Yes, but he also wasn't working with type theories derived from toposes when I saw him last. On the other hand, type theories derived from programming languages are right up his alley, so I should have paid more attention to that aspect. Sorry, Mike!

]]>Mike Stay is not just some random Mike S, Toby—he’s your academic brother!

]]>Mike *Shulman* wrote in part:

too many Mikes

Yeah, when I saw a paper by Mike S about type theories derived from categories, I think that I just jumped to the conclusion that it was you and skipped over the rest of the name. (Otherwise, I wouldn’t have asked about this, because I wouldn’t have expected Christian to know.)

]]>Note that embedding a topos in its (2,1)-topos of stacks to get stack semantics is very similar to what Christian and Mike Stay (eh, too many Mikes) are doing here, embedding the category of a higher-order algebraic theory in its 1-topos of presheaves to get native type theory. (Maybe this is closer to the question you meant to ask?) A HOAT could also be embedded in a higher topos of stacks, generalizing the native type theory to a “native stack semantics” that would also contain a univalent universe of small types, corresponding to the representable objects (the original types of the HOAT). A strictified version of that universe of representables is also visible in the ordinary native type theory, though of course it is not univalent there. Christian and I discussed this a bit, though I think eventually they decided not to include this universe of representables in the paper.

]]>I have been summoned! First let me link to these slides that summarize my current point of view on stack semantics, which has evolved somewhat over the past decade. Briefly, the stack semantics as I presented it 10 years ago is a first-order fragment of the internal dependent type theory of the (2,1)-topos of stacks of groupoids over the topos.

The latter DTT differs from what Jacobs calls FhoDTT mainly in its universes. FhoDTT contains a universe of propositions, which is a type, but no universes of types. (Jacobs uses the words “type” and “kind” instead of “proposition” and “type”.) In the FhoDTT of a topos, the types are the objects of the topos, and the propositions turn out to be identifiable with those types that are h-propositions in the sense of HoTT, i.e. have at most one element: the subobject fibration in the topos, with the universe of propositions being the subobject classifier. But syntactic FhoDTT as Jacobs presents it does not mandate that all the propositions are h-propositions or that all the h-propositions are propositions (although one could add these as axioms).

By contrast, the DTT of stacks contains a universe of small *types*. In the stack model over a topos, the types are stacks of groupoids while the small types are objects of the topos, and the universe is the (core of the) self-indexing of the topos. One then has both the “large” h-propositions, which are closed sieves, and the subcollection of small h-propositions, which are subobjects in the topos. The latter can be classified by a universe of propositions which is a small type (the subobject classifier), although the stack semantics also works over e.g. pretoposes that don’t have a subobject classifier. The former could be classified by a large universe of large propositions, but we don’t as often need to do that.

Neither of these constructions assume that the topos has any *internal* universes (other than the universe of propositions given by its subobject classifier). If it does (e.g. it is a Grothendieck topos and there exist inaccessible cardinals in the category Set), they can be added to the syntax as types of small types (in FhoDTT) or small types of smaller types (in stack DTT). But unlike the universes mentioned above, these will not generally be univalent if the original topos was only a 1-topos. The stack semantics can also add more larger “outer” universes if there are inaccessible cardinals larger than the size of the topos, although to have more than one of these that are univalent one needs to move to stacks of higher groupoids.

I hope this answers the question!

]]>Very interesting question. It looks like Mike is focused on foundational issues like unbounded quantification in a topos (which I guess I’m using in the monoid example) to strengthen the categorical bridge between material and set theory. It seems that he only uses dependent types when necessary.

Since I don’t know how to mentally translate between the Kripke-Joyal-style forcing relation and type theory, it’s not totally clear. I’m sure he’ll give a good answer — I could send him a message.

]]>Hi Toby, yes I was suppressing those dependencies only because it gets fairly long. You’re right; we should at least have . Thanks.

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