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
In category theory
An equalizer is a limit
over a parallel pair i.e. of the diagram of the shape
(See also fork diagram).
This means that for and two parallel morphisms in a category , their equalizer is, if it exists
an object ;
- pulled back to both morphisms become equal:
- and is the universal object with this property.
The dual concept is that of coequalizer.
In type theory
In type theory the equalizer
is given by the dependent sum over the dependent equality type
In Set the equalizer of two functions of sets is the subset of elements of on which both functions coincide.
For a category with zero object the equalizer of a morphism with the corresponding zero morphism is the kernel of .
For the given diagram, first form the pullback
This gives a morphism into the product.
Define to be the further pullback
One checks that the vertical morphism equalizes and and that it does so universally.
If a category has products and equalizers, then it has limits; see there.