Preserving (co)equalizers #
Constructions to relate the notions of preserving (co)equalizers and reflecting (co)equalizers to concrete (co)forks.
In particular, we show that equalizerComparison f g G
is an isomorphism iff G
preserves
the limit of the parallel pair f,g
, as well as the dual result.
The map of a fork is a limit iff the fork consisting of the mapped morphisms is a limit. This
essentially lets us commute Fork.ofι
with Functor.mapCone
.
Instances For
The property of preserving equalizers expressed in terms of forks.
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The property of reflecting equalizers expressed in terms of forks.
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If G
preserves equalizers and C
has them, then the fork constructed of the mapped morphisms of
a fork is a limit.
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If the equalizer comparison map for G
at (f,g)
is an isomorphism, then G
preserves the
equalizer of (f,g)
.
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If G
preserves the equalizer of (f,g)
, then the equalizer comparison map for G
at (f,g)
is
an isomorphism.
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The map of a cofork is a colimit iff the cofork consisting of the mapped morphisms is a colimit.
This essentially lets us commute Cofork.ofπ
with Functor.mapCocone
.
Instances For
The property of preserving coequalizers expressed in terms of coforks.
Instances For
The property of reflecting coequalizers expressed in terms of coforks.
Instances For
If G
preserves coequalizers and C
has them, then the cofork constructed of the mapped morphisms
of a cofork is a colimit.
Instances For
If the coequalizer comparison map for G
at (f,g)
is an isomorphism, then G
preserves the
coequalizer of (f,g)
.
Instances For
If G
preserves the coequalizer of (f,g)
, then the coequalizer comparison map for G
at (f,g)
is an isomorphism.
Instances For
Any functor preserves coequalizers of split pairs.