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
The dependent sum is a universal construction in category theory. It generalizes the Cartesian product to the situation where one factor may depend on the other. It is the left adjoint to the base change functor between slice categories.
The dual notion is that of dependent product.
Let be a category and a morphism in such that pullbacks along exist in . These then constitute a base change functor
between the corresponding slice categories.
The dependent sum or dependent coproduct along the morphism is the left adjoint of the base change functor
This is directly seen to be equivalent to the following.
The dependent sum along is the functor
given by composition with .
Relation to the product
Assume that the category has a terminal object . Let be any object and assume that the terminal morphism admits all pullbacks along it.
Notice that a pullback of some along is simply the product , equipped with its projection morphism . We may regard as the image of the base change functor , but this is not quite just the product in , which instead corresponds to the terminal morphisms . But we have:
The product is, if it exists, equivalently the dependent sum of the base change of along :
Here we write “” also for the morphism .
Relation to type theory
Under the relation between category theory and type theory the dependent is the categorical semantics of dependent sum types .
Notice that under the identification of propositions as types, dependent sum types (or rather their bracket types) correspond to existential quantification .
The following table shows how the natural deduction rules for dependent sum types correspond to their categorical semantics given by the dependent sum universal construction.
|type theory||category theory|
|natural deduction||universal construction|
|dependent sum type||dependent sum|
Relation to some limits
For a category with finite limits and an object, then dependent sum
preserves and reflects fiber products.
For a category with finite limits and any object, the naturality square of the unit of the -adjunction on any morphism in
is a pullback.
By prop. 2 it suffices to see that the diagram is a pullback in under , where, by Frobenius reciprocity, it becomes
For a category with finite limits and any object, the naturality square of the counit of the -adjunction on any morphism in
is a pullback.
By Frobenius reciprocity the diagram is equivalent to
Standard textbook accounts include section A1.5.3 of
and section IV of