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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• added to composition a new section with trivial remarks on composition in enriched category theory.

• created geometric fibre. Can someone lease check these algebraic geometry entries as that area is quite far from my safety zone! so I will get some things wrong.

• added to free module and to submodule a remark on the characterization of submodules of free modules.

• I finally started linear equation. But am too tired now to really do it justice…

• In stratified space, many of the references had page numbers given as if 123 { 234, rather than 123 - 234. This is probably a paste from somewhere else, but I was wondering how it happened so as to avoid it myself. I changed it. (Might it be a strange font?)

• I have touched quasi-isomorphism, expanded the Idea-section and polished the Definition-section, added References

• Urs had a framework at deduction and I put in something very brief. Also disambiguation at derivation.

• For some text I need to explain the relation between sequents in the syntax of dependent type theory and morphisms in their categorical semantics.

I wanted to explain this table:

$\,$ types terms
(∞,1)-topos theory $\;\;\;\;X \stackrel{\vdash \;\;\;\;E}{\to} \;\;\Type$ $\;\;\;\;X \stackrel{\vdash \;\;\;t}{\to} {}_X \;\;E$
homotopy type theory $x : X \vdash E(x) : Type$ $x : X \vdash t(x) : E(x)$

So I was looking for a place where to put it. This way I noticed that sequent used to redirect to sequent calculus. I think this doesn’t do justice to the notion and so I have

• split off a new entry sequent

leaving the whole entry in genuinely stubby state. But no harm done, I think, if we compare to the previous state of affairs.

• splitt off an entry over-(infinity,1)-topos with material that had been scattered elsewhere and needed to be collected in order to allow referencing it

• I have been adding various entries to various categories such as infinity groupoid was added to category:∞-groupoid, as it was not there! This is partially for my information as I have forgotten what entries there are on things of current interest to me, but it will explain why there seem to be a lot of entries changed by me but not in substance.

• When making inhabitant redirect to term a few minutes back I also found the entry term to be in an unfortunate state. I tried to improve it a bit by giving it more of an Idea section, and at least a vague indication of the formal definition.

• at implication there is currently the statement

$q \to r \vdash (p \to q) \to (q \to r)$,

That’s a typo, right?

• I hope to be adding bits and pieces to an article real coalgebra, which I’ve started. (In some sense it might fit better on my web, but for some reason I’m placing it on the main nLab.)

• I ended up spending some time with expanding extension of scalars. Towards the end I had more plans, but I’ll stop now, need to do something else.

• created four lemma (should still state the dual version, will do so later)

• I have added some links to preprint on the entry Lascar group. I do not understand the model theory, but its link with Galois theory may be of use to someone looking at model theory and type theory elsewhere on the Lab, so I hope it is useful.

• I took simple function out of measure space, putting there abstract definitions up through the integral on $L^1$.

• creatd connecting homomorphism with (just) the pedestrian description.

(Relation to snake lemma and more generally to fiber sequences not there yet…)

• As I said in another thread, I would like to see the $n$Lab entries related to universes be somehow better, more organized, more comprehensive.

In order to get a handle on it I decided, as so often, to tabulate what we have and what we should have, so I am creating:

universe - contents

• at inductive reasoning it says

Induction here is not to be confused with mathematical induction.

We should point out that, however, there is a close relation:

one can see this still in the German tem for, “induction over the natural numbers” which is not Induktion, but vollständige Induktion: meaning ” complete induction” !

I guess the reasoning is clear, mathematical induction (at least that over the natural numbers) is a special case of inductive reasoning, namely that where we can be sure that we are inducing from a complete set of instances of the general rule.

Does anyone feel like touching the entry accordingly to clarify this?

• turns out plenty of entries were asking for quotient group. I created something. But am running a bit out of steam for tonight.

• 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?

• 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.

• I ended up polishing type theory - contents (which is included as a floating table of contents in the relevant entries):

1. expanded and re-arranged the list under “syntax”, created stubs for the missing items definition and program

2. 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 $B \to A$ 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?

• I added a sentence to fundamental group which contains a link to an example for a fundamental group of an affine scheme.
• 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.