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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.

Beware that the term “sum” in “dependent sum” is really referring to **coproducts . The dual notion is that of dependent product.**

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

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

between the corresponding slice categories.

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

$(\sum_f \dashv f^* )
\colon
\mathcal{C}_{/A}
\stackrel{\overset{\sum_f}{\to}}{\underset{f^*}{\leftarrow}}
\mathcal{C}_{/I}
\qquad.$

This is directly seen to be equivalent to the following.

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

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

given by composition with $f$.

Assume that the category $\mathcal{C}$ has a terminal object $* \in \mathcal{C}$. Let $X \in \mathcal{C}$ be any object and assume that the terminal morphism $f \colon X \to *$ 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 $X \times A \to X$. We may regard $X \times A \to X$ as the image of the base change functor $f^* \colon \mathcal{C}_{/*} \to \mathcal{C}_{/X}$, but this is not quite just the product in $\mathcal{C}$, which instead corresponds to the terminal morphisms $X \times A \to *$. But we have:

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

$\sum_{X} X^* A \simeq X \times A \in \mathcal{C}
\quad .$

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

Under the relation between category theory and type theory the dependent sum 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 $\exists x\colon X, P x$.

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.

The inference rules for dependent pair types (aka “dependent sum types” or “$\Sigma$-types”):

For $\mathcal{C}$ a category with finite limits and $X \in \mathcal{C}$ an object, then dependent sum

$\underset{X}{\sum}: \mathcal{C}_{/X} \longrightarrow \mathcal{C}$

By this proposition limits over a cospan diagram in the slice category are computed as limits over the cocone diagram under the cospan in the base category. By this proposition this inclusion is a final functor, hence preserves limits. Since the dependent sum of the diagram is the restriction along this final functor, the result follows.

For $\mathcal{C}$ a category with finite limits and $X\in \mathcal{C}$ any object, the naturality square of the unit of the $(\underset{X}{\sum} \dashv X^\ast)$-adjunction on any morphism $(f \colon A \to B)$ in $\mathcal{C}_{/X}$

$\array{
A &\longrightarrow& X^\ast \underset{X}{\sum} A
\\
\downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{X^\ast \underset{X}{\sum} f}}
\\
B &\longrightarrow& X^\ast \underset{X}{\sum} B
}$

is a pullback.

By prop. it suffices to see that the diagram is a pullback in $\mathcal{C}$ under $\underset{X}{\sum}$, where, by Frobenius reciprocity, it becomes

$\array{
\underset{X}{\sum} A &\stackrel{(A,id)}{\longrightarrow}& X \times \underset{X}{\sum} A
\\
\downarrow^{\mathrlap{\underset{X}{\sum}f}} && \downarrow^{\mathrlap{(id, \underset{X}{\sum} f)}}
\\
\underset{X}{\sum} B &\stackrel{(B,id)}{\longrightarrow}& X \times \underset{X}{\sum} B
}
\qquad .$

For $\mathcal{C}$ a category with finite limits and $X\in \mathcal{C}$ any object, the naturality square of the counit of the $(\underset{X}{\sum} \dashv X^\ast)$-adjunction on any morphism $(f \colon A \to B)$ in $\mathcal{C}$

$\array{
\underset{X}{\sum} X^\ast A &\longrightarrow& A
\\
\downarrow^{\mathrlap{\underset{X}{\sum}X^\ast f}} && \downarrow^{\mathrlap{f}}
\\
\underset{X}{\sum} X^\ast B &\longrightarrow& B
}$

is a pullback.

By Frobenius reciprocity the diagram is equivalent to

$\array{
X\times A & \longrightarrow& A
\\
\downarrow^{\mathrlap{(id,f)}} && \downarrow^{\mathrlap{f}}
\\
X \times B &\longrightarrow& B
}
\qquad.$

Let $C$ be an (∞,1)-category. We still want to define the dependent sum as the functor $C_{/X} \to C_{/Y}$ given by composing with the functor $X \to Y$, but the details are more complex.

The codomain fibration $tgt : C^{[1]} \to C$ is a cocartesian fibration classified by a functor $L : C \to (\infty,1)Cat : X \mapsto C_{/X}$.

Then for any $f : X \to Y$, we can define $\Sigma_f = L(f)$ to be the dependent sum.

Since the morphisms $[z \to x] \to [z \to y]$ induced by composition are cocartesian morphisms, we see that $\Sigma_f$ is indeed given by composition with $f$.

By proposition 6.1.1.1 and the following remarks of Lurie, if $C$ has pullbacks, then this is also a cartesian fibration, and is classified by a map $R : C^{op} \to (\infty,1)Cat$. Then $f^* = R(f) : C_{/Y} \to C_{/X}$ is the functor computing pullbacks along $f$, and we have the adjunction $\Sigma_f \dashv f^*$.

Standard textbook accounts include section A1.5.3 of

and section IV of

Other references:

Only fairly indirectly related is the notion of “dependent sums” in

- Peter Bonart,
*Towards a General Theory of Dependent Sums*(2024). [arXiv:2404.08139]

Last revised on April 15, 2024 at 13:28:54. See the history of this page for a list of all contributions to it.