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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeOct 25th 2017

the entries dependent type and indexed set did not know of each other.

I have now cross-linked them minimally in their “Related entries”-sections. But this would deserve to be expanded on for exposition…

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMay 21st 2022
• (edited May 21st 2022)

made “type family” (requested elsewhere) redirect here

1. added a section on the relation between families of types, families of elements, and functions, and how that results in the relation between type theory and category theory inside of dependent type theory itself.

Anonymous

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJan 21st 2023
• (edited Jan 21st 2023)

you wrote (here):

Given a family of types, $x:A \vdash B(x)$, one could construct a family of elements $z:\sum_{x:A} B(x) \vdash \pi_1(z):A$ via the rules for dependent sum types. Conversely, given a family of elements $x:A \vdash f(x):B$ one could construct a family of types $y:B \vdash \sum_{x:A} f(x) =_B y$ as the family of fiber types of $x:A \vdash f(x):B$.

$\,$

This seems confusing to the point of being notationally wrong.

I guess you mean to talk about these two functions:

$\array{ \underset{ Y \,\colon\, Type }{\sum} \big( Y \to X \big) &\longrightarrow& \underset{x \,\colon\, X}{\prod} Type \\ (Y,f) &\mapsto& \big( x \colon X \;\vdash\; fib_x(f) \big) }$

and

$\array{ \underset{x \,\colon\, X}{\prod} Type &\longrightarrow& \underset{ Y \,\colon\, Type }{\sum} \big( Y \to X \big) \\ \big( x \,\colon\, X \;\;\vdash\;\; E(x) \big) &\mapsto& \Big( \pi_1 \,\colon\, \big( \underset{x \,:\, X}{\sum} E(x) \big) \to X \Big) }$

?

2. no, I mean syntactically, not relative to a universe.

Anonymous

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJan 21st 2023

I see, I was misled by expecting to read something else.

Do you mind if we write “one can” or “there is” instead of “one could”?

3. one could -> one can

Anonymous

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeJan 21st 2023

Thanks.

In addition, I have taken the liberty of adding whitespace and parenthesis to the formulas (here) in order to optically resolve the syntax tree.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeJan 25th 2023
• (edited Jan 25th 2023)

I have added (here) the analogous discussion reflected in a type universe, with a reference to Theorem 4.8.3 in the HoTT book.

Then I added (here) a joint lead-in paragraph for this and the previous discussion