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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMay 15th 2012

now I have finally the time to come back to this, as announced, and so I am now starting an entry:

So far there is just some literature collected. I now plan to extract the essence of Seely’s artice into the entry in some technical detail.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMay 15th 2012
• (edited May 15th 2012)

it won’t come as a surprise to you that I am creating a table.

Currently it looks like this:

flavor of type theory $\phantom{\rule{thickmathspace}{0ex}}$equivalent to$\phantom{\rule{thickmathspace}{0ex}}$ flavor of category theory
first-order logic hyperdoctrine (Seely 1984a)
dependent type theory locally cartesian closed category (Seely 1984b)
intensional type theory with identity types locally cartesian closed (∞,1)-category (conjectural)
homotopy type theory with higher inductive types elementary (∞,1)-topos (conjectural)

But that made me wonder: what’s the best terminology for the entries on the left? What should be the canonical term for that part of homotopy type theory which does not assume univalence and hence is supposedly the language of locally cartesian closed $\left(\infty ,1\right)$-categories?

• CommentRowNumber3.
• CommentAuthorMike Shulman
• CommentTimeMay 15th 2012

I dunno. Just “homotopy type theory” maybe?

I think there’s also an equivalence between some kind of type theory and elementary 1-toposes somewhere in the literature…

1. One could add in the entry that the morphisms in the categories in the left column of the table are what Seely calls ”derivations”.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeMay 16th 2012
• (edited May 16th 2012)

have been working on HoTT methods for homotopy theorists. Mainly added a bunch of translation tables, also reorganized a bit. Still far from what it should eventually be, but at least it is beginning to look like an entry.

A point to be amplified more, also with respect to the $n$Café-discussion, is how in the internal HoTT language statements about fibers simplify, in that the naive external statements which are generally wrong become true internally: “A morphism is $n$-truncated/$n$-connected if all fibers are so.”

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeMay 16th 2012

the morphisms in the categories in the left column of the table are what Seely calls ”derivations”.

That seems like a weird name to me; I’d be more inclined to call them something like “translations”.

2. it won’t come as a surprise to you that I am creating a table.

I like the encyclopedic flavor of your tables!

have been working on HoTT methods for homotopy theorists. Mainly added a bunch of translation tables,

In the basic dictionary table I missed that a morphism $f:X\to Y$ which is not necessarily a fibration corresponds to a typing judgement $x:X⊢f\left(x\right):Y$. Is this omission intended?

I’d be more inclined to call them something like “translations”.

This sounds fairly descriptive. Can you explain why this is a good name?

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeMay 16th 2012

Mike writes:

I dunno. Just “homotopy type theory” maybe?

It’s just that eventually we need two terms, one for the internal language of locally cartesian closed $\left(\infty ,1\right)$-catgegories in general, and one for that of elementary $\left(\infty ,1\right)$-toposes.

I think there’s also an equivalence between some kind of type theory and elementary 1-toposes somewhere in the literature…

Scott and Lambek in their section II have an adjunction between toposes with NNO and type theories with product types and natural numbers. Is that maybe what you have in mind? If not, I’d be very interested in finding the reference that you do have in mind.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeMay 16th 2012
• (edited May 16th 2012)

Concerning “derivations” and “translations”: I think you are maybe talking about different tables each?

Seely calls a term of a type a “derivation”, because it can be regarded a proof of a proposition. What should be called “translations” are the morphisms in the category of type theories, since they translate one internal language into another)

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeMay 16th 2012
• (edited May 16th 2012)

Stephan writes:

I like the encyclopedic flavor of your tables!

Thanks for letting me know. I am certainly benefitting from making such tables explicit. It helps me organize my thoughts and see clearly.

In the basic dictionary table I missed that a morphism f:X→Y which is not necessarily a fibration corresponds to a typing judgement x:X⊢f(x):Y. Is this omission intended?

It’s not really omitted, as this is a special case of the clause for a term (forth item in the table). There also the type $Y$ may depend on $x$. Your typing judgement is obtained from this by assuming that $Y$ does not in fact depend on $X$, hence by assuming that the fibration in question is the projection $Y×X\to X$.

Maybe I should add a remark about that (as soon as the $n$Lab reacts again).

• CommentRowNumber11.
• CommentAuthorDavid_Corfield
• CommentTimeMay 16th 2012

Is the relationship between the hyperdoctrine approach and the syntactic category approach to logic known? We were talking about it here.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeMay 16th 2012

We were talking about it here.

Hm, interesting that slide 96 of Hirschowitz.

I guess the devil is in the details somewhere, but Seely seems to have made this kind of adjunction an equivalence

$\mathrm{FirstOrderTheories}\simeq \mathrm{Hyperdoctrines}$

already in 1987 (references at relation between type theory and category theory).

I haven’t looked through all the details. One thing that seems noteworthy is that in his proof Seely in fact considers hyperdoctrines whose fibers are not required to be posets. Maybe that makes the difference. But I’d say the equivalence proven justifies whatever fine-tuning on $\mathrm{Hyperdoctrines}$ made it work.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeMay 16th 2012

Hm, Seely’s article contains a mistake, in that it is not dealing with the non-uniqueness of pullbacks in the category correctly.

In

there is at least a partial fix. Though I need to figure out if that fix retains the equivalence of categories $\mathrm{DependentTypeTheories}\simeq \mathrm{LocallyCartesianClosedCategories}$ that Seely claimed.

• CommentRowNumber14.
• CommentAuthorMike Shulman
• CommentTimeMay 16th 2012

• CommentRowNumber15.
• CommentAuthorMike Shulman
• CommentTimeMay 16th 2012

I think you are maybe talking about different tables each?

Ah, indeed we were! I misinterpreted Stephan’s remark #4. But in that case, I think the morphisms should just be called “terms”.

• CommentRowNumber16.
• CommentAuthorMike Shulman
• CommentTimeMay 16th 2012

It’s just that eventually we need two terms, one for the internal language of locally cartesian closed (∞,1)-catgegories in general, and one for that of elementary (∞,1)-toposes.

“homotopy type theory” and “univalent (homotopy) type theory”?

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeMay 16th 2012

“homotopy type theory” and “univalent (homotopy) type theory”?

But what about the higher inductive types?

Currently in the entry I have (I think) “homotopy type theory” for the language of LCC $\infty$-categories and “homotopy type theory with higher inductive types and univalence” for the language of elementary $\infty$-toposes.

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeMay 16th 2012

Ah, great! Thanks, Mike. And what a coincidence that we posted #13 and #14 in the very same minute! :-)

• CommentRowNumber19.
• CommentAuthorTobyBartels
• CommentTimeMay 16th 2012

@Mike #15: Whether morphisms should be called ‘terms’ depends on whether objects are called ‘types’. Are the objects here types or contexts?

• CommentRowNumber20.
• CommentAuthorUrs
• CommentTimeMay 16th 2012

Toby: types.

The article under discussion is this one.

• CommentRowNumber21.
• CommentAuthorUrs
• CommentTimeMay 16th 2012
• (edited May 16th 2012)

re 18: I have worked this and related references into the entry.

When I started this quest, I had no idea that what I am after was still not settled by December last year! I thought this is ancient stuff. Interesting to see that it is not only in physics that some statements become so much folk lore that people forget that they haven’t proven them yet ;-)

• CommentRowNumber22.
• CommentAuthorMike Shulman
• CommentTimeMay 17th 2012

I think we can be flexible with terminology and not demand a concise term for every possible variation on a theme.

• CommentRowNumber23.
• CommentAuthorMike Shulman
• CommentTimeMay 24th 2012

Is it really a good idea to include the detailed constructions of the functors $\mathrm{Con}$ and $\mathrm{Lan}$ in the particular case of DTT vs LCCCs on the page relation between type theory and category theory? The construction of $\mathrm{Con}$ is already described at (now) syntactic category; why duplicate it elsewhere? If we want to include the proof that $\mathrm{Con}\left(T\right)$ is LCCC when $T$ is a DTT, it seems more natural to me to put it at syntactic category or at dependent type theory. Similarly for $\mathrm{Lan}$.

• CommentRowNumber24.
• CommentAuthorUrs
• CommentTimeMay 24th 2012

That sounds reasonable. I didn’t finish what had planned to write there anyway.

Right now I don’t have the time to look into this. But if you have a second to do so, feel free. Otherwise I can try to look into it later.

• CommentRowNumber25.
• CommentAuthorMike Shulman
• CommentTimeMay 24th 2012

I think there is a lot of duplication and confusing organization right now. (Nobody’s fault, surely, it just happened organically.) The page internal logic discusses, for first-order theories, exactly the sort of thing that categorical semantics currently does for dependent type theories. A lot of the same thing is also at the introductory section on categorical semantics at type theory, as well as at relation between type theory and category theory.

I suppose it makes sense to keep an intuitive discussion at type theory for people who are just stumbling on that page. But maybe the rest of it can be unified and better organized. What about merging relation between type theory and category theory with categorical semantics? Maybe a descriptive pagename would be categorical semantics of type theory. Then we can try to centralize constructions of the functor $\mathrm{Lang}$ on the page internal logic, construction of the functor $\mathrm{Syn}$ on the page syntactic category, and discussion of the adjunction/equivalence between them at the merged page.

• CommentRowNumber26.
• CommentAuthorUrs
• CommentTimeMay 25th 2012

Yes, I feel precisely the same way. The current status of the pages is the result of several attempts to start with the same topic, each of them unfinished.

I’d certainly have no objections against merging pages.