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- Discussion Type
- discussion topick-ary exact category
- Category Latest Changes
- Started by Mike Shulman
- Comments 2
- Last comment by Urs
- Last Active Sep 5th 2012

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.

- Discussion Type
- discussion topicChan-Paton bundle
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 5th 2012

expanded at

*Chan-Paton bundle*the Idea-section and added two pointers to lecture notes. Also expanded at*Freed-Witten anomaly*a little.

- Discussion Type
- discussion topictype theory - contents
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 4th 2012

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.

- Discussion Type
- discussion topicaxiom of choice in type theory
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Sep 4th 2012

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?

- Discussion Type
- discussion topicpseudocircle
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Sep 4th 2012

stub for

*pseudocircle*(a finite topological space)

- Discussion Type
- discussion topicfundamental group
- Category Latest Changes
- Started by Stephan A Spahn
- Comments 8
- Last comment by Tim_Porter
- Last Active Sep 3rd 2012

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

- Discussion Type
- discussion topictotal complex
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Sep 3rd 2012

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.

- Discussion Type
- discussion topicinterval object in chain complexes
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Tim_Porter
- Last Active Sep 3rd 2012

- Discussion Type
- discussion topicStructural rules.
- Category Latest Changes
- Started by TobyBartels
- Comments 9
- Last comment by Mike Shulman
- Last Active Sep 3rd 2012

Stubs for structural rule, weakening rule, contraction rule, and exchange rule.

- Discussion Type
- discussion topicdichotomy
- Category Latest Changes
- Started by Todd_Trimble
- Comments 2
- Last comment by Urs
- Last Active Sep 2nd 2012

Speaking of philosophy, I added a little to dichotomy between nice objects and nice categories.

- Discussion Type
- discussion topicAn Introduction to Homological Algebra
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Sep 2nd 2012

- Discussion Type
- discussion topictruncation of a chain complex
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 2nd 2012

it seems that a few days back I had created a note

*truncation of a chain complex*

- Discussion Type
- discussion topicsuspension of a chain complex
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 2nd 2012

I felt like having a simple stand-alone entry

*suspension of a chain complex*, so I created one.

- Discussion Type
- discussion topicDedekind completion of quasiorders
- Category Latest Changes
- Started by TobyBartels
- Comments 2
- Last comment by TobyBartels
- Last Active Sep 2nd 2012

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.

- Discussion Type
- discussion topicBinary functions
- Category Latest Changes
- Started by TobyBartels
- Comments 3
- Last comment by TobyBartels
- Last Active Sep 2nd 2012

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.

- Discussion Type
- discussion topicReal number object
- Category Latest Changes
- Started by SridharRamesh
- Comments 6
- Last comment by TobyBartels
- Last Active Sep 2nd 2012

- Am I correct in supposing that the first definition of Dedekind cuts at real numbers object is missing an openness condition (as given in the later, power object-using definition on the same page)?

- Discussion Type
- discussion topicchain map
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 1st 2012

some trivial additions to

*chain map*.

- Discussion Type
- discussion topiccoefficient
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Sep 1st 2012

created an entry

*coefficient*and linked it withas well as with

- Discussion Type
- discussion topicMod
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 31st 2012

at

*Mod*I started a new subsection*RMod is an abelian category*, spelling out some of the details of this statement.

- Discussion Type
- discussion topicbilinear function
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 31st 2012

*bilinear map*(redirecting*multilinear map*)

- Discussion Type
- discussion topiccommutator subgroup
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 31st 2012

- Discussion Type
- discussion topicgroup commutator
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 31st 2012

- Discussion Type
- discussion topicprojective presentation
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 30th 2012

felt the need to have a quick entry

*projective presentation*.

- Discussion Type
- discussion topicgroup theory - contents
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Aug 30th 2012

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)

- Discussion Type
- discussion topicUniversal coefficient theorems
- Category Latest Changes
- Started by Andrew Stacey
- Comments 22
- Last comment by Andrew Stacey
- Last Active Aug 30th 2012

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).

- Discussion Type
- discussion topicSequence spaces
- Category Latest Changes
- Started by TobyBartels
- Comments 9
- Last comment by Tom Leinster
- Last Active Aug 30th 2012

Another new article: sequence space. I await the inevitable report that this term is also used for other things.

- Discussion Type
- discussion topicBanach coalgebras
- Category Latest Changes
- Started by TobyBartels
- Comments 26
- Last comment by TobyBartels
- Last Active Aug 28th 2012

New page: Banach coalgebra.

Hopefully you all know that $l^1$ is a Banach

*algebra*under*convolution*, but did you know that $l^\infty$ is a Banach*coalgebra*under*nvolution*? (Actually, they are both Banach bialgebras!)

- Discussion Type
- discussion topiccompactly generated triangulated category
- Category Latest Changes
- Started by Karol Szumiło
- Comments 4
- Last comment by Mike Shulman
- Last Active Aug 28th 2012

- I have created compactly generated triangulated category and most probably violated many of your formatting and typesetting conventions...

- Discussion Type
- discussion topicelement in an abelian category
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by IngoBlechschmidt
- Last Active Aug 28th 2012

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.

- Discussion Type
- discussion topicchain homology
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 27th 2012

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*.

- Discussion Type
- discussion topicThompson's group F
- Category Latest Changes
- Started by Mike Shulman
- Comments 3
- Last comment by Mike Shulman
- Last Active Aug 27th 2012

stub for Thomspon’s group F.

- Discussion Type
- discussion topicchains and cochains - table
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 26th 2012

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 Type
- discussion topicunimodular lattice
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 25th 2012

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.

- Discussion Type
- discussion topicHeisenberg Lie n-algebra
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Aug 25th 2012

I am splitting off

*Heisenberg Lie n-algebra*from*Heisenberg Lie algebra*.

- Discussion Type
- discussion topicrelations - contents
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 24th 2012

I have created a table

*relations - contents*and added it as a floatic TOC to the relevant entries.

- Discussion Type
- discussion topicsnake lemma
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Aug 24th 2012

statement of the

*snake lemma*

- Discussion Type
- discussion topiccohomology theory
- Category Latest Changes
- Started by Mike Shulman
- Comments 25
- Last comment by TobyBartels
- Last Active Aug 24th 2012

Is there a reason for the page cohomology theory to exist independently, rather than as a redirect to generalized (Eilenberg-Steenrod) cohomology?

- Discussion Type
- discussion topicdelta-functor
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 24th 2012

Created

*delta-functor*.(also touched

*Tohoku*, adding hyperlinks and “the”-s)

- Discussion Type
- discussion topicbornological set
- Category Latest Changes
- Started by Todd_Trimble
- Comments 3
- Last comment by TobyBartels
- Last Active Aug 23rd 2012

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.

- Discussion Type
- discussion topic(infinity,n)-category with adjoints
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 23rd 2012

I have started an entry

*(∞,n)-category with adjoints*, prompted by wanting to record these slides:- Nick Rozenblyum,
*Manifolds, Higher Categories and Topological Field Theories*, talk Northwestern University (2012) (pdf 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.- Nick Rozenblyum,

- Discussion Type
- discussion topicDual spaces
- Category Latest Changes
- Started by TobyBartels
- Comments 11
- Last comment by TobyBartels
- Last Active Aug 22nd 2012

New page dual space with redirect from dual basis.

- Discussion Type
- discussion topic2-plectic geometry
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 22nd 2012

I am splitting off

*2-plectic geometry*from*n-plectic geometry*.

- Discussion Type
- discussion topicadditive and abelian categories - contents
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Aug 20th 2012

I created another floating TOC

*additive and abelian categories - contents*and added it to the relevant entries

- Discussion Type
- discussion topicduality - contents
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 20th 2012

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.

- Discussion Type
- discussion topicproduct-preserving functor
- Category Latest Changes
- Started by Todd_Trimble
- Comments 10
- Last comment by DavidRoberts
- Last Active Aug 19th 2012

Created product-preserving functor.

- Discussion Type
- discussion topicDirects sums of Banach spaces
- Category Latest Changes
- Started by TobyBartels
- Comments 12
- Last comment by TobyBartels
- Last Active Aug 18th 2012

New article: direct sum of Banach spaces. These come in even more variety than I originally thought!

- Discussion Type
- discussion topicShort linear maps
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Aug 16th 2012

I got tired of writing ‘short linear map’, so now we have short linear map.

- Discussion Type
- discussion topicfunction application
- Category Latest Changes
- Started by Mike Shulman
- Comments 10
- Last comment by Todd_Trimble
- Last Active Aug 16th 2012

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.

- Discussion Type
- discussion topicdual object in a closed monoidal category
- Category Latest Changes
- Started by Todd_Trimble
- Comments 16
- Last comment by TobyBartels
- Last Active Aug 14th 2012

- Discussion Type
- discussion topicZ_infinity-modules
- Category Latest Changes
- Started by Zhen Lin
- Comments 4
- Last comment by Urs
- Last Active Aug 13th 2012

I started the article Z-infinity-module. Hopefully someone here can say something more interesting about them!

- Discussion Type
- discussion topicadjoint equivalence
- Category Latest Changes
- Started by Mike Shulman
- Comments 6
- Last comment by Mike Shulman
- Last Active Aug 13th 2012

I added some remarks to adjoint equivalence about improving equivalences to adjoint ones, with links to&from equivalence and equivalence in a quasicategory.

- Discussion Type
- discussion topicRiesz representation theorems
- Category Latest Changes
- Started by TobyBartels
- Comments 2
- Last comment by Urs
- Last Active Aug 12th 2012

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.

- Discussion Type
- discussion topicSpam and other types of 'interference'
- Category Latest Changes
- Started by Tim_Porter
- Comments 8
- Last comment by Tim_Porter
- Last Active Aug 8th 2012

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.

- Discussion Type
- discussion topicProblems with all pages on dependent type theory
- Category Latest Changes
- Started by Harry Gindi
- Comments 10
- Last comment by Mike Shulman
- Last Active Aug 7th 2012

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).

- Discussion Type
- discussion topicPreduals
- Category Latest Changes
- Started by TobyBartels
- Comments 9
- Last comment by TobyBartels
- Last Active Aug 6th 2012

Not much here, but: predual.

- Discussion Type
- discussion topicInclusion-exclusion
- Category Latest Changes
- Started by TobyBartels
- Comments 5
- Last comment by Mike Shulman
- Last Active Aug 6th 2012

New page: inclusion-exclusion principle

- Discussion Type
- discussion topicDuals
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Aug 6th 2012

- Discussion Type
- discussion topicmonomorphism in an (oo,1)-category
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Mike Shulman
- Last Active Aug 5th 2012

- Discussion Type
- discussion topicPositive cones.
- Category Latest Changes
- Started by TobyBartels
- Comments 26
- Last comment by TobyBartels
- Last Active Aug 5th 2012

New page: positive cone, including the extended positive cone of a

*W*^{*}-module.

- Discussion Type
- discussion topicLambert W function
- Category Latest Changes
- Started by Todd_Trimble
- Comments 3
- Last comment by TobyBartels
- Last Active Aug 2nd 2012

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 $n$-element set (which is $n^{n-2}$).