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it seems that a few days back I had created a note truncation of a chain complex
I felt like having a simple stand-alone entry suspension of a chain complex, so I created one.
Dedekind completions of quasiorders (not just linear orders) may now be found at Dedekind completion. Example: the lower Dedekind completion of the quasiorder of continuous functions is the quasiorder of lower semicontinuous functions.
I put a bunch of stuff there that might be of interest to the logicians and foundationalists among us, although it’s still pretty trivial.
some trivial additions to chain map.
created an entry coefficient and linked it with
as well as with
at Mod I started a new subsection RMod is an abelian category, spelling out some of the details of this statement.
bilinear map (redirecting multilinear map)
felt the need to have a quick entry projective presentation.
I have started a floating table of contents group theory - contents, and started adding it to some relevant entries
(the toc is neither meant to already be complete nor to be optimally organized, please expand and polish as you see the need)
I was reading Adams’ lectures on generalised cohomology theories and added some stuff from there to universal coefficient theorem about the more general case (including the Kunneth theorem).
Another new article: sequence space. I await the inevitable report that this term is also used for other things.
New page: Banach coalgebra.
Hopefully you all know that is a Banach algebra under convolution, but did you know that is a Banach coalgebra under nvolution? (Actually, they are both Banach bialgebras!)
I added an Idea-section to element in an abelian category and added a reference by George Bergman.
This links back to the new Idea section at abelian categories - embedding theorems. Check if you agree with the wording.
I have added to the entry chain homology and cohomology an actual Idea-section and an actual definition. The material that used to be there I have moved into a section Chain homology – In the context of homotopy theory.
stub for Thomspon’s group F.
created a little table: chains and cochains - table and included it into the relevant entries (some of which still deserve to be edited quite a bit).
discussion over in the thread on modular lattice keeps making me create stub likes unimodular lattice and Leech lattice. No genuine content there yet, though. It just seems necessary to have these entries at all.
I am splitting off Heisenberg Lie n-algebra from Heisenberg Lie algebra .
I have created a table relations - contents and added it as a floatic TOC to the relevant entries.
statement of the snake lemma
Is there a reason for the page cohomology theory to exist independently, rather than as a redirect to generalized (Eilenberg-Steenrod) cohomology?
Created delta-functor.
(also touched Tohoku, adding hyperlinks and “the”-s)
I added a few observations under a new section “Results” at bornological set. Bornological sets form a quasitopos. I don’t have a good reference for the theorem of Schanuel.
Related is an observation which hadn’t occurred to me before: the category of sets equipped with a reflexive symmetric relation is a quasitopos. I’d like to return to this sometime in the context of thinking about morphisms of (simple) graphs.
I have started an entry (∞,n)-category with adjoints, prompted by wanting to record these slides:
If anyone can say more about the result indicated there, I’d be most grateful for a comment.
Also, I seem to hear that at Luminy 2012 there was some extra talk, not appearing on the schedule (maybe by Nick Rozenblyum, but I am not sure) on something related to geometric quantization. If anyone has anything on that, I’d also be most grateful.
New page dual space with redirect from dual basis.
I am splitting off 2-plectic geometry from n-plectic geometry .
I am starting a table of contents, to be included as a floating TOC for entries related to duality:
But it’s a bit rough for the time being. I haven’t decided yet how to best organize it and I am probably still lacking many items that deserve to be included. To be developed. All input is welcome.
Created product-preserving functor.
New article: direct sum of Banach spaces. These come in even more variety than I originally thought!
I got tired of writing ‘short linear map’, so now we have short linear map.
I created function application, so as to be able to link to it from fixed-point combinator. While adding links, I was motivated to expand a bit on function.
I started the article Z-infinity-module. Hopefully someone here can say something more interesting about them!
I added some remarks to adjoint equivalence about improving equivalences to adjoint ones, with links to&from equivalence and equivalence in a quasicategory.
I'm putting all the big duality theorems from measure theory at Riesz representation theorem. Only a couple are filled in so far, but I'm out of time for today.
There have been several entries with no math content recently. What is the procedure? I am meaning: Search results for definion and The Enemy of my Enemy is not my Friend.
Heya. I haven’t actually made the necessary changes, but the various pages on dependent type theory make the statement that every DTT or MLTT is the internal logic of an LCCC and every LCCC is the categorical semantics of some DTT/MLTT. However, this is extremely confusing (it took me 2 or 3 hours to find a page where it was made completely clear), since it makes explicit use of super-strong extensionality (I think this is called beta-translation), that is to say, it is a theorem about extensional DTTs/MLTTs.
It’s not even totally clear to me that every intensional type theory actually has an (∞,1)-categorical semantics without the consideration of the univalence axiom. I would make this clearer, but I am really out of my depth with type theories, so I’m just alerting you to the fact that this is stated confusingly almost everywhere (the only place where it’s clear is in the page on identity types).
Not much here, but: predual.
New page: inclusion-exclusion principle
New page: positive cone, including the extended positive cone of a W*-module.
Wrote Lambert W function. It was an excuse to record Joyal’s proof of Cayley’s theorem on the number of tree structures one can put on an -element set (which is ).
I created a stub on excision, but this is just a link to the Wikipedia page for the moment.
Concrete, abstract: group actions, groups; concrete categories, categories; Cartesian spaces, vector spaces; von Neumann algebras, -alebras; material sets, structural sets; etc. At concrete structure.
I wrote an overview over some constructions on- and examples of group schemes.
as some of you will have seen, I had spent part of the last week with attending talks at String-Math 2012 and posting some notes about these, to the Café (here). For many of these notes I added material to existing Lab entries (mostly just references) or created Lab entries (mostly just stubs).
But since at the same time I was also finalizing the writup of an article as well as doing yet some other things, the whole undertaking was a bit time-pressured. As a result, I decided it would be too much to announce every single Lab edit that I did here on the Forum.
So I ask you for understaning that hereby I just collectively announce these edits here: those who care should please scan through the list of blue links here and see if they spot pointers to Lab entries where they would like to check out the recent edits.
I think I can guarantee, though, that in all cases I did edits that should be entirely uncontroversial, their main defect being that in many cases they leave one wish for more exhaustive discussion.
added some references by Catahrina Stroppel at the end of categorification in representation theory
(also added the words “representation theory” to the entry itself :-)
I've been meaning to write this for a while. Now I need to look at Bourbaki this weekend to explain their approach.
New page: Radon–Nikodym derivative.
I moved some material from state to create pure state (redirect from mixed state).
I have created a stub quantum affine algebra as a means to collect some references, alluded to here.
If there is any expert on the matter around, he or she should please feel invited to add an illuminating Idea-section to the entry.
I created types and calculus and seven trees in one. Both entries as yet contain just references.
It would be nice to have more articles expanding on the reltion of calculus and (higher) category theory /type theory.