Together with my colleague Ph.D. student Mattia Coloma and with Domenico Fiorenza we have written a short note on the graded Hori transform following the latest paper of Fei Han and Varghese Mathai. We show how, in the rational setting, the graded Hori map can be naturally seen as a “pull-iso-push” transform.

You can find it here, any comment before we upload it on arxiv is welcome.

]]>I would like to know the great nForum community of "categorical physics" would be interested in contribute. Give it a chance and take a look into the project https://github.com/gcarmonamateo/GeomFormes and hopefully caught your interest in it.

(Sorry for the imprecisions in the English language). ]]>

I´m Software Architect experienced in Optimization Algorithms and Distributed Expert Systems. I have recently developed a technique which breaks limitations on Neural Persistence, on which I want to release my research article. However, I think that it is recomendable firstly to introduce appart the philosophical proceeding using fibred categories as a powerfull innovation in research level. I want to review and discuss this publicly for a better acceptance before resarch article comes out.

Please find out below and give me your feedback:

http://ixilka.net/publications/innovations_in_maths.pdf

]]>Hi all!

Please somebody review my recently officially published (INFRA-M publisher) open access book:

http://www.mathematics21.org/binaries/volume-1.pdf

Even a very short review matters.

The book contains a very wide generalization of general topology and (surprise!) a (generalized) limit of arbitrary (discontinuous) function. As a bonus I can send you a secret draft of my article about properties of generalized limits of discontinuous functions and a definition of generalized solutions of differential equations.

I need this very much to have the right to create my own articles in Wikipedia about my research (because I already have one independent source, I need one more). Please, please!

]]>There is a new preprint by Kraus & Raumer.

Here is the abstract:

The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove a theorem about equality types of coequalizers and pushouts, reminiscent of an induction principle and without any restrictions on the truncation levels. This result makes it possible to reason directly about certain equality types and to streamline existing proofs by eliminating the necessity of auxiliary constructions. To demonstrate this, we give a very short argument for the calculation of the fundamental group of the circle (Licata and Shulman ’13), and for the fact that pushouts preserve embeddings. Further, our development suggests a higher version of the Seifert-van Kampen theorem, and the set-truncation operator maps it to the standard Seifert-van Kampen theorem (due to Favonia and Shulman ’16). We provide a formalization of the main technical results in the proof assistant Lean.

From what I have skimmed:

There are some interesting techniques used in this paper. For example using ’wild categories’ (untruncated categories) to approximate (oo,1)-categories.

There is also some interesting discussion of a higher van Kampen at the end.

In type theory, coinductive types are used to represent processes, and are thus crucial for the formal verification of non-terminating reactive programs in proof assistants based on type theory, such as Coq and Agda. Currently, programming and reasoning about coinductive types is difficult for two reasons: The need for recursive definitions to be productive, and the lack of coincidence of the built-in identity types and the important notion of bisimilarity. Guarded recursion in the sense of Nakano has recently been suggested as a possible approach to dealing with the problem of productivity, allowing this to be encoded in types. Indeed, coinductive types can be encoded using a combination of guarded recursion and universal quantification over clocks. This paper studies the notion of bisimilarity for guarded recursive types in Ticked Cubical Type Theory, an extension of Cubical Type Theory with guarded recursion. We prove that, for any functor, an abstract, category theoretic notion of bisimilarity for the final guarded coalgebra is equivalent (in the sense of homotopy type theory) to path equality (the primitive notion of equality in cubical type theory). As a worked example we study a guarded notion of labelled transition systems, and show that, as a special case of the general theorem, path equality coincides with an adaptation of the usual notion of bisimulation for processes. In particular, this implies that guarded recursion can be used to give simple equational reasoning proofs of bisimilarity. This work should be seen as a step towards obtaining bisimilarity as path equality for coinductive types using the encodings mentioned above.

]]>*C’est au cours de ce travail [“La Longue Marche”] aussi (mais développé dans des notes distinctes) qu’apparaît le thème central de la géométrie algébrique anabélienne, qui est de reconstituer certaines variétés X dites “anabéliennes” sur un corps absolu K à partir de leur groupe fondamental mixte, extension de Gal(K̅/K) par π1(XK̅); c’est alors que se dégage la “conjecture fondamentale de la géométrie algébrique anabélienne”, proche des conjectures de Mordell et de Tate que vient de démontrer Faltings. Esquisse d’un programme*

I am currently working in the transcription (with the collaboration of M. Künzer) of this “Note”. The project is open source and I would like to invite you to contribute.

]]>I recently put on the arXiv this preprint:

https://arxiv.org/abs/1704.00303

that stemmed from a question I posed on MathOverflow a few months ago (the title is the same, googling gives both the arxiv preprint and the MO-thread), and that received some attention and positive comments (I hope).

We authors are in the phase of polishing some details, and improving the clarity of the discussion. Once this process is finished, I’d like to have it published: what is, in your opinion, a good journal where to send the preprint?

Thanks!

]]>I’m on the fine-tuning phase of a short note about coend calculus. The discussion is still in a sketchy form, in particular

- There are some subtleties I’d like to fix; not sure that all the constructions I make are possible, remaining in a single universe $\mathbf{U}$;
- I feel I’m waving hands too vigorously in a couple of arguments (section 3.3 and the “introduction” to operads are extremely unsatisfying and messy);
- Several references are still missing or can be extremely improved to clarify the discussion;
- I’m still unable to retrieve the original paper by Yoneda
*On Ext and exact sequences*. Jour. Fac. Sci. Univ. Tokyo 8 (1960), 507 - 576. which introduced the integral notation.

…plus several other mistakes which you will certainly notice if you begin reading! As always, any kind of advice, comment or criticism is welcome: the only reason why the note lacks an acknowledgements section is that the list will certainly grow bigger in the semi-public phase.

]]>The preprint The homotopy theory of coalgebras over a comonad by Hess and Shipley looks interesting. Among other things it contains

- a theorem about replacing a model structure by a Quillen equivalent one in which the cofibrations are the monomorphisms
- a theorem about when the category of coalgebras for a comonad inherits a model structure by “left transfer”.

I wonder whether there could be a model structure on coalgebraically cofibrant objects?

]]>I'm one of the founders of Arbital, a website for crowdsourced, intuitive math explanations. We are currently doing a collaborative project to explain the Universal Property concept in Category Theory. If you'd like to check it out, take a look here: https://arbital.com/project/

Once we are done, I'll post a link here so people can read the explanation. :) ]]>

I just submitted the following to “The Journal of Mathematics and Music”.

“Using the traditional accidentals of western music theory, a musical space dubbed accidental space is introduced in three contexts. The first is as an algebra reminiscent of that used in quantum mechanics; this version places special significance on palindromic modes such as Dorian. The second is as a network reminiscent of that used in graph theory; this version shows clear patterns regarding chord quality clustering. The final is as a category with modes as objects and accidentals as morphisms; this version provides a singular context which encompasses both algebra and network. “

**I’m fairly certain this is a mathematical something, but I’m not a professional mathematician; I’m looking forward to being proven right or wrong by others outside of me.**

Hey all,

I’m writing a roughly 20 page paper for submission to the Journal of Mathematics and Music and I was curious if these boards are an appropriate place to post and get feedback.

As I am no affiliated with any university, I don’t have access to professional feedback on CT nor can I upload to Arxiv (I’m a physicist, not a mathematician).

If this is not the place and/or there is a better place and you could guide me there I’d be very thankful.

On the other hand if this is the place to engage in discussion about the topics I bring up in the abstract, I’ll post the draft and I’d be thankful as well!

Sincerely,

Ricardo Javier Rademacher Mena

www.linkedin.com/in/ricardo

www.thevniversity.com

“Using the traditional accidentals of western music theory, a musical space dubbed accidental space is introduced in three contexts. The first is as an algebra reminiscent of what is used in quantum mechanics; this version places special significance on palindromic modes such as Dorian. The second is as a network reminiscent of what is used in graph theory; this version shows clear patterns regarding chord quality clustering. The final context is as a category with modes as objects and accidentals as morphisms; this view has the advantage of providing a singular context with which to understand the two former views. “

]]>I got an email which is really more for Mike than for me. Maybe Mike got it too, but in case not, I repost it here:

I teach an introductory axiomatic set theory class at the University of Warwick. We cover the basics of Zermelo set theory. In particular, this is material set theory. Over the years I've become acquainted with alternate structural presentations such as ETCS. Now and again I have looked at the SEAR page on nLab. Some students from my class have approached me in connection with our undergraduate summer research programme. It occurred to me that it might be instructive for them to see what would be involves in redeveloping the matierial from our course from such an alternative viewpoint. I don't believe there is an ETCS based text, although Lawvere-Rosebrugh's Sets for Mathematics is a step in that direction. Perhaps SEAR would represent a more comfortable middle ground. I've been unable to locate any SEAR materials beyond what I find at nLab. Is there any existing work that my students should be made aware of?

The course in question seems to be this one; in any case, the email came from the instructor listed there, from the email address given on the page for that instructor.

]]>With Urs Schreiber and Alessandro Valentino we are finalizing a short note on central extensions of mapping class groups from characteristic classes.

A preview of the note is available here: *Higher extensions of diffeomorphism groups (schreiber)*

Any comment or criticism is most welcome

]]>Domenico Fiorenza and I are completing a paper about hearts of t-structures in stable $\infty$-categories, which shows that in the $\infty$-categorical setting semiorthogonal decompositions on a stable $\infty$-category $\mathcal{C}$ arise decomposing morphisms in the *Postnikov tower* induced by a chain of t-structures, regarded (thanks to our previous work) as multiple factorization systems on $\mathcal{C}$.

A slightly unexpected result is that t-structures having stable classes, i.e. those $(\mathcal{C}_{\ge 0}, \mathcal{C}_{\lt 0})$ such that both classes are stable $\infty$-subcategories of $\mathcal{C}$, are precisely the *fixed points* for the natural action of $\mathbb{Z}$ on the set of t-structures, given by the shift endofunctor.

As always, any comment, suggestion, criticism is welcome.

]]>With Alessandro Valentino we have now written a short note on anomalous tqfts and projective representations. In case you’d like to have a preview of it before we post it to the arXiv, any suggestion, comment or criticism is welcome.

]]>With Fosco Loregian we are now fine tuning a short note on t-structures and factorization systems in $\infty$-stable categories. In case you’d like to have a preview of it before we post it to the arXiv, any suggestion, comment or criticism is welcome.

]]>This paper: http://arxiv.org/abs/1402.3280, says roughly that for every (locally compact Hausdorff) group satisfying the Baum-Connes conjecture with coefficients (e.g. every a-T-menable group), acting on a space, if the associated action groupoid $\Gamma$ carries a $[0,1]$-family of gerbes $\mathcal{G} \to \Gamma\times [0,1]$, then the maps of twisted $C^\ast$-algebras $C^*_r(\Gamma\times[0,1],\omega) \to C^*_r(\Gamma,\omega_t)$ $\forall t\in [0,1]$, induce an isomorphism on K-theory, where $\omega$ is the cocycle classifying the gerbe.

This to me sounds similar to the question in the title of the thread, at least in the special case considered in the paper, which is expected to be true for locally compact groupoids more generally.

Thoughts?

]]>Hi all,

I decided to release the short paper *WISC may fail in the category of sets*, as I feel it’s a nice construction, even if it didn’t achieve my original goal of showing WISC independent of ZF (and Karagila beat me to it anyway). Rather, I get the result that WISC can fail in a well-pointed topos with nno (and assuming at most ETCS as base set theory). In fact I suspect my proof is constructive, but I’d like some outside perspective on that (Mike? Toby?)

I still need to fix a couple of references (chapter and verse of an example from SGA IV, a specific statement of Mike and Benno vdB’s paper with Moerdijk rather than his unpublished note), but it feels reasonably close to final. However, I wouldn’t mind comments (any sort) if people are so inclined. EDIT: reading it again now that I’ve exposed it to the world has made me pick up on a few things that I want to change. This is precisely the sort of response I wanted from myself, at least.

Also, I’m terrible at choosing journals (I’d be inclined to stick everything I can in TAC, but I get the feeling this is not, for better or worse, the best career move), so if anyone has any suggestions, I’m all ears.

]]>For those who are not yet aware, all of *K-theory* is now available online again. Unfortunately one needs to be a Portico subscriber, but my library is, if anyone is stuck ;-)

Has anyone here had a look at the two preprints (arXiv:1302.3684 and arXiv:1302.5325) and how they interact with the ideas on infinity categories? The basic idea seems reasonably natural and nice but I have not looked further than the start.

]]>