dependent sum

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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.

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.

type theory | category theory | |
---|---|---|

syntax | semantics | |

natural deduction | universal construction | |

dependent sum type | dependent sum | |

type formation | $\frac{\vdash\: A \colon Type \;\;\;\;\; x \colon A \;\vdash\; B(x)\colon Type}{\vdash \; \left(\sum_{x \colon A} B\left(x\right)\right) \colon Type}$ | $\left(A \in \mathcal{C}, \array{ B \\ \downarrow^{\mathrlap{p_1}} \\ A} \; \in \mathcal{C}/A \right) \Rightarrow \left( B \in \mathcal{C}\right)$ |

term introduction | $\frac{\vdash\: a \colon A \;\;\;\;\; \vdash\; b \colon B(a)}{\vdash (a,b) \colon \sum_{x \colon A} B\left(x\right) }$ | $\array{ Q &\stackrel{(a,b)}{\to}& B \\ & {}_{\mathllap{a}}\searrow & \\ && A }$ |

term elimination | $\frac{\vdash\; t \colon \left(\sum_{x \colon A} B\left(x\right)\right)}{\vdash\; p_1(t) \colon A\;\;\;\;\; \vdash\; p_2(t) \colon B(p_1(t))}$ | $\array{ Q &\stackrel{t}{\to}& B \\ & & \downarrow^{\mathrlap{p_1}} \\ && A }$ |

computation rule | $p_1(a,b) = a\;\;\;\; p_2(a,b) = b$ | $\array{ Q &\stackrel{(a,b)}{\to}& B \\ & {}_{\mathllap{a}}\searrow & \downarrow^{\mathrlap{p_1}} \\ && A }$ |

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: