Contents
Context
Limits and colimits
limits and colimits
1-Categorical
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limit and colimit
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limits and colimits by example
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commutativity of limits and colimits
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small limit
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filtered colimit
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sifted colimit
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connected limit, wide pullback
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preserved limit, reflected limit, created limit
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product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum
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finite limit
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Kan extension
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weighted limit
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end and coend
2-Categorical
(∞,1)-Categorical
Model-categorical
Contents
Definition
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
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an object ;
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a morphism
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such that
- 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
Examples
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In Set the equalizer of two functions of sets is the subset of elements of on which both functions coincide.
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For a category with zero object the equalizer of a morphism with the corresponding zero morphism is the kernel of .
Properties
Proposition
A category has equalizers if it has binary products and pullbacks.
Proof
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.
Proposition
If a category has equalizers and (finite) products, then it has (finite) limits.
For the finite case, we may say equivalently:
Proposition
If a category has equalizers, binary products and a terminal object, then it has finite limits.
Proposition
Let be an arrow in a category which is an equaliser of a pair of arrows of . Then is a monomorphism.
Proof
If are arrows of such that , then it follows immediately from the uniqueness part of the universal property of an equaliser that .