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As a physicist, I have been influenced by the program of symmetric monoidal categories since they seem to capture many relevant aspects of quantum theory. That said, I would prefer to see all my structure presented “internally”. For this reason, I am intrigured by the similarity of monoid axioms and monad axioms especially monads in a bicategory where we have can present the monads axioms in terms of string diagrams. I am interested in comonoids as they seem to axiomatise classical structures. More importantly, comonoids are internal to the symmetric monoidal category which reflects the notion of an interface to an underlying structure. At the moment, I would like to dispense with the monoidal structure as I see it as excessive, and yet retain something like classical interfaces. Thus the interest in the comonad axioms (in a bicategory). Can we have internal categories with just a bicategory and a monad? CAT is a bicategory, so this would be like living in CAT instead of a monoidal category with tons of extra axioms.
What do you mean ’have internal categories’? Internal to which category?
Do you mean instead of defining internal categories using the bicategory of spans in a category? CAT is a Cartesian symmetric monoidal bicategory, so unless you never want to have products of categories (which I don’t think you want), you need some sort of monoidal structure around.
To add to what David said: a small category is (essentially) a monad internal to the bicategory of spans of sets. Similarly, a category object in a finitely complete category is a monad in the bicategory . An internal presheaf on a category object is a left -algebra (left or right, depending on convention) whose underlying span goes from to , where is terminal in .
The apparatus of string diagrams extends without difficulty from monoidal categories to bicategories. (Bicategories can be thought of as “many-object monoidal categories”, analogous to how categories are “many-object monoids”. They aren’t any simpler than monoidal categories, in terms of “excessive structure”. But the “excessive structure” can be ameliorated by passing to equivalent strict monoidal categories or to strict 2-categories, which are the proper environments for string diagram interpretations.) In the extended interpretation, nodes are labeled by 2-cells, strings by 1-cells, and planar regions in the complement of the string diagram are labeled by 0-cells. Then data of monads in bicategories, and of morphisms of monads in bicategories, can be represented by string diagrams as you note.
One could go on, but I can’t figure out from your comment (Ben) what you are really up to, so I’ll stop here.
Hi Todd, David,
I want to thank you both for your input here and your patience. As usual it would be difficult to explain where I am going with this because A) its based in some abstruse, new physics and B) I have no idea what I’m doing….so let’s start!
In 2006, Rob Spekkens published a paper in which he “derives” many quantum mechanical effects by positing a situation where one may receive answers to only half as many questions as would be required to specify the state of a system. This restriction can be seen as a kind of interface to an underlying system. I wrote about that here.
Since then, the idea has taken hold with the category folks as we can see here. Of course, the category guys had already axiomatized the category of finite dimensional Hilbert spaces.
What I am objecting to is the idea of having a “quantum category”, or perhaps the idea of presentation of the axioms themselves as straight-up axioms of some category. If you look at my paper, you can kind-of see that the interface is the comonoid structure internal to the monoidal category. With that said, let’s recall that Spekkens is pointing out that simply the quality of having a limited interface produces quantum theory (almost). Thus, it is the quality of having an interface that gives quantum theory.
This does not jive with offering up a single category with some axioms and saying the objects are quantum in nature.
I am, therefore, left feeling that quantumness should be offered up by a quality, not a category, and that quality is of having an interface. The solution, so far, has been to follow the beaten path and use monoidal structure with many excess axioms (the cat FDHilb) and have comonoids internally and even cats internally. Don’t get me wrong. That has been awesome. Still, it is clear to me that we need to internalize all this.
I am now going to say some things that concern, to a degree, the presentation of structure, and to a degree, the apprehension of structure.
Suppose that I could, in some way, provide you with a monad without giving you the underlying category. So, I have given you an endofunctor and even a few rules for its composition. These would be the monad axioms. I mean, I could just as well give you an algebraic theory, but concretely presented just enough for you to do basic calculation or term rewriting.
At first, you are happy to get your hands on the device since, from your perspective, you only see it as a calculational tool. Eventually, however, you start to wonder about how this system is capable of computing the way it does. Thus, you start to wonder about the underlying category. Meanwhile, all you have access to is the monad: you only have access to the interface.
I quickly checked over at n-category cafe and have found a post by Mike Schulman about this very subject.
I will finish off with a plug for myself. This paper basically describes what physics would look like if you based it on these ideas. It’s pretty bad and contains way too much stuff in too little detail. Enjoy! I am working on a second which could focus more intently on this internalization stuff. Clearly, I could use some help.
This is what it’s like to be a physicist.
I don’t believe this sentence.
I think what you are trying to understand is better understood without reference to being or not being a physicist (what does that even mean?). If you just leave away these comments, your discussion loses nothing. But you might gain some focus on the subject, I’d think.
Hi Urs,
I don’t mind editing the post. I agree it was not very good language.
Hi Ben,
you have a dodgy link: http://www.math.ntnu.no/~stacey/Mathforge/nForum/www.cs.mcgill.ca/~bsprot1/EvolvingUniverseFeb24.pdf which should probably be http://www.cs.mcgill.ca/~bsprot1/EvolvingUniverseFeb24.pdf
I think what you are trying to understand is better understood without reference to being or not being a physicist (what does that even mean?).
Urs, I might have misunderstood you, but if it is what I understood I strongly agree with you. I like that you promote common intellectual sense here.
I often hear around often strange arguments that some reasoning can be weakened just because the calculation is produced by remote usage of physical intuition. Any intuition is basically learned behaviour from vague internal representation of a previous experience; if one extends the experience to new examples it may or may not apply. For this we have logics etc. In one case we had some identity which we had to prove for every positive integer . For some essential features were there. The identity followed from the two separate identities. For bigger one the two identities generalized to a very complex conjectural identity, of the form where are itself quite complicated (and depending on additional parameters in the game) and which are given by a formula of the same shape. So they said that it MUST be that . Of course if it is we have a proof that the sum is . But the sum can be zero even if the summands are not. For there is only one term and that is the case which we knew to be zero. So they said that it must be so for general . I said look there is no logic, you reverse the direction of the implication, there may be some proof that the sum is zero without requiring that for all . Yes, but they started appealing to “physics”, that such virtual situations are mathematical nonsenses and the “physics” won’t take such (what ? we had to prove an identity ? why would physics make some mathematical proofs and calculational mechanisms of a mathematical identity appearing in the game not allowed ?). But physical application, even if true, would need only . Still, they insisted that my picking up the logic is too mathematical and that I need to trust physics and search for proof of as the “intuition” suggests that strongly. So they started working on and spent another month, until I produced a computer generated counterexample of and the research went back to proving , which we eventually succeeded. So the “physics intution” should not be overstretched to a religion, beyond the area of good experience.
I am going to qualify my statement because it concerns a fairly deep aspect of physics. As for this whole debate about physics intuition, you will see there is nothing too philosophically objectionable about my ideas.
It is my contention that the internalization which we are probing out here reflects the visceral experiences of being in a lab. Even moreso, it reflects the experience of being in contact with a system for which you have limited understanding. We sometimes take for granted that just about every apparatus these days comes with a rich and nearly perfectly accurate theory which describes it.
Physically speaking, we discover a system by interacting with it and watching it react. We do this in the presence of some kind of theory. In mathematics, it is not clear how we discover structure. What is clear, is that I can give the axioms of an algebraic structure and then computation in that structure is nearly mechanical, and this then transcends to nicely algebraic logics. Sometimes when we learn or discover mathematics, we play with things. For instance, we might multiply a vector by a matrix, and then multiply the same vector by the adjoint of the same matrix. Then we could do a test like looking at the inner product of the two resulting vectors. This tells us something about adjoints of operators.
This is already pretty deep because, if we buy this notion of the interactive apprehension of structure, we could imagine updating a category which we think encodes the system behaviour. Category update like that would be a novel mathematical discipline. I should mention that I would like to start that discussion on this forum at some time in the future, but that’s another story.
I am trying to bring these ideas to physics, and perhaps it is jumping the gun, as Urs has suggested.
Let me just finish. It is the case that internalization captures a relatively new axiom which people like Lee smolin, Fotini Markopolou and Lucien Hardy are trying to add to physics. This axiom regards the fact that physics is done decidedly within a universe. A statement like that is very frustrating to deal with mathematically and it has taken me a long time to find something that reflects it in the foundations of mathematics.
The category theory guys have been working on this but its possble that they don’t know it. The idea of interfaces through which we discover structure is related to the Holographic principle. There is a categorical abstraction of the holographic principle(HP) but I just can’t tell if it is talking about the same thing.
My work concerns the experiences of a scientist in his laboratory. Normally the HP is applied to the situation of observing a black hole and so the interface is around the black hole as if it were in a box. This is the pacman version. Alternatively, though equivalently, we can imagine that, instead, it is we who are in the box and the surface is what we are using to “see” the entire universe. This is the telescope that I talk about in “Probing of structure”.
It could be a goal to fit all these pieces together under a nice mathematical method. That method would then be based on some very intuitive beliefs about doing physics.
I didn’t complain anout using physical intuition. What I said is that it is a distraction to go on about what it means to be or not to be a physicist.
Didn’t you also complain on the qualification on what it means or not to apply physical intuition (as opposed to any kind of vague reasoning, theoretical reasoning, extension of reasoning etc.) about which I think there is lots of fetishism.
In mathematics, it is not clear how we discover structure. What is clear, is that I can give the axioms of an algebraic structure and then computation in that structure is nearly mechanical, and this then transcends to nicely algebraic logics.
Why do you limit reasoning in mathematics to axiomatic analysis ????????????
Why do you limit reasoning in mathematics to axiomatic analysis ????????????
Well… Ultimately…
=)
I would like to restart this thread eschewing all the physics talk. This will be an honest attempt to express an idea with purely mathematical value. I am trying to reverse the roles of two categories in a particular construction. In particular, we start with a construction of internal categories as we find here. Next, we suppose that I can give you one, most, or all the internal cats in a monoidal cat. Each of these is different, and should lead to different results, so for now, let’s say I can give them all to you. What I do not give you are the axioms and morphisms of the monoidal category. That is a question of how to present a structure. This post is about the apprehension of structure because my question to you, after giving you the internal cats, is about the underlying monoidal category. I am now trying to think of a good question to ask you, but let’s take this to a kind of quick conclusion. I want you to somehow present to me a theory of the underlying category using only the internal cats I have given you. Perhaps this can be done inductively starting with the internal cat with only one morphism. I am definitely thinking of a categorification, or perhaps just an abstraction of structural induction. The categories are the elements of the ordered set and functors between them….
Maybe the last post should be its own thread. Let me know if anyone finds it interesting.
Maybe you are asking: given the 2-category of categories internal to a monoidal category , can one reconstruct from it?
The question of Urs in 15 is interesting. As a simple test case, one can take to be cartesian monoidal (i.e., finitely complete). Here I think it’s easy, because we retrieve as equivalent to the homotopy category of the sub-2-category of whose objects are “discrete” internal categories. (By “homotopy category” of a 2-category, I mean we take objects to be the 0-cells, and morphisms to be isomorphism classes of 1-cells.) What do I mean by “discrete”, in terms of pure 2-categorical concepts? The idea is that for any 2-category with comma objects, we define for any object to be the comma object , and then define to be (essentially) discrete if the canonical map is an equivalence. Observe that if is finitely complete, then has comma objects.
I’m not sure about the case for other monoidal categories .
@Todd - if you define discrete objects representably, then I believe you recover the ex/reg completion of . Also, when taking the equivalence in your definition, do you mean for it to be an equivalence in qua 2-category, or just some sort of weak equivalence (fully faithful, essentially surjective after some fashion)?
You could be right, David, although after a few glasses of wine I’m not prepared to say. :-) Am I getting the wrong answer in the case ? As for the second question, I had in mind qua 2-category.
If you know an answer to Urs 15 in the case of finitely complete, please go ahead.
The representable definition of discrete object is equivalent to Todd’s, if powers by the interval category exist (and the latter can be constructed from comma objects as he says). But I think David is right that just looking at the 2-categorically discrete (i.e. 0-truncated) objects in is going to get you more than — it’ll be something like the category of internal equivalence relations in with equivalence-respecting morphisms. That’s a subcategory of the ex/lex completion of , but it’s not (in general) all of it; also contains “pseudo-equivalence-relations”. But if is regular, then I think you do get all the ex/lex completion this way. However, is not equivalent to unless satisfies the axiom of choice — so if satisfies AC then you do get it back this way.
Finally, if is regular, then I think its ex/reg completion can be identified with the subcategory of discrete objects in . So if is exact, then you can recover it from — and therefore also from if you remember which functors are “weak equivalences”.
But I think David is right that just looking at the 2-categorically discrete (i.e. 0-truncated) objects in is going to get you more than
Okay. I admit I was uneasy with my idea in the first place. Does anyone have a better idea of how to get at from ( merely finitely complete), or is it the case that we can’t retrieve just from as a 2-category? That would also be interesting.
What if we take those objects for which is a set for all , and not just equivalent to a set, then we should get the discrete objects, and this doesn’t require anafunctors or anything. For anafunctors you need at least the canonical singleton pretopology (the smallest pullback-stable class of regular epis) or the choice of some singleton pretopology, and for merely finitely complete ambient categories this is extra structure.
By the way, I realized that the definition given in 16 doesn’t actually get you the representably discrete objects, only the groupoidal objects; for discreteness you need to also say something about parallel arrows being equal.
@David #21: Yes, I think that should work. Of course it requires as a strict 2-category.
Here’s a different idea. First consider the (non-strictly) 0-truncated objects of an a 1-category. Now consider the objects of this category which are projective with respect to regular epis. I think this is close to giving you back . On the one hand, every 0-truncated object of is a coequalizer of some equivalence relation between strictly-discrete categories; thus if it is regular-projective, this epi splits, so it is a retract of an object of , hence itself an object of . On the other hand, it’s a fact that can be identified with the regular projectives in ; thus every object of (regarded as a strictly-discrete category) is projective with respect to those regular epis in which remain regular epi in . The maybe-catch is that I don’t think is closed in under coequalizers (if it were, it would be the whole thing), and it might happen to randomly have some coequalizers that are different from those in .
On the other hand, I think that if we look at the 0-truncated objects in the tricategory of internal bicategories in , we get exactly . Thus, can be recovered as the regular-projective 0-truncated objects of .
By the way, I realized
Bah! Yes, of course. I think I took temporary leave of my senses back there.
To complete my sense of irritation, not only am I having some bad insomnia, but I have a really bad earworm. Right now I’m ready to kill Hall and Oates. :-)
Something like this is the subject of Bourn and Penon’s 2-catégories réductibles (TAC reprint), in which they show that a finitely (2-)complete (strict) 2-category K satisfies their exactness axioms if and only if it is comonadic (via a 2-lex functor) over some Cat(C). The lex category C is then recovered as the category of codiscrete objects (objets grossiers) in K – those X for which is an isomorphism.
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