If I recall correctly that book uses the Withney embedding theorem to show that any vector bundle can be complemented to a trivial one plus the fact that the global sections functor is additive. Looks like it doesn‘t require compactness.

]]>I added a link to a recent lecture series by Nicolas Orantin available online.

]]>created a stub entry *topological recursion* in order to record some references, and added cross-links with various related entries

Would it be appropriate to add to the nLab?

It sounds good. You should start somewhere and then we can give more substantial feedback.

]]>My aim is to translate the logic used into an equivalent topos theory of finite logic, or even do away with logic altogether and directly map complexity classes with a corresponding category. The goal of this is to reframe questions in complexity theory as questions in category theory.

For example: we can describe the existential second order quantifier as the left adjoint of the diagonal functor (context extension?) from $P(P(X)) \to P(P(X))\times P(P(X))$. I'm not sure whether $X$ is the ccc with all the propositional formulas, or if it's the finite model. I'm still learning.

Would it be appropriate to add to the nLab? I'm autodidactic with category theory, so I'm not sure if I can accurately describe things, but I'd still like a space to share my ideas. ]]>

I have a question about the definitions of star-autonomous category. The proof given there shows that the two definitions are equivalent as *properties* of a symmetric monoidal category. (Actually, it doesn’t quite finish the proof by showing that $d_A$ is an isomorphism for $\bot\coloneqq I^*$ in the second definition, but I can believe that that works.) However, are they equivalent as *structure*? Or, technically, I suppose, as *stuff*?

The first definition equips a symmetric monoidal category with the stuff of an object $\bot$, the (property-like) structure of being closed, and the property of the double-dualization map $d_A$ being an isomorphism. The second definition equips it with the stuff of a functor $(-)^*$, the structure of a natural isomorphism $\mathcal{C}(A\otimes B,C^*) \cong \mathcal{C}(A,(B\otimes C)^*)$, and… no properties. This immediately makes me suspicious: nearly always non-property-like structure has to be equipped with coherence properties to be “correct”. And I can easily imagine coherence properties that the second definition could satisfy, like the equality of two isomorphisms $\mathcal{C}(A\otimes (B\otimes C),D^*) \cong \mathcal{C}(A,(B\otimes (C\otimes D))^*)$ — and it seems likely that when the data is constructed from the first definition then it would satisfy these properties. Are these properties automatically *always* satisfied by the second definition? Or (as seems more likely to me) does passing back and forth from the second definition to the first and back again “coherentify” the isomorphism $\mathcal{C}(A\otimes B,C^*) \cong \mathcal{C}(A,(B\otimes C)^*)$ into a possibly-different one that does satisfy the coherence axiom, analogous to replacing an equivalence by an adjoint equivalence?

Well, it’s partly my fault for taking the thread in a different (though related) direction in #2.

]]>Re #6: oh, so it is! Helps to look at the thread title now and then. :-)

]]>Actually, I now realized that there are actually *three* different notions of functor between linearly distributive category that appear in the literature: the Cockett-Seely “linear functors”, the linear functors whose $\otimes$-part and $\parr$-part agree as functors (which I’m tempted to call “Frobenius linear functors”), and the ones that moreover are *strong* monoidal for both structures. The “$\ast$-autonomous nucleus” of a linearly distributive category uses “linear functors”, whereas the free linearly distributive category on a polycategory uses strong Frobenius linear functors. Which notion of functor is the free $\ast$-autonomous category on a linearly distributive category (#16 above) free with respect to? I’m guessing also the strong Frobenius ones?

(Or, more precisely, a linear bicategory with linear adjoints for all morphisms.)

]]>it seems one is invited to ’oidify’ (horizontally categorify)

That’s the name of this thread: linear bicategory. (-:

]]>True. I need to find time to think about this with more leisure.

]]>Why assume that this only involves natural language? Why can’t “Until we first get clear on this relationship” include the use of Bohr toposes?

]]>Maudlin would surely know of this.

Curious then how he advertizes this as his personal way of thinking about the situation. His case might be helped by pointing to the established history of the concept.

The interesting issue is the one Tanona raises

Not sure. At this point I feel natural language is beyond its range of applicability. I’d rather we phrase the coordination business more formally, say as in the paragraph on Bohr toposes, maybe with some linear logic thrown in (as we have discussed elsewhere) and then see what really remains of the measurement problem.

]]>I’m not sure it was really a ’choice’, in the sense that the author held the symmetric notion in one hand and the non-symmetric one in the other, and then weighed in favor of the symmetric notion. As far as I know, the symmetric notion has been studied much more (for example, from the perspective of studying coherence problems, I’m much more at home with the symmetric notion).

I for one have no objection to making adjustments to the page to redress this. But once one has the non-symmetric notion, it seems one is invited to ’oidify’ (horizontally categorify); has an appropriate 2-dimensional notion been considered in the literature?

]]>Is there an AT category-like structure on Fam(Ab) such that the exactness properties force the A side (or an analogue) to be dominant?

]]>Maudlin would surely know of this. The interesting issue is the one Tanona raises towards the end of our page in the context of QM:

…the characterization of collapse as a separate physical process is misguided because the phenomenon which collapse is supposed to address concerns not an actual process within quantum mechanical theory but rather the coordination between empirical measurements and representations of quantum systems. Until we first get clear on this relationship, it is premature to propose new processes to account for features of that relationship.

Then coordination can be seen as more fundamental than interpretation, in the sense of interpretations of QM.

]]>I added to star-autonomous category a mention of “$\ast$-autonomous functors”.

]]>I would like to see an explanation for why $Fam(Ab)$ fails to be a topos, so we can see why it doesn’t apply to parametrized spectra.

]]>I added a sketch definition of the 2-category of linearly distributive categories, and a remark that the forgetful functor from $\ast$-autonomous ones to linearly distributive ones also has a *right* adjoint. I also uniformized the notation on the page to $(\otimes,\top)$ and $(\parr,\bot)$ for the monoidal structures; I hope no one objects to that.

Thanks!

I have tried again to point Tim Maudlin to this here.

]]>Well, I think it’s more that the *original* notion was symmetric, with the non-symmetric case introduced rather later. My impression (although I’m not an expert in this field) is that recently people have been pushing to make the general term win out, as you say. But whoever wrote our page chose to talk only about the symmetric case, so I didn’t want to unilaterally make the change in case they had a good reason for that choice.

I have made the two parts of the statement at *smooth Serre-Swan theorem* more explicit.

I have been just citing this from Nestruev’s book. That book never requires the base manifold to be compact, it seems. I need to think about this again. Do we need to require a compact base manifold for the second part of the statement to be true?

]]>I added to dualizable object a section about duals in linearly distributive categories.

]]>