/sci/ - Science & Math

>>11430729
I haven't heard of that. Do you mean topological insulators? I have only glanced at these, but I would love to learn more about them.

Salut dear chanlings.

What are you studying right now? What ideas have captured your fascination?

Also, does anyone know how to get a hold of Nikolai Durov? I really need to speak to him about vectoids.

>>9432435
What do you mean by /mg/? I'm so out of the loop. Lol

>>9432421
If you want deets, feel free to ask questions and engage rather than sit around sounding puzzled. I'm not here to publish a paper, but to slosh around the collective brain juice, Writing these helps me more than anybody else, frankly.

I guess I need to be a bit less cryptic to be taken seriously here, which is fine and good. All of this is completely formalized in terms of higher algebra and enriched categories. I'm happy to offer explicit descriptions!

None of this is crank-tier; I use language carefully.

Hello /sci/,

I wanted to share recent ideas of mine that will help illuminate my prior writings here.

Enriched categories are where the meat is at. It's all generalized linear algebra: enrich things over modules over a monoid in a bicategory. Hom profunctors are categorified rank-(1,1) tensor fields. Enriched categories are generalized modules, and enriched topoi (free examples being given by enriched presheaf topoi) are generalized algebras.

Here is what's going on with the Goodwillie calculus: infinity-groupoids are presented simplicially by presheaves on the simplex category. The simplex category categorifies the natural numbers, and pointed sets are the free 2-module over F1 on one object. The category of pointed simplicial sets presents homotopy theory for pointed infinity-groupoids, with smash product arising as Day convolution. Thus, pointed homotopy types are presented by the cofree 2-algebra over F1 on one generator. This in fact stabilizes to the full infinite case: pointed homotopy types are elements of the cofree infinity-algebra over F1. This is why they behave as formal power series.

I've been studying differential geometry on moduli spaces lately, especially pertaining to the process of quantization in physics. Any cool things that you guys can share with me? I'm always looking to learn stuff!

Best,
OHP

>>9235979
I listened to a talk by Smale at the University of Michigan, and he discussed his current work trying to study the dynamics of genomes in essentially the space you describe.

Hey there /sci/chonauts!

I was hoping to kick up a conversation about information theory in category theory. My end goal is to be able to prove certain statements by showing that there is only enough "entropy" in an expression to allow for a small set of possibilities. I have finally collected some of my preliminary ideas on my latest blog post. Take a look and share your thoughts!

https://psychicapparatuses.wordpress.com/2017/08/23/categorical-information-and-fractals/

>>9113884
Fair Silke-chan, you are a witch and should already be acquainted with categorical mysticism. But, imagine that you could step outside of a universe of things and examine them globally? What if you could compare entirely different theories by constructing universal bridges (AKA adjunctions) between them? You would be a happy witch indeed. That's what category theory is good for.

Hello /sci/,

I have written a new post at Psychic Apparatuses (https://psychicapparatuses.wordpress.com/2017/06/11/action-complex-of-homotopy-groups/)) detailing the construction of a complex of commuting actions of lower homotopy groups on higher homotopy groups. I haven't found anything in the literature discussing these higher actions, but it's all quite natural since all of the actions satisfy nice universal properties with respect to one another (the group structure on connected components of iterated loop spaces is just a special subsystem of this complex).

Any thoughts on what can be done with this? How do commuting actions restrict the structure of the groups involved?

>>8928657
You're certainly right, HoTT is a lovely field with lots to offer in this direction. This construction fits into the overarching framework of HoTT, which is quite handy. I'm seeing now that this construction is the nerve associated to a very natural cosimplicial monoid, but where I would really place this construction is near the Dold-Kan correspondence. It's part of the Baez's "cosmic cube," so it fits into the higher category theory/HoTT picture quite snugly!

Hey guys, I have a new construction to share. I think it is something both topologists and algebraists can appreciate. It's related to the bar construction and nerve construction for categories, and basically gives you a way to examine algebraic stuff in terms of homotopy theory. A run-down is given at Psychic Apparatuses: https://psychicapparatuses.wordpress.com/2017/05/22/partition-fibrations-of-monoids/

Any ideas on where to expand this idea to? For example, passage to rings means supplying an action of the multiplicative monoid on the bundle. I was hoping for a good discussion on these concepts.

The cool thing about logarithmic spirals is the relationship between the area up to a certain point, the radius at that point, and the curvature at that point. The existence of a shape with these properties demonstrates a lot of structural balance in analysis.

>>8861406
Just do it around each individual. Cut to the chase.

>>8857446
Topkek, pretty much. This can be said for just about any advancement in any particular field, though; if someone proved the Riemann Hypothesis most people wouldn't honestly care. Regarding homotopy types, I think they are cool as fuck, and if I can share the love and educate other people on them, then at least they will have had adequate exposure to decide whether or not they care about them.

If the false vacuum spontaneously decayed and the decay reached us, everything that gives our existence meaning would be destroyed with us. It wouldn't be so tragic; indeed, nobody would exist long enough to observe the event and reflect on this "tragedy." There would be no loss because there would be no observers to appreciate the feeling of loss. Adding to this, the probability of this occurring seems very small, far less than the probability of a quasar blasting us into nothing. So, why worry about it? It's unlikely to occur, and if it does then nobody will care.

Hey /sci/, everyone's least favorite namefag, here to offer some guiding wisdom to those pursuing an understanding of category theory and homotopy theory!

I am starting to realize just how powerful the general notion of nerve and realization is (https://ncatlab.org/nlab/show/nerve+and+realization).). Special cases of this glorious construction include the simplicial nerve system used in algebraic topology, the Dold-Kan correspondence of homological algebra and homotopy theory, and even some cool stuff tying dynamical systems and homotopy theory via an emphasis on spaces of orbits.

The idea is quite simple: you give me a cocomplete category (roughly, a category where we can glue stuff together along any maps we like), and you give me a second category (a category of shapes), with a functor that encodes how the objects in the second category can be viewed as objects of the first (the archetypal example is the functor telling us how to view abstract simplices as topological simplices). Then there is a canonical way to take a presheaf on the category of "shapes" and turn that thing into an object of the cocomplete category, and this functor is left adjoint to a nerve operation, which gives you an abstract object built out of the "shapes" that contains all of the information about how those shapes probe the object in the cocomplete category.

Right now I am working out an even more general framework, letting one start with arbitrary (locally-small) categories and an arbitrary functor, to yield a very powerful construction in topos theory. But if these ideas interest you, check out the nLab link provided above. Once my new idea is fleshed out I plan to make a post about it on my blog (https://psychicapparatuses.wordpress.com/).). Happy learning, friends!

>>8856811
I think monoids and categories are more fundamental. You can't develop the notion of algebras internal to a monoidal category if you take groups as fundemantal. If you start with monoids and categories, then you can always localize and take maximal invertible subobjects to get groups and groupoids. The reason the latter are important, particularly in geometry and topology, is because the space we work with are undirected and enjoy internal symmetry.

>>8851170
That's awesome dude! Pointset topology is very different than the flavour of topology that I do, so I cannot speak much about locales. They are most interesting to me as they fit in to the various periodic tables of higher categories; perhaps a homotopy theory of locales will tell us how to generalize to a homotopy theory of (n,r)-categories. Maybe we can finally figure out what the "right" definition is for an (infinity, infinity)-category.

>>8850354
Category theory is more of a structural meta-theory than a theory itself. We prove theorems for increasingly broad structures so that they apply to any suitable category theory; this is really what started driving the march into higher category theory.

>>8850325
We have* Atiyah* homotopy* (sorry, I'm under time pressure)

>>8850302
We quite a bit of work ahead of us to fully generalize and collate all of the results that relate geometry of spaces to their topology (we have Atiya-Singer, de Rham's theorem, and Grothendieck-Riemann-Roch relating algebraic data to topological data, et cetera), but it does seem that some of the most powerful theorems out there are either of the flavour relating these two sorts of information or theorems in model theory that prove powerful existence results based on pretty mild criteria (such as the compactness theorem and Loewenheim-Skolem). It would be quite lovely if we could find nice proofs of this latter breed in terms of topology... imagine a dictionary letting us translate between model theory and homotopty type theory that shows some of these finishing moves are in fact equivalent statements in different languages! The consequences would be quite far-reaching.

>>8848518
Hey Animenon!

Life is great. I recently celebrated my twenty-first birthday, and my research is going very well. I've been making some mad progress on tying F1 into linear algebra and homotopy theory; right now I am developing a general nerve-realization setup to show that homotopy theory is the "proper" notion of geometry locally modeled on F1 vector spaces, just as the theory of manifolds is locally modeled on real vector spaces. I'm hoping to get a better understanding of the situation over finite fields as well.

That localic homotopy theory sounds pretty interesting. I'm wondering if the adjunction relating Top to Locale is actually a Quillen adjunction relating these homotopy theories? Unfortunately I'm working a pretty long shift today, but I may look into that further tomorrow. Thanks for sharing! How is your thesis coming along?

>>8847910
Oh, sorry about that. A stack is just an infinity presheaf satisfying descent conditions, right? I've always just conflated stacks with algebraic stacks in my head, I guess. How do algebraic stacks differ in their construction? Thanks for pointing this out to me!

>>8847709
>>8847553
>>8847686
Haha, sorry for letting my autism spill out with so little restraint, but it's not every day that I get to share the beauty of homotopy with you anons!

(>>8847377)
My personal favorite combinatorial model for homotopy theory is the theory of symmetric simplicial sets, which are like standard simplicial sets without orderings on cells. These objects relate homotopy theory to linear algebra, because a symmetric simplicial set is to the field on one element as a finite-dimensional manifold is to the real numbers. From here one can find more ties back to algebraic geometry, differential geometry, and even knot theory.

Important algebraic facts can be derived using homotopy theory, such as the link between Frobenius' theorem and the Hopf fibrations.

Differential geometry can be described in terms of homotopy theory, a la synthetic differential geometry. Things such as de Rham's theorem are quite trivial to prove in this setting, as they use more simple constructions in geometric/classical homotopy theory and show that they are facts in all homotopy theories. We use modalities to relate certain theorems and to show that they are true in any infinity topos.

Homotopy theory is used in quantum field theory. Cobordism cohomology theories can be interpreted as placing limits on particle interactions; Witten et alii have used the theory of genera on cobordism cohomology rings to derive physical laws. Physicists also use vector bundles everywhere, and homotopy theory gives the tools for classifying such bundles. This leads to higher Chern-Weil theory and its relatives.

Homotopy Type Theory, which is the abstract language of any flavour of homotopy theory, has applications in computer science and logic, as well as philosophy. Voevodsky's univalence axiom has Leibniz's identity of indiscernibles as a corollary, for example. The homotopy theory of topological spaces is universal amongst homotopy theories.

Homotopy theory is used in number theory, where we want to find nice homotopical descriptions of higher stacks, which vastly generalize Grothendieck's scheme theory.

Just peruse the nLab, friend.

>>8847307
I think topology is coolest from a homotopical perspective. Homotopy theory distills topology from the (ugly, in my opinion) pointset stuff to the abstract concept of connectivity and interactions between paths of different dimension. Things start behaving very, very nicely after you mod everything out by coherent homotopy.

First off, the homotopy hypothesis: infinity groupoids are models for homotopy types (spaces up to homotopy equivalence). An infinity groupoid has 0-cells (points), 1-cells connecting them (paths), 2-cells between these (homotopies), and you can compose all of these cells to get new ones (for example, you can whisker a 2-cell and a 1-cell to get a new 2-cell, which is "degenerate" along the 1-cell). The fundamental operations in homotopy are suspensions, deloopings, and the smash product (after passage to pointed homotopy types). There is a process of stabilization, producing things called spectra. Spectra can be thought of as higher algebraic things, though: they are infinity modules over the sphere spectrum, which has the integers for all of its positive homotopy groups. The sphere spectrum is thus the Eilenberg-MacLane spectrum for the integers, and the statement that spectra are modules over this thing is essentially equivalent to Brown representability: spectra and generalized cohomology theories can be weakly identified (the caveat being that cohomology cannot detect some maps, called phantom maps).

This algebraic approach gets cooler, though: there is a notion of "good" cohomology theories, called complex orientable cohomology theories. There is a universal one, called MU, which is a spectrum representing complex cobordism cohomology theory. This is called the Thom spcetrum, and its cohomology ring calssifies formal group laws. This ties chromatic homotopy theory to algebraic geometry in some nifty ways.

Then you have a ton of methods for modelling homotopy theory, many of which have combinatorial flavours. (continuing)

Heyo /sci/, I'm tripping balls right now. I had to share some ideas I had today looking at the existence of adjoints to the canonical embedding of an enriching category into enriched categories of presheaves (you can take it all to be enriched over sets if you want).

The bare minimum of information needed to do enriched category theory is basically a monoidal category V, called a cosmos or something similar. The idea is that the object of morphisms from A to B in some V-category C is an object of V, and functors work by supplying morphisms in V between hom objects.

So, say you have V. Now, you take some V-category C*, and then you look at [C*,V] the "V-presheaves" on C. The enriched Yoneda lemma, just proven in full generality last year, says that C sits inside of [C*,V] fully and faithfully, even if all we have is a monoidal category to enrich over (not necessarily closed or symmetric!).

But notice that V sits inside of [C*,V] by sending objects to the constant functor on them. I was looking at homs into enriched constant presheafs, and I proved that natural transformations into the constant functor on v are just lifts along the forgetful functor V/v->V. Now, if you are familiar with the process of internalization in category theory, a lift along a forgetful functor encodes the structure that is forgotten onto the presheaf being lifted.

This is mostly useless until you realize that v could be some kind of moduli object. In this case, V/v is the category of structures that v modulates, and a lift along this is an internalization of this structure into another category. So, enriched presheaves constant on moduli objects are themselves global moduli objects!! This is profound to me.