nLab
dependent sum

Idea

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.

Definition

Let $𝒞$ be a category and $f:A\to I$ a morphism in $𝒞$ such that pullbacks along $f$ exist in $𝒞$. These then constitute a base change functor

$$f^{*}:{𝒞}_{/I}\to {𝒞}_{/A}$$f^* \colon \mathcal{C}_{/I} \to \mathcal{C}_{/A}

between the corresponding slice categories.

Definition

The dependent sum or dependent coproduct along the morphism $f$ is the left adjoint ${\sum }_f:{𝒞}_{/A}\to {𝒞}_{/I}$ of the base change functor

$$(\sum _f⊢f^{*}):{𝒞}_{/A}\stackrel{\stackrel{\sum _f}{\to }}{\underset{f^{*}}{\leftarrow }}{𝒞}_{/I}.$$(\sum_f \vdash f^* ) \colon \mathcal{C}_{/A} \stackrel{\overset{\sum_f}{\to}}{\underset{f^*}{\leftarrow}} \mathcal{C}_{/I} \,.

This is directly seen to be equivalent to the following.

Definition

The dependent sum along $f:A\to I$ is the functor

$$\sum _f≔f\circ (-):{𝒞}_{/A}\to {𝒞}_{/I}$$\sum_f \coloneqq f\circ (-) \colon \mathcal{C}_{/A} \to \mathcal{C}_{/I}

given by composition with $f$.

Properties

Relation to the product

Assume that the category $𝒞$ has a terminal object $*\in 𝒞$. Let $X\in 𝒞$ be any object and assume that the terminal morphism $f:X\to *$ admits all pullbacks along it.

Notice that a pullback of some $A\to *$ along $X\to *$ is simply the product $X\times A$, equipped with its projection morphism $X\times A\to X$. But if we regard this as the image of the base change functor $f^{*}:{𝒞}_{/*}\to {𝒞}_{/X}$ then it is not quite just the product in $𝒞$. Instead we have:

Proposition

The product $X\times A\in 𝒞$ is, if it exists, equivalently the dependent sum of the base change of $A\to *$ along $X\to *$:

$$\sum _X{X}^{*}A\simeq X\times A\in 𝒞.$$\sum_{X} X^* A \simeq X \times A \in \mathcal{C} \,.

Here we write ”$X$” also for the morphism $X\to *$.

Relation to type theory

Under the relation between category theory and type theory the dependent is the categorical semantics of dependent sum types .

Notce that under the identification of propositions as types, dependent sum types (or rather their bracket types) correspond to existential quantification $\exists x:X,Px$.

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
	syntax	semantics
	natural deduction	universal construction
	dependent sum type	dependent sum
type formation	$\frac{⊢X:\mathrm{Type}x:X⊢A(x):\mathrm{Type}}{⊢\left({\sum }_{x:X}A\left(x\right)\right):\mathrm{Type}}$
term introduction	$\frac{x:X⊢a:A(x)}{⊢(x,a):{\sum }_{x\prime :X}A(x\prime )}$
term elimination	$\frac{⊢t:\left({\sum }_{x:X}A\left(x\right)\right)}{⊢p_1(t):X⊢p_2(t):A(p_1(t))}$
computation rule	$p_1(x,a)=x{p}_2(x,a)=a$

Related concepts

• product, coproduct

• Kan extension

• base change

  • dependent sum, dependent product

  • dependent sum type, dependent product type

References

Standard textbook accounts include section A1.5.3 of

• Peter Johnstone, Sketches of an Elephant

and section IV of

• Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic