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I’ve modified types and logic - table to show that type theory ’substitution’ is not only category theory ’composition’ but category theory ’pullback’ as well. In doing so, I added a footnote linking to a post explaining this. However, although the footnote itself shows up on both the types and logic - table page and the relation between type theory and category theory page, the superscript ’1’ in the table doesn’t show up on the latter, only on the former.
Thanks, good point.
I don’t know how to deal with that technical issue, but I have a suggestion that would get rid of the problem,too:
we should have an nLab page that discusses this, not just a pointer to an external source.
In particular that nLab page should mention that if we consistently think of dependent types in terms of their classifying maps, then “substitution is composition”, after all. :-)
Fair enough! I’ll leave that task to someone with more knowledge than me. :-)
However, the bug I’ve mentioned is mysteriously affecting at least one other page as well; the footnote I created is included on the Burali-Forti’s paradox page, despite no footnote reference on that page!
Okay, at types and logic - table I have removed the footnote and expanded the entry which used to say “composition” to read “composition of classifying maps / pullback of display maps”.
Notice that at substitution there is a section on categorical semantics. If anyone wants to further expand on “substitution is pullback/is composition”, then this would be the canonical place to do so.
Ah, and so the effect of that “bug” has also disappeared: the table types and logic - table is included in the “floating table of context” type theory - contents which in turn is included in all entries related to type theory. Hence making the first table have a footnote apparently propagates that footnote to the bottom of every page into which it is included (which does actually make sense, in a way, were it not for the fact that one of the including stages is a “pulldown” inclusion).
Anyway, the footnote is gone.
I did some general further cross-linking of the entries related to this issue. In particular I added pointers to classifying morphism and added there a brief comment on classifying dependent types and pointing back to substitution.
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