence relation s on X X , subquotients of X X are in correspondence with partial equivalence relation s on X X .
Examples
Last revised on January 1, 2011 at 09:33:03.
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