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I have touched cokernel, briefly adding some basics. More needs to be done here.
Using the LaTeX macro package TikZ, I’ve redrawn most of the SVGs on the knots and links pages. I hope that I haven’t trodden on any toes in so doing! I may have missed a few diagrams as well.
I’ve shifted the actual SVGs to pages of their own. This makes it easier to edit the pages with them on - TikZ’s SVG export isn’t as compact as the inbuilt SVG editor - and easier to include on other pages. For example, I can imagine that the trefoil knot is going to appear again and again!
(Incidentally, are the two trefoils distinct? If so, which have I drawn at trefoil knot - SVG).
I’ve named the pages with - SVG in their name, though for the moment I’ve also put in redirects to the name without the SVG. When actually including the diagram, one should always use the canonical name (ie with the - SVG) since it may be that we actually write a page about the trefoil knot one day. But I thought that for the moment, a nice aspect of hyperlinks is that if we mention the trefoil knot in a page then we can put in a link to an actual picture.
Diagrams done so far:
Pages with includes include: link, Reidemeister move, colorability, bridge number.
What would be fantastic here is if the “source” link took one to the actual LaTeX/TikZ source! I do intend to put that up on the nLab, but I need to clean it up a little as it depends on some customised style files that have a lot of crud in them.
I have been attacking some of the grey links in knot theory and the related pages. If someone has the time (and the patience) adding a few more links would be a good thing. I have added Crowell, Fox, Dehn, Alexander, Louis Kauffman, plus some non-people pages such as Alexander polynomial. That needs some diagrams if it is to do what it ’should’ and my svg skills are too slight to attempt that today. :-)
I finally gave the poor entry physics a bit of text.
Made some changes at logic and started inductive inference and George Polya.
There are still things to change at logic
As a discipline, logic is the study of methods of reasoning. While in the past (and often today in philosophical circles), this discipline was prescriptive (describing how one should reason), it is increasingly (and usually in mathematical circles) descriptive (describing how one does reason).
Could whoever wrote it explain what they meant? Seems odd to me.
Also I don’t think that category-theoretic logic should be there. Should it not appear in mathematical logic, or be a new page?
Expanded second-countable space and started first-countable space.
To Toby, mainly, of course also to anyone interested:
We have a page-“category” foundational axiom, but we have no entry of that name. We should start one! foundational axiom.
(We do have axiom of foundation. A bit of a different thing.)
I’ve been editing second order arithmetic (I usually write “second-order arithmetic”, with a hyphen). I would appreciate someone taking a look and making corrections if necessary. There are probably some hyperlinks which could be added.
Created κ-ary site.
I renamed familial regularity and exactness to k-ary exact category (with redirects from k-ary regular category), and updated some of the statements to match my exact completions paper a bit better.
I’m planning to create k-ary site as well, and add a statement of the main theorem of the paper somewhere on the nLab. But I’m undecided as to where that statement should go and how it should interact with the existing pages regular and exact completions and pretopos completion. Opinions are welcome.
I ended up polishing type theory - contents (which is included as a floating table of contents in the relevant entries):
expanded and re-arranged the list under “syntax”, created stubs for the missing items definition and program
expanded the (logic/type theory)-table to a (logic/category theory/type theory)-table and subsumed some of the items into it that were floating around elsewhere.
at axiom of choice into the section In dependent type theory I have moved the explicit statement taken from the entry of dependent type theory (see there for what I am talking about in the following).
One technical question: do we need the
: true
at the very end of the formal statement of the theorem?
One conceptual question: I feel inclined to add the following Remark to that, on how to think about the fact that the axiom of choice is always true in this sense in type theory. But please let me know what you think:
Heuristically, the reason that the axiom of choice is always true when formulated internally this way in dependent type theory is due to the fact that its assumption thereby is stated in constructive mathematics:
Stated in informal but internal logic, the axiom of choice says:
If is a map all whose fibers are inhabited, then there is a section.
But now if we interpret the assumption clause
a map all whose fibers are inhabited
constructively, we have to provide a constructive proof that indeed the fibers are inhabited. But such a constructive proof is a choice of section.
So constructively and internally the axiom is reduced to “If there is a section then there is a section.” And so indeed this is always true.
Would you agree that this captures the state of affairs?
stub for pseudocircle (a finite topological space)
split off total complex from double complex. Let the Definition-section stubby, as it was, but added a brief remark on exactness and on relation to total simplicial sets, under Dold-Kan and Eilenberg-Zilber. More to be done.
Stubs for structural rule, weakening rule, contraction rule, and exchange rule.
Speaking of philosophy, I added a little to dichotomy between nice objects and nice categories.
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.