Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex-geometry computable-mathematics computer-science connection constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry goodwillie-calculus graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory history homological homological-algebra homology homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie lie-theory limit limits linear linear-algebra locale localization logic mathematics measure-theory modal-logic model model-category-theory monoidal monoidal-category-theory morphism motives motivic-cohomology nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics planar pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthoratmacen
    • CommentTimeApr 27th 2019

    Although Nuprl’s logic (henceforth simply “Nuprl”; the proof assistant itself is not relevant) is not based on category theory, you can still do the usual trick of considering it as a category by looking at its closed types and unary functions. I’ve been thinking a little about Nuprl’s structure as a category. It’s a variant of extensional type theory, and includes dependent function types, unit, nullary and binary coproducts, integers, and quotients. So I figure that makes it an LCCC with a NNO and finite limits and colimits. An interesting difference from many other extensional, set-level, constructive foundations is that it’s not exact.

    So when reasoning internally, it matters whether you take propositions to be arbitrary subsingletons, or regular subsingletons. The Nuprl preference definitely seems to be essentially to regard the regular subsingletons as the propositions. Regular images are always available, and they’re well behaved from a proof assistant implementation standpoint. (Not-necessarily-regular images are always available too, but seem rarely used.)

    I’m trying to get a sense of how (un)common it is to reason using just the regular subsingletons from a categorical perspective. And whether it’s considered a good/bad/interesting idea.

    More info: Nuprl is based on a meaning explanation (but not that one), and is a category of PERs. But the way universes are added to the computation system makes them behave like plain sets within a category of assemblies. (They are uniform. Their subobjects are also uniform.)

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeApr 27th 2019

    I’ve mainly encountered this in the context of quasitoposes. However, the resulting lack of PUC makes the resulting internal logic rather impractical and unfamiliar to most mathematicians, even constructive ones; and so the internal logic of quasitoposes is used much less than that of toposes. I once tried asking Bob Harper how NuPRL deals with the lack of PUC, but I think I didn’t end up with an answer that really helped me.

    • CommentRowNumber3.
    • CommentAuthoratmacen
    • CommentTimeApr 28th 2019

    Thanks for telling me you think this is uncommon. And thanks for the link to your MathOverflow question.

    It seemed odd to me to say that lack of PUC makes the internal logic impractical and unfamiliar. PUC is just telling you about functions, not the bare logic. But then I remembered that in category theory, “internal logic” means something very specific, and is all based on morphisms, which are the functions internally.

    Maybe it helps to think of the regular subsingletons as an internal reflection of the external logic? I’m pretty sure that’s the case with assemblies and PERs. If not, my original question was mistaken. Perhaps lack of PUC is a good reason not to think of arbitrary subsingletons as propositions.

    I don’t know the real reason Nuprl people didn’t/don’t add PUC. It didn’t have it from the start, because it was based on Martin-Löf’s meaning explanation. Also, programmers are not in the habit of using unique choice. They couldn’t be, since most programming languages can’t even express the principle.

    I personally wouldn’t want PUC added to Nuprl unless it’s done in a careful way to retain the current regular subobjects, and their implementation. Like how assemblies are a full subcategory of the realizability topos.

    An answer to your MO question reminded me that Bishop’s foundation had an intensional layer and an extensional layer. This seems to come up a lot. nLab talks about presets vs. sets. With HoTT we consider two-level systems. I think Nuprl is most appropriate as an intensional layer. If this surprises you, it’s probably because “extensional type theory” was that bad a name.

    • CommentRowNumber4.
    • CommentAuthoratmacen
    • CommentTimeApr 28th 2019

    I’m confused. I followed links to Andrej’s blog where he says PUC is true for “numbered sets” (which I think is equivalent to PERs over number realizability). Thinking about it, this seems right, if you consistently use props-as-subsingletons, but not if you use props-as-regular-subsingletons. In a later comment, he says “the usual interpretation of first-order logic in a category validates unique choice”.

    It sounds like the version of PUC involved which is actually missing sometimes is the “epic monos are isos” one, right? Is that the one you’re talking about?

    Oh I see. On the quasitopos page, it says the internal logic is usually taken to be the strong subobjects, which is the reason function comprehension fails. I guess that explains it.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeApr 28th 2019

    Right, if propositions are strong (i.e regular) subobjects, then PUC fails, whereas if they are arbitrary subobjects it holds. I think something of the same sort is true in NuPRL; you can choose what kind of subobjects you want to consider “propositions”, and thereby get or not get PUC.

    IMHO the best way to deal with these things is to take arbitrary subobjects as the propositions and then regard the regular subobjects as the \sharp-modal ones for an appropriate modality \sharp. That way the fundamental logic satisfies PUC, but you can also express statements about the logic of regular subobjects if you want to.

    • CommentRowNumber6.
    • CommentAuthoratmacen
    • CommentTimeApr 28th 2019
    Yeah, I used to know that, in the case of Nuprl. I forgot. Nuprl is a type system, so both kinds of subobjects are modalities.
    • CommentRowNumber7.
    • CommentAuthoratmacen
    • CommentTimeApr 28th 2019

    I don’t know the real reason Nuprl people didn’t/don’t add PUC…

    I personally wouldn’t want PUC added to Nuprl unless it’s done in a careful way…

    What I meant here was making Nuprl into a pretopos.

    • CommentRowNumber8.
    • CommentAuthorspitters
    • CommentTimeApr 30th 2019

    I believe Jon Sterling once worked on the categorical structure of NuPrl. Quasitoposes also show up in the type theory of Coq; see Arnaud Spiwack’s thesis.

    • CommentRowNumber9.
    • CommentAuthoratmacen
    • CommentTimeMay 2nd 2019

    I really like Arnaud’s argument against adopting PUC. It starts on page 22 (or 28) of his thesis at:

    Functions which are both injective and surjective are called bijections. Hence we can state the the principle of unique choice as follows: unique choice: bijections have an inverse.

    I’m not sure it’s OK to copy too much of the text, but you can download the PDF and find what I’m talking about.

    The gist, as I understand it, is that you want computations to be efficient, but proofs shouldn’t have to be (that is, in terms of their realizers), so a principle conflating them should not be used.

    I wonder what realizability topos fans would say to that. Shoot, I wonder what Martin-Löf would say to that. But I completely agree.

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeMay 2nd 2019

    Here is a link that works.

    I think that that might be a reasonable argument if one were trying to develop an “effective mathematics” in which all definable functions were efficient. (Although another totally reasonable approach to such a thing would be to require proofs to also be efficient, thereby maintaining PUC. He claims that requiring proofs to be efficient would prevent automation, but that wouldn’t be the case if the automation were correct for a formal system in which all proofs were efficient.) But while developments like soft linear logic hold out some hope that such an effective mathematics might exist, MLTT is not it, since it is trivial therein to define functions with extremely high complexity. So I don’t see how that is an argument against PUC for MLTT.

    The most I can see that argument saying is that when trying to manually control the complexity of the functions you define, it’s handy to have an “abstraction barrier” behind which you can put proofs of arbitrarily high complexity without worrying that they might pollute your functions. As far as I can see, this barrier doesn’t actually mean anything formally (since the ambient theory is strong enough to allow you to define unfeasible functions), it’s just a “tag” with a user-defined significance. Which is totally reasonable, but it makes more sense to me to include “both worlds”, where the naturally defined propositions inside MLTT satisfy PUC and the tag is a modality. This approach has the additional benefit that a (higher) modality can also be applied to types that are not propositions, allowing you to mark not just proofs but also functions as places where you’re not going to worry about complexity.

    • CommentRowNumber11.
    • CommentAuthoratmacen
    • CommentTimeMay 2nd 2019

    Efficiency in programming is a multi-faceted thing. The kind of languages I see where everything definable is “efficient”, efficiency is taken to be some well-behaved complexity class, like polynomial time. This is too coarse for algorithm design, which tends to use big-O-style asymptotic analysis. But in practice big-O is just a ballpark estimate of efficiency, since it’s definitely not OK to have a program that’s 100 times slower than it should be, although that would not register in big-O.

    So I don’t believe there is any language where everything definable is as efficient as it needs to be, for practical programming. Also, algorithm design traditionally relies on Turing completeness, so ironically, I suspect that intrinsically efficient languages don’t have enough efficient programs. (I’m almost certain they don’t, in theory. I mean in practice.) I don’t think Arnaud meant efficiency in some precise, theoretical sense like that; he meant it the way a programmer does. So your second paragraph is the one I think is on topic.

    So we’re talking about the difference between the functions, and the things that are functional relations according to regular subobject logic. Since arbitrary subobject logic evidently satisfies PUC, this is also the difference between regular subobject “functional relations” and arbitrary subobject “functional relations”. I’ll call the former just “functional relations”, and the latter “functional programs”. So the PUC missing from Nuprl, and Arnaud’s setup, is for regular subobjects, and would collapse the distinction between functional relations and programs.

    There is definitely a formal difference between them, in general. If you assume excluded middle restricted to regular subobjects, you get uncomputable functional relations, but no uncomputable functional programs. Maybe that’s not what you had in mind, but it’s completely reasonable to use this sort of system to reason classically about programs. (A classical “program logic”.) I understood his argument as promoting, even in the intuitionistic case, that you manually manage efficiency of functional programs and not functional relations.

    I don’t know what you mean by “the naturally defined propositions inside MLTT”, and including “both worlds”.

    • CommentRowNumber12.
    • CommentAuthoratmacen
    • CommentTimeMay 2nd 2019
    Oh, well he actually does say “computational complexity” in addition to “efficiency”. So maybe he’s not as fussy as me. :)
    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeMay 2nd 2019

    The naturally defined propositions are the subobjects, in HoTT they are sometimes called h-propositions. By both worlds I meant that you have a logic that can talk about both subobjects and regular subobjects and use either or both as appropriate or even mix them, rather than forcing yourself to choose one of them as “the logic” and discard the other. This is naturally achieved by adding a “regular-subobject reflection” modality to the logic of arbitrary subobjects, and in many cases such a modality also extends to a higher modality on all types (not just propositions).

    • CommentRowNumber14.
    • CommentAuthoratmacen
    • CommentTimeMay 2nd 2019

    I’m not sure we’re on the same page. In the kind of system we’re talking about, you can already do all that. It’s a type system. Types (families) are generally not subsingletons (subobjects). But you can quotient a type down to a subsingleton. That’s propositional truncation. In these systems, you can also take a “subset” (equalizer!) of unit to get a regular subsingleton. This seems like the “regular-subobject reflection” you want. In Nuprl, subset is a primitive. In Arnaud’s setup, it uses Coq’s Prop-based subset. These are two different ways to enforce computational irrelevance of the proofs. Both kinds of truncation are idempotent monadic modalities.

    • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeMay 2nd 2019

    Yes – my point is just that we should treat it like a modality, rather than declaring that only the modal propositions are to be used in “logic” and that therefore PUC “fails”, when in fact PUC is true for the non-modal fragment of the logic.

    • CommentRowNumber16.
    • CommentAuthoratmacen
    • CommentTimeMay 2nd 2019
    Oh I see.
    • CommentRowNumber17.
    • CommentAuthorjonsterling
    • CommentTimeMay 3rd 2019

    Hi all, the lack of unique choice and the lack of effective quotients in Nuprl is a huge source of pain for people who wanted to use it. I don’t think anyone really ever engaged with the unique choice issue, but the lack of effectivity for quotients was so devastating that Nuprl users generally try to avoid quotienting anything ever (somewhat ironic, considering that Nuprl was for many years one of the few implementations of type theory that boasted quotient types). Zooming out, my understanding (I rely on Mike to correct it!) is that this kind of situation is typical of working in either categories of PERs or categories of assemblies, and that the realizability construction corrects it — for instance, by redefining the notion of a function in terms of its graph.

    To type theorists like myself who had grown up in the tradition of Nuprl-style realizability, one of the big selling points of HoTT/UF in contrast to what we were used to was to obtain a constructive and computational account of unique choice and effective quotients, which didn’t involve getting utterly owned by manually working inside an exact completion. We haven’t looked back…

    • CommentRowNumber18.
    • CommentAuthoratmacen
    • CommentTimeMay 4th 2019

    Hi Jon,

    It seems like there’s a single cause in PERs and assemblies for both of: quotients generally aren’t effective (not exact), and not all monos are regular (not balanced). It’s that equality proofs are computationally irrelevant. The “fix” in general is exact completion, as you seem to know.

    Classically, realizability topoi are equivalent to exact completions of categories of (partitioned; or not) assemblies. The impression I’ve gotten is that without choice, the traditional construction of a realizability topos may not be exact. So I don’t think that style of realizability construction (realizing truth rather than objects, and encoding functions as relations) is the actual essence of the solution.

    As Mike confirmed in #5, the arbitrary-subobjects internal logic already satisfies unique choice, so its failure in the regular-subobjects logic (in assemblies-like situations) is just a symptom of not being balanced. What’s really missing is exactness, not unique choice.

    It seems like there’s a cultural divide on whether the use of computationally-relevant proofs is actually a good idea. It’s definitely interesting, and was a key motivation for MLTT. But like Arnaud was saying, for programs, efficiency matters, and for proofs, it oughtn’t. At least not for the same sense of “efficiency”. I get the impression this is a programmers vs. constructive mathematicians disagreement. Do you reason about programs, or reason using programs? It seems like Arnaud and I are “programmers”, and you and Mike are “mathematicians”.

    If your equivalence relations are computationally irrelevant, Nuprl’s quotients are already effective. So if you keep your programs and proofs straight, Nuprl has no serious problems that I know of. Computational relevance of proofs is reflected in the difference between regular subobjects, which are irrelevant, and others, which can be relevant. With exact completion, you sweep this real and important difference under the rug. That’s fine, as a high-level approach. But I appreciate this low-level aspect of Nuprl. Nuprl encourages you to care how your computations are implemented. Programming via constructive reasoning is a fascinating paradigm, but it’s too avant-garde for me. And anyway, that’s not what Nuprl’s good for.

    So to you type theorists who’ve moved to HoTT and aren’t looking back, good for you; you’re probably getting closer to what you want. For other type theorists, it’s heading in the wrong direction. Nuprl is still king for combining dependent types and program logics.

    • CommentRowNumber19.
    • CommentAuthorjonsterling
    • CommentTimeMay 4th 2019
    • (edited May 4th 2019)

    BTW, I am also a programmer ;-) I have a couple followup questions and comments:

    1. I have also heard about the difficulty with choice in the ex-reg completion, but I don’t know the specifics. Does anyone know if there is an alternative version which can be made constructive?

    2. Indeed, if your equivalence relation is itself computationally irrelevant, it is ok—the issue is that in practice, this is not usually what happens. A lot of these difficulties are chronicled in a paper from Nogin a long time ago, and he came up with some practical ways to mitigate them, but I don’t think the issue is resolved by just “keeping things straight”.

    3. I am quite a bit less pessimistic about combining dependent types and programs than to say that Nuprl is still the king — any king, for instance, whose utility relies on the odious “pointwise functionality” construct ought to be toppled. I am in very strong agreement about the importance of (strict) proof irrelevance (as opposed to just “homotopy-irrelevance”) but I note that this is different from Nuprl’s computational irrelevance.

    Because we still care about programming, but want a system that is both syntactically and semantically more well-behaved, we have been working to port ideas from HoTT to lower dimensions in a way that is compatible with the (perennially hypothetical) use-cases of Nuprl, if not exactly the spirit of Nuprl. Our first contribution in that area is the XTT type theory, a version of cubical type theory with judgmental UIP: Cubical Syntax for Reflection-Free Extensional Equality. We still have to investigate things like propositions and quotient types, but I think we can do something that is quite well-behaved and efficient to program with / extract.

    • CommentRowNumber20.
    • CommentAuthorMike Shulman
    • CommentTimeMay 4th 2019

    I think the issue with PERs/assemblies/NuPRL is best expressed by saying that you can’t have both unique choice and effective quotients with the same logic. If you use arbitrary subobjects, you get PUC but not exactness, while if you use regular subobjects you get effective quotients but not PUC.

    I feel in this discussion a bit of the sense of the dichotomy between nice objects and nice categories. Years and years of experience in category theory and related fields of mathematics teach us that nice categories are usually the way to go. But I’ve given up hope of ever convincing Matt that category theory has anything useful to do for him. (-:O

    • CommentRowNumber21.
    • CommentAuthorjonsterling
    • CommentTimeMay 4th 2019
    • (edited May 4th 2019)

    Mike, thanks for that remark about both unique choice and effective quotients simultaneously, it really helps and clarifies! :-)

    And btw, I hope this is not too harsh a thing for me to say, but I got convinced about the importance of (1) having good categorical tools, and (2) working in nice categories, just about the moment that I wanted to start to use type theory to actually do something, instead of just theorize about doing that and stroke my beard – and I am saying this both as a programmer and as a “mathematician” (maybe this label should be reserved for others, since I feel more like a dilettante); I note that serious attempts to apply computational type theory to programming have been ironically stillborn in contrast to the massive body of work applying type theory to programming in the context of Idris/Agda/Coq (in spite of the obvious shortcomings of these systems!) — it is one thing to be “meant for programming” and another to be “used for programming”. It is all well and good to study type theory because it is intrinsically beautiful, and maybe doing that does not lead one into contradiction with the ideological tenets of the Nuprl doctrine, but the type theoretic tradition draws its strength not from its intrinsic beauty but from the fact that it is useful to a diverse spread of scientists, from its (often unpredictable) applications.

    • CommentRowNumber22.
    • CommentAuthoratmacen
    • CommentTimeMay 4th 2019

    Mike, I like categories with nice morphisms. ;o)

    Jon, I wonder: does the lack of effective quotients also bother you when you write ML? I had imagined one does not even think that way, when thinking like a programmer. By the way, I was mainly trying to emphasize the distinction between programming and constructive math, not to label people. I just pointed out the side of this disagreement people fell on. I know you program. Probably more than me, lately.

    • CommentRowNumber23.
    • CommentAuthorMike Shulman
    • CommentTimeMay 4th 2019

    Going back to #18 and 19, can you give any details or references for this issue with choice in exact completions / realizability toposes? I’m not bringing anything to mind right now.

    • CommentRowNumber24.
    • CommentAuthoratmacen
    • CommentTimeMay 4th 2019

    Re #19:

    Re 1 (and #23): I don’t know much about it. But my impression was that the problem is specifically with getting a subobject classifier. Why do all subsingletons have small codes? But maybe I’m confused, and there’s no problem.

    Re 2: Do you remember the name of the paper?

    Re 3: I can’t follow the grammar in the beginning. It looks like you don’t like Nuprl because of some peculiarity of its semantics? By “pointwise functionality”, do you mean that thing in the meaning of the hypothetical judgment forms? I also found that weird. I came up with something different.

    I’ll take a look at this XTT. Should I take it as an example of what you mean by “(strict) proof irrelevance”? I’m skeptical there are three different kinds of irrelevance visible when working up to isomorphism. (“homotopy”, “strict”, “computational”)

    • CommentRowNumber25.
    • CommentAuthoratmacen
    • CommentTimeMay 5th 2019

    Should I take it as an example of what you mean by “(strict) proof irrelevance”?

    Another thing I’d like to know, which maybe wasn’t clear, is whether XTT is intended to be balanced. The original topic of this thread was how category theorists deal with reasoning in a non-balanced, non-exact categories that are otherwise pretoposes.

    Nuprl and Coq have a special class of propositions, which I referred to as computationally irrelevant. This was true of OTT as well. But what I’m focusing on is the implementation-independent fact that (once you have extensionality and quotients) these systems are non-balanced.

    I’m skeptical there are three different kinds of irrelevance visible when working up to isomorphism. (“homotopy”, “strict”, “computational”)

    Never mind, you didn’t claim that. Sorry.

    • CommentRowNumber26.
    • CommentAuthorspitters
    • CommentTimeMay 5th 2019

    Strict proof irrelevance has been added to both Coq and agda.

    • CommentRowNumber27.
    • CommentAuthorjonsterling
    • CommentTimeMay 5th 2019

    Another thing I’d like to know, which maybe wasn’t clear, is whether XTT is intended to be balanced. The original topic of this thread was how category theorists deal with reasoning in a non-balanced, non-exact categories that are otherwise pretoposes.

    We haven’t checked this yet, but I think there is hope — if it ends up working, it will be for roughly the same reasons that sets in HoTT are balanced. But let me not talk more about it, since in order to even state this theorem (much less prove it), there are things that need to be added to XTT.

    • CommentRowNumber28.
    • CommentAuthormbid
    • CommentTimeMay 5th 2019

    Regarding the discussion about the classicality of the fact that the effective topos is the ex/lex completion of the partitioned assemblies: AC is used in the proof that every partitioned assembly is projective, see Van Oosten’s book Prop 3.2.7. Basically a projective object of the effective topos is precisely a partitioned assembly X whose underlying set |X| is projective, and one needs AC to get rid of the latter condition.

    There is a constructive version of this fact, however, which is in Hofstra’s Relative Completions. Van Oosten also writes briefly about this in section 3.2.4 of his book. The idea appears to be to complete PAss relative to Set instead of globally.

    • CommentRowNumber29.
    • CommentAuthoratmacen
    • CommentTimeMay 5th 2019

    Re #26: From the cited paper: “Proof-irrelevance, the principle that any two proofs of the same proposition are equal…”

    I don’t like that terminology. Why not call it “proof uniqueness”? Irrelevance means, or ought to have meant, something else. (Just a gripe. No technical complaints.)

    Re #28: Thanks! That’s not nearly as bad as I thought.

    • CommentRowNumber30.
    • CommentAuthorMike Shulman
    • CommentTimeMay 6th 2019

    Ah, ok, so the problem isn’t with the construction of realizability toposes via (say) triposes, it’s that they may not be ex/lex completions, because ex/lex completions have a lot of projective objects. That makes perfect sense; ex/lex completions are pretty special. I assume that they can still be obtained as an ex/reg completion of a suitable subcategory, however.

    • CommentRowNumber31.
    • CommentAuthoratmacen
    • CommentTimeMay 8th 2019

    I thought maybe I should share more of my thoughts about exactness/balancedness, and why I think it’s not ideal for the strict/non-fibrant level of a proof assistant.

    On the HoTT list, not so long ago, I shared my opinion that “subsumptive equality” is a very good thing, since it gives you equations that you can rewrite with without adding a coercion operation to your term. Definitional equality is subsumptive, but you can’t reason about it, so this avoids coercions only in a few lucky cases. I like type systems with a subsumptive equality type. Equality reflection is one way to get that. Probably the best way, since it seems like the alternatives (subsumptive equality type but not reflective equality) necessarily lack function extensionality.

    Note that avoiding coercions is not about coherence. Coercive identity types with UIP have no coherence issues, they’re just a pain in the butt. Avoiding coercions is about convenience. But this is formal proofs we’re talking about; you need all the convenience you can get. In principle, avoiding coercions should also improve proof checking performance, since the terms will be smaller without coercions. (Or with fewer coercions, if you also have a coercive equality.)

    What happens if we take the subsumptive equality type to mean equality of morphisms? It seems like there are three possibilities:

    1. Derivations are computationally relevant; not just terms.
    2. You don’t have canonicity, because you relied on something that should’ve been computationally relevant, but wasn’t implemented.
    3. You’re not balanced, because some things are computationally relevant, but subsumptive equality witnesses aren’t.

    Well, so it seems like (1) is silly: you obviously just picked the wrong term language. (2) is not so bad, honestly. The HOL proof assistants don’t have canonicity, and are still used for program extraction. But then you have to wonder what’s the best fragment with canonicity. In other words, first find the best system with canonicity, then ask if it’s worth it to break canonicity for some reason (like a quotient effectivity axiom). (3) is Nuprl, basically.

    So then it seems it’s not a good idea, mathematically, to consider subsumptive equality to be equality of morphisms. I’m completely OK with that! But I want a subsumptive equality type to be there anyway, and you can still analyze the more-intensional category where morphisms are defined to be terms up to subsumptive equality.

    I started by saying that the strict layer shouldn’t be balanced. That’s because I want a subsumptive equality type, and I want to be in class (3) above. So if there’s also going to be a mathematical equality type, like to provide exactness, it looks like you need a two-level type system. Even if the mathematical equality still satisfies UIP!

    Way back in #3, I insinuated that two-level type theory is basically Bishop’s distinction between an intensional layer (types and operations) and an extensional layer (sets and functions). In the not-so-unlikely event that Nuprl is deemed too intensional because it’s not exact, then please do consider it for the intensional layer. What extensionality it does have should be understood as technology to make the subsumptive equality richer than definitional equality. Mathematical equality can be richer still.

    • CommentRowNumber32.
    • CommentAuthorMike Shulman
    • CommentTimeMay 8th 2019

    The real problem with NuPRL is not so much its non-exactness but the fact that it lacks general categorical semantics, being inextricably tied to a particular computational model.

    • CommentRowNumber33.
    • CommentAuthoratmacen
    • CommentTimeMay 8th 2019

    If there was a variant of Nuprl that was a conservative extension of intrinsic extensional type theory, would that solve the problem?

    • CommentRowNumber34.
    • CommentAuthorMike Shulman
    • CommentTimeMay 9th 2019

    Maybe. I’m a little dubious that such a thing could be called “a variant of NuPRL” however.

    Actually though probably saying “the real problem” was too strong; I think Jon in #21 is describing a purely syntactic problem with NuPRL, independently of its lack of models.

    • CommentRowNumber35.
    • CommentAuthoratmacen
    • CommentTimeMay 9th 2019

    If you take variants of Nuprl and ETT without universes, it’s not clear to me that Nuprl isn’t already a conservative extension. Do you know some incompatibility besides unusual types and funky type equality? (Which do not seem to affect conservativity for small types.)

    Regarding #21, Nuprl is a proof assistant based on programming ideas. It’s not a conventional programming language. It has nothing that seriously competes with Coq/Agda/Idris’ program extraction feature. My interest in Nuprl is based on my belief that in the long run, its approach can get us to a more convenient and practical proof assistant than anything remotely like Coq/Agda.

    • CommentRowNumber36.
    • CommentAuthorMike Shulman
    • CommentTimeMay 10th 2019

    I don’t know much about NuPRL, but my understanding is that it is based on a particular underlying operational semantics, consisting of PERs on some particular reduction system, and that everything that is true about that semantics is provable in NuPRL. In particular, this means that facts like strong Church’s thesis and Brouwer’s continuity principle are true in NuPRL, which they certainly are not in ETT.

    • CommentRowNumber37.
    • CommentAuthoratmacen
    • CommentTimeMay 10th 2019

    No. I wish I could point you to a nice small set of official rules, but I don’t think there is such a thing. The formal system is fairly similar to other dependent type systems, but with some definitely unusual aspects, not all of which are equally important.

    You must be thinking of the syntactic PER semantics, which I think satisfies Church’s thesis and continuity. The formal system almost certainly doesn’t prove those. At least not without deliberately adding them, which they might’ve done. I think Jon knows more about that. And it’s definitely not the case that all the true semantic judgments are formally derivable, due to Gödel’s incompleteness.

    The operational semantics is the reduction system. The PER semantics is a layer on top of that, which culminates in definitions of the semantic judgments. Aspects of this semantics are engineered to justify the desired formal rules, which correspond to selected facts about the semantic judgments.

    • CommentRowNumber38.
    • CommentAuthoratmacen
    • CommentTimeMay 10th 2019

    OK, terminology:

    Following (I think) Robert Harper (in lectures) “computational type theory” refers to a style of semantic type system along the lines of realizability, syntactic logical relations, PER semantics, and meaning explanations. These systems are not formal systems, but they have types, objects, and judgments, so can be thought of as type systems. Often, the types and objects are syntax. (So the objects are terms.) Often, the judgments are not effectively axiomatizable.

    Nuprl-style” type systems are formal dependent refinement type systems with equality reflection and few/no type annotations, whose closed terms have values according to some computable operational semantics.

    I’m emphasizing this partly to avoid confusion, since the way to justify a Nuprl-style system is a computational type system, but they’re not the same. But also, I’m primarily interested in Nuprl-style formal systems, whereas I get the impression Harper is primarily interested in computational type theory.

    • CommentRowNumber39.
    • CommentAuthorMike Shulman
    • CommentTimeMay 10th 2019

    Well, that is different from what other people have told me about “NuPRL”, so I guess not everyone uses words the same way. Whatever it is that you want to talk about, if you can provide or point to a precise definite mathematical definition of it, we can continue the conversation; otherwise it seems we don’t have anything to talk about.

    • CommentRowNumber40.
    • CommentAuthoratmacen
    • CommentTimeMay 11th 2019

    The formal system I’m really interested is the one I’m working on. I will try putting up those draft rules, soonish. In the mean time, if you want, you can consider the rules in the Nuprl 5 manual, section A.3. They depend on the term language and maybe (I forget) evaluation semantics described earlier in appendix A, but not on the judgment semantics. They are an old version of the Nuprl rules. Multiple features have since been added.

    • CommentRowNumber41.
    • CommentAuthorMike Shulman
    • CommentTimeMay 14th 2019

    Thanks for the link. Section A.2.2 defines the “precise meaning” of the judgments in terms of untyped reduction, which afaik is problematic for a categorical semantics. How is this related to the inference rules in section A.3? I generally think of inference rules as what defines (by induction) the “meaning” of the judgments.

    • CommentRowNumber42.
    • CommentAuthoratmacen
    • CommentTimeMay 14th 2019

    Right, A.2.2 is a somewhat precise summary of the PER semantics. It’s pretty precise by programmer standards; probably not by mathematician standards. I’m saying to ignore A.2.2 and read A.3 as an inductive definition. The intent though is that A.3 is sound with respect to A.2.2.

    • CommentRowNumber43.
    • CommentAuthoratmacen
    • CommentTimeMay 15th 2019

    The draft formal rules of my system are up at prototype CLF.

    • CommentRowNumber44.
    • CommentAuthorMike Shulman
    • CommentTimeMay 15th 2019

    Can you save me some time by pointing out exactly how the inductive definition of A.3 differs from, say, MLTT+UIP, or whatever intrinsically typed system it is easiest to compare it to?

    In your prototype CLF, to start with I have no idea what “CompComp” would mean in categorical semantics.

    • CommentRowNumber45.
    • CommentAuthoratmacen
    • CommentTimeMay 16th 2019

    Maybe we should continue this discussion at the discussion thread for prototype CLF, since it’s not about regular subsingletons anymore.

    • CommentRowNumber46.
    • CommentAuthoratmacen
    • CommentTimeJun 14th 2019

    I thought of another argument against balanced constructive systems:

    You could say most math is rigorous but informal and that most programming is formal but unrigorous. In order to use program extraction to obtain runnable programs from constructive mathematics, the computationally relevant stuff must be fully formalized. In a balanced system, that seems to mean that all proofs must be fully formalized. In a non-balanced system, by controlling which proofs are computationally relevant, you can do rigorous but informal reasoning about runnable (fully formalized) programs.

    This is similar to Arnaud’s argument (see #9): both arguments are observing that there are different standards for programs and proofs, so conflating them leads to systemic practical problems.

    • CommentRowNumber47.
    • CommentAuthorMike Shulman
    • CommentTimeJun 14th 2019

    And my response in #10 applies also equally well.

    • CommentRowNumber48.
    • CommentAuthoratmacen
    • CommentTimeJun 15th 2019

    I don’t think so. I thought you later clarified (in #15) that #10 was in favor of making general subobject logic available in a non-balanced system. I’m OK with that. But equality should be computationally irrelevant. #10 doesn’t seem to address that.

    Arnaud was arguing against PUC for regular subobjects, because it would make the system balanced, which would mean that equality would generally not be computationally irrelevant.

    • CommentRowNumber49.
    • CommentAuthorMike Shulman
    • CommentTimeJun 15th 2019

    I thought you later clarified (in #15) that #10 was in favor of making general subobject logic available in a non-balanced system.

    No, it was saying that a “non-balanced system” should be something defined inside a balanced system using a modality. Just as you can use a modality to tag the “computationally irrelevant” things, you should be able use it to assume axioms and tag them as “informally verified”.

    I’m not sure what makes you think that equality should be computationally irrelevant…

    • CommentRowNumber50.
    • CommentAuthoratmacen
    • CommentTimeJun 15th 2019

    Do you already know the rules for the modality you’re recommending?

    Program extraction doesn’t work on informal proofs, so informal proofs should only be allowed for computationally irrelevant propositions. If those do not include some equality form, I figure you wouldn’t get anywhere. Are we on the same page that the non-balanced subsystem is those types for which equality is computationally irrelevant?

    • CommentRowNumber51.
    • CommentAuthorMike Shulman
    • CommentTimeJun 15th 2019

    Well, there are basic rules satisfied by any modality. If your modality has extra properties then you can add them as axioms. Any type, including equality, can be made into a modal version by applying the modal operator, e.g. (x=y)\bigcirc(x=y). You could restrict to the types whose equality is modal (generally called the “separated” types for the modality), but if your modality is axiomatic rather than defined (as in this case) those won’t include any naturally-occurring types. So I think a better approach is to just use the same types but modally reflect any of the “propositions” that you want to consider “computationally irrelevant”.

    FWIW, this isn’t just something I made up; Agda already has something like an irrelevance modality. You might also be interested in this if you haven’t seen it yet.

    • CommentRowNumber52.
    • CommentAuthoratmacen
    • CommentTimeJun 15th 2019

    It seems we are not on the same page after all. I sort of know about Agda’s irrelevance features. I’ve seen Coq’s sProp thing, since (Bas?) Spitters cited it in #26. An interesting point in that paper, if I’m remembering correctly, was that to be compatible with HoTT, sProp doesn’t have an equality type.

    On the prototype CLF page, I wrote about two modalities: T\lfloor T \rfloor and T\lceil T \rceil. (In “other semantic judgments” and “constructive propositions”.) T\lfloor T \rfloor is always computationally irrelevant. This “squash” modality seems to be the modality you’re thinking of, based on those citations. But its availability is independent of whether the system is balanced. The sign that CLF is not balanced is that equality types are always “squashed” (squash-stable), and simultaneously, T\lceil T \rceil (propositional truncation) is generally not.

    The modality I thought you’re getting at is one such that (t=tTt = t' \in T) is generally not squashed, but (t=tTt = t' \in \bigcirc T) always is. Since T\bigcirc T is in the non-balanced subsystem.

    • CommentRowNumber53.
    • CommentAuthorMike Shulman
    • CommentTimeJun 16th 2019

    Well, I’m glad we cleared that up then. It is true that equality types of modal types are always themselves modal; I’m just saying that you don’t generally want to apply an “irrelevance” modality to the types themselves that you’re doing computation with, since you don’t want them to be irrelevant. I suppose you could imagine a modality \bigcirc such that P\bigcirc P is irrelevant when PP is a proposition but not otherwise, but that doesn’t seem to make much sense to me.

    I don’t remember anything about sPropsProp not having an equality type. How would you prevent it from having one? I’m sure there is no two-level business going on.

    • CommentRowNumber54.
    • CommentAuthoratmacen
    • CommentTimeJun 16th 2019
    • (edited Jun 16th 2019)

    No wait, I don’t think we’ve cleared it up. Let’s talk about a realizability topos. Assemblies are the non-balanced subsystem. As I understood your proposal, T\bigcirc T should be the closest assembly to TT. T\lceil T \rceil is the propositional truncation of TT, which is always an assembly. There is a codiscrete modality (is that the term?), which gets you a plain set (which is always an assembly). T\lfloor T \rfloor is the codiscrete propositional truncation, I think.

    I am still not clear whether the thing we’re trying to argue about is how to deal with the assemblies, or whether it’s worthwhile to go beyond assemblies.

    sPropsProp doesn’t necessarily have equality because it doesn’t have inductive families as a primitive. The ones it has are defined by large elimination. Anyway, it should be clear, independent of implementation, that sPropsProp should not have equality for HoTT, or even for a realizability topos, because

    • sPropsProps are computationally irrelevant, while in a constructive HoTT or topos, equality can be relevant.
    • I agree that it’s not two-level, so you don’t get two kinds of equality.
    • CommentRowNumber55.
    • CommentAuthorMike Shulman
    • CommentTimeJun 16th 2019

    I don’t mean the same thing by a “non-balanced subsystem” that you do. As I said in #51, I mean the same types with a different “logic” on top of them. I don’t think there is such a thing as a “closest assembly to TT”.

    Oh, by “sProp has equality” you meant what I would say as “all types have sProp-valued equality”. Yes, of course that’s not true, or at least not the ordinary equality; I would expect it’s reasonable to have every type have a propositionally-truncated sort of equality that’s valued in sProp.

    • CommentRowNumber56.
    • CommentAuthoratmacen
    • CommentTimeJun 16th 2019

    So then I don’t understand what it is you’re recommending. Maybe you could say what all the available modalities are, what their informal semantics are, which ones are computationally irrelevant, and which ones affect an equality type?

    You can squash an equality type down into sProp, if that’s what you mean. By doing so, you generally lose the ability to transport using it.

    • CommentRowNumber57.
    • CommentAuthorMike Shulman
    • CommentTimeJun 17th 2019

    There’s no complete list of “all the available modalities”. That would be like trying to make a list of “all the monads in the world”. I’m proposing to use HoTT augmented by a modality whose informal semantics is “computationally irrelevant”. I don’t have a particular concrete set-theoretic semantics in mind for it, although perhaps it could be something like the indiscrete objects in a realizability topos. I don’t know what you mean by “which ones affect an equality type” – any modality can be applied to any type, including an equality type.

    • CommentRowNumber58.
    • CommentAuthoratmacen
    • CommentTimeJun 17th 2019

    Of course I meant the ones featured in the rules. :|

    Regarding “indiscrete objects in a realizability topos”, so you mean the modality would be something like what realizability topos calls Γ\Del \Gamma?

    • CommentRowNumber59.
    • CommentAuthoratmacen
    • CommentTimeJun 17th 2019

    (By “which ones affect an equality type” I meant which modalities MM it’s not the case that M(t1= Tt2)(t1= Tt2)M(t1 =_T t2) \cong (t1 =_T t2).)

    • CommentRowNumber60.
    • CommentAuthorMike Shulman
    • CommentTimeJun 18th 2019

    Which rules are “the” rules?

    Yes, Γ\nabla\Gamma is the modality for the indiscrete objects. But it could be that that doesn’t do what you want, I haven’t thought about it very much.

    I can’t think offhand of any nontrivial modalities for which all equality types are automatically modal.

    • CommentRowNumber61.
    • CommentAuthoratmacen
    • CommentTimeJun 18th 2019

    Which rules are “the” rules?

    The rules you’d propose to use.

    Yes, Γ\nabla\Gamma is the modality for the indiscrete objects. But it could be that that doesn’t do what you want, I haven’t thought about it very much.

    This is an interesting idea. It does something I want, but I’m pretty sure it’s not a complete replacement for squash. I suppose the indiscrete objects ought to be considered the computationally irrelevant ones, and the subsingletons are “mathematically irrelevant” ones, and squashed types are irrelevant in both senses. For program extraction, I think it’s the indiscrete types whose elements can be erased. But while still working in the proof assistant, to reason about the elements, which are generally not mathematically irrelevant, they cannot yet be erased. And elements of subsingletons generally cannot be erased or you lose canonicity. But elements of squashed types can be erased right after you check them. So I think you can think of squash as an optimized implementation of making something completely irrelevant.

    Coq’s sProp is less inclusive than squash, for reasons of decidable type checking. (If you want decidable type checking, I suppose you could say that even squashed types may not be completely irrelevant, since their closed proof terms can witness totality.)

    In HoTT with this new indiscrete modality, are the modal types necessarily hSets?

    I can’t think offhand of any nontrivial modalities for which all equality types are automatically modal.

    Yeah probably not, for HoTT. Back when I asked that, I didn’t remember you wanted this to be a HoTT.

    • CommentRowNumber62.
    • CommentAuthorMike Shulman
    • CommentTimeJun 18th 2019

    I’m not proposing any particular rules. I’m proposing that you, who are the one who knows what you want to achieve (I don’t), consider obtaining it by formulating rules (I can’t guess what they might be) for a modality inside an ambient balanced system, rather than making the system as a whole unbalanced.

    • CommentRowNumber63.
    • CommentAuthorMike Shulman
    • CommentTimeJun 18th 2019

    (There’s no reason the indiscrete objects would be hSets, although the correct notion of “realizability \infty-topos” is still up in the air.)

    • CommentRowNumber64.
    • CommentAuthoratmacen
    • CommentTimeJun 18th 2019

    Two responses to #62: One, I’m still intent on doing a non-balanced system first, since I’m almost positive there’s no benefit to CLF being balanced.

    Two, from your remark from #55 ‘I don’t think there is such a thing as a “closest assembly to TT”.’, it sounds like being an assembly is not something to be characterized with an additional modality. But I do probably want a lot of structure on the assemblies to be exposed. So I actually can’t think of any modalities that I wouldn’t have anyway, with just assemblies.

    For generalizing to higher dimensions, it seems right to think of realizability topos objects as 0-truncated 1-types. By that, I mean that only two dimensions of cell are implemented: elements and equality proofs. But the equality proofs are unique. So the objects are 0-truncated and “1-squashed”. Assemblies are 0-squashed, and therefore 0-truncated automatically. So it would be some kind of puzzling out what to do about n-squashed and/or n-indiscrete types.

    It’s not intuitive to me that there’s no closest assembly to TT. Why doesn’t 0-squashing internalize, like usual (-1)-squashing does?

    • CommentRowNumber65.
    • CommentAuthorMike Shulman
    • CommentTimeJun 19th 2019

    Actually I take that back; assemblies are the modal types for a modality, in fact they are the ¬¬\neg\neg-separated objects in the effective topos. Sorry!