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- Discussion Type
- discussion topicbraided 3-group
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jul 21st 2021

had created

*braided 3-group*a good while back. Now I have added the example of Brauer/Picard/Unit-3-groups and cross-linked with*Brauer group*.

- Discussion Type
- discussion topicfusion category
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jul 21st 2021

briefly added something to fusion category. See also this blog comment.

- Discussion Type
- discussion topicsymmetric 2-group
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jul 21st 2021

- Discussion Type
- discussion topicEvan Jenkins
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 21st 2021

removed the line

I’m a graduate student at the University of Chicago.

since it seems to be outdated, and instead added a pointer to the personal webpage.

Also added pointer to:

- Evan Jenkins,
*Extensions of groups by braided 2-groups*(arXiv:1106.0772)

Has this ever been “published”, in any form?

- Evan Jenkins,

- Discussion Type
- discussion topiccorecursion
- Category Latest Changes
- Started by David_Corfield
- Comments 27
- Last comment by TobyBartels
- Last Active Jul 21st 2021

I thought it better to use $pred(n)$ rather than $n -1$ in the addition, since it’s supposed to apply to $\infty$.

- Discussion Type
- discussion topicmodel structure on chain complexes
- Category Latest Changes
- Started by Tim_Porter
- Comments 8
- Last comment by Urs
- Last Active Jul 21st 2021

At model structure on chain complexes, an ’anonymous editor’ suggests that a line saying ’blah blah’ should be completed to something more illuminating!

- Discussion Type
- discussion topicabelian category
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Jul 21st 2021

I am hereby moving an old query-box discussion from

*abelian category*to here. I suggest that to the extent this reached a conclusion, that conclusion should be moved to the Properties-section of the entry

[begin forwarded discussion]

The following discussion is about whether a pre-abelian category in which (epi,mono) is a factorization system is necessarily abelian.

+–{: .query} Mike: In Categories Work, and on Wikipedia, an abelian category is defined to be (in the terms above) a pre-abelian category such that every monic is a kernel and every epi is a cokernel. This implies that (epi, mono) is an orthogonal factorization system, but I don’t see why the converse should hold, as this seems to assert.

Zoran Skoda It is very late night here in Bonn, so check on my reasoning, but I think that the answer is simple. Let $f: A\to B$. The canonical map $\coker(\ker f)\to \ker (\coker f)$ exists as long as we have additive category admitting kernels and cokernels. The arrow from A to coker (ker f) is epi as every cokernel arrow, and the arrow of $\ker(\coker f) \to B$ is mono. Now canonical arrow in between the two is automatically both mono and epi. For all that reasoning I did not yet assume the axiom on uniquely unique factorization. Now assume it and you get that the canonical map must be isomorphism because it is the unique iso between the two decompositions of $f$: one in which you take epi followed by (the composition of) two monics and another in which you have (the composition of) two epis followed by one monic. Right ?

Now do this for $f$ a monic and you get a decomposition into iso iso kernel and for $f$ an epi and you get the cokernel iso iso as required.

Mike: Why is the canonical comparison map mono and epi? It’s late for me too right now, but I think that maybe a counterexample is the “multiplication by 2” map $\mathbb{Z}\to \mathbb{Z}$ in the category of torsion-free abelian groups.

However, if you assume explicitly that that comparison map is always an isomorphism, then I believe it for the reasons that you gave.

Zoran Skoda I do not see this as a counterexample, as this is not a pre-abelian category, you do not have cokernels in this category ? In a pre-abelian category always the canonical map from coker ker to ker coker has its own kernel 0 and cokernel 0.

Mike: Torsion-free abelian groups are reflective in abelian groups, and therefore cocomplete. In particular, they have cokernels, although those cokernels are not computed as in Ab. In particular, the cokernel of $2:\mathbb{Z}\to\mathbb{Z}$ is 0.

Zoran Skoda Yes, I was thinking of this reflection argument (equivalence of torsion and localization argument), that is why I put question mark above. Now I tried to prove the assertion that in preabelian cat the canonical map has kernel 0 and cokernel 0 and I can’t for more than an hour. But that would mean that for example Gelfand-Manin book is wrong – it has the discussion on A4 axiom and it says exactly this. Popescu makes an example of preabelian category where canonical map is not iso, but emphasises in his example that it is bimorphism. On the other hand, later, he says that preabelian category is abelian iff it is balanced and the canonical map is bimorphism, hence he requires it explicitly. Let me think more…

Zoran Skoda I have rewritten in minimalistic way, leaving just what I can prove, and assuming that you are right and Gelfand-Manin book has one wrong statement (that the canonical map in preabelian category is mono and epi). But let us leave the discussion here for some time, maybe we can improve the question of the difference between preabelian with factorization and abelian.

Mike: I refactored the page to make clear what we know and what we don’t, and include some examples. Maybe someone will come along and give us a counterexample or a proof. I wonder what the epimorphisms are in the category of torsion-free abelian groups, and in particular whether it is balanced (since if so, it would be a counterexample).

Mike: Okay, it’s obvious: the epimorphisms in $tfAb$ are the maps whose cokernel (in $Ab$) is torsion. Thus $2:\mathbb{Z}\to\mathbb{Z}$ is monic and epic, so $tfAb$ is not balanced. And since $2:\mathbb{Z}\to\mathbb{Z}$ is its own canonical map, that canonical map

*is*monic and epic in $tfAb$, so this isn’t a counterexample.*Zoran*: http://www.uni-trier.de/fileadmin/fb4/INF/TechReports/semi-abelian_categories.pdf says at one place that Palamodov’s version of semi-abelian category is preabelian + canonical morphism is epi and mono. =–[end forwarded discussion]

- Discussion Type
- discussion topicdeterministic random variable
- Category Latest Changes
- Started by nLab edit announcer
- Comments 10
- Last comment by Dmitri Pavlov
- Last Active Jul 21st 2021

- Discussion Type
- discussion topicreflective subcategory
- Category Latest Changes
- Started by Urs
- Comments 22
- Last comment by varkor
- Last Active Jul 20th 2021

edited reflective subcategory and expanded a bit the beginning

- Discussion Type
- discussion topicbornological topos
- Category Latest Changes
- Started by David_Corfield
- Comments 1
- Last comment by David_Corfield
- Last Active Jul 20th 2021

Added reference

- F. William Lawvere, Section 2 of
*Toposes generated by codiscrete objects in combinatorial topology and functional analysis*, Reprints in Theory and Applications of Categories, No. 27 (2021) pp. 1-11, pdf.

- F. William Lawvere, Section 2 of

- Discussion Type
- discussion topicWilliam Lawvere
- Category Latest Changes
- Started by Urs
- Comments 42
- Last comment by David_Corfield
- Last Active Jul 20th 2021

In the category:people-entry “William Lawvere” I have created a subsection “Motivation from foundations of physics” where I want to collect pointers to where and how Lawvere was/is motivated from finding foundations for (classical continuum) physics.

Explicit evidence for this that I am aware of includes notably the texts

*Toposes of laws of motion*and the introduction to the book*Categories in Continuum Physics*.The Wikipedia entry has this about motivation from physics:

Lawvere studied continuum mechanics as an undergraduate with Clifford Truesdell. He learned of category theory $[...]$ found it a promising framework for simple rigorous axioms for the physical ideas of Truesdell and Walter Noll. $[...]$ meeting on “Categories in Continuum Physics” in 1982. Clifford Truesdell participated in that meeting, as did several other researchers in the rational foundations of continuum physics and in the synthetic differential geometry which had evolved from the spatial part of Lawvere’s categorical dynamics program). Lawvere continues to work on his 50-year quest for a rigorous flexible base for physical ideas, free of unnecessary analytic complications.

Question: Can anyone point me to more on this early phase of the story (graduate student is supposed to start to look into continuum mechanics, starts to wonder “What is a vector field, really?, what a differential equation?” and ends up revolutionizing the foundations of differential calculus)?

- Discussion Type
- discussion topicaxiom of replacement
- Category Latest Changes
- Started by TobyBartels
- Comments 2
- Last comment by Mike Shulman
- Last Active Jul 20th 2021

- Discussion Type
- discussion topicIntroduction to Homotopy Theory
- Category Latest Changes
- Started by Oscar_Cunningham
- Comments 4
- Last comment by nLab edit announcer
- Last Active Jul 20th 2021

- Discussion Type
- discussion topicJan Spalinski
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 20th 2021

brief

`category:people`

-entry for hyperlinking authors at*Homotopy theories and model categories*

- Discussion Type
- discussion topicHomotopy theories and model categories
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 20th 2021

- Discussion Type
- discussion topicalgebraic set theory
- Category Latest Changes
- Started by Keith Harbaugh
- Comments 1
- Last comment by Keith Harbaugh
- Last Active Jul 20th 2021

- Discussion Type
- discussion topicopposite adjunction
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jul 20th 2021

- Discussion Type
- discussion topicKönig's theorem
- Category Latest Changes
- Started by Richard Williamson
- Comments 4
- Last comment by Jem Lord
- Last Active Jul 20th 2021

- Discussion Type
- discussion topicmodel structure on pointed objects
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jul 20th 2021

am giving this its own little entry, for ease of hyperlinking (currently most of the paragraphs here are copied over from the corresponding section at

*category of pointed objects*)

- Discussion Type
- discussion topicslice model structure
- Category Latest Changes
- Started by Urs
- Comments 28
- Last comment by Urs
- Last Active Jul 20th 2021

- Discussion Type
- discussion topicopposite model structure
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jul 20th 2021

added pointer to

- Philip Hirschhorn, Proposition 7.1.10 of:
*Model Categories and Their Localizations*, AMS Math. Survey and Monographs Vol 99 (2002) (ISBN:978-0-8218-4917-0, pdf toc, pdf)

- Philip Hirschhorn, Proposition 7.1.10 of:

- Discussion Type
- discussion topiclaw of a random variable
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 20th 2021

A bare minimum – for the moment just to make requested links work at

*Dirac measure*and at*deterministic random variable*

- Discussion Type
- discussion topicDirac measure
- Category Latest Changes
- Started by PaoloPerrone
- Comments 3
- Last comment by Urs
- Last Active Jul 20th 2021

- Discussion Type
- discussion topictoo simple to be simple
- Category Latest Changes
- Started by Oscar_Cunningham
- Comments 39
- Last comment by Hurkyl
- Last Active Jul 20th 2021

(Hi, I’m new)

I added some examples relating too simple to be simple to the idea of unbiased definitions. The point is that we often define things to be simple whenever they are not a non-trivial (co)product of two objects, and we can extend this definition to cover the “to simple to be simple case” by removing the word “two”. The trivial object is often the

*empty*(co)product. If we had been using an unbiased definition we would have automatically covered this case from the beginning.I also noticed that the page about the empty space referred to the naive definition of connectedness as being

“a space is connected if it cannot be partitioned into disjoint nonempty open subsets”

but this misses out the word “two” and so is accidentally giving the sophisticated definition! I’ve now corrected it to make it wrong (as it were).

- Discussion Type
- discussion topicMarcus Spradlin
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 19th 2021

brief

`category:people`

-entry for hyperlinking references at*twistor string theory*

- Discussion Type
- discussion topictwistor string theory
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jul 19th 2021

added pointer to:

- Nathan Berkovits,
*An Alternative String Theory in Twistor Space for $N = 4$ Super-Yang-Mills*, Phys. Rev. Lett. 93 (2004) 011601 (arXiv:hep-th/0402045)

- Nathan Berkovits,

- Discussion Type
- discussion topicVelayudhan Parameswaran Nair
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 19th 2021

brief

`category:people`

-entry for hyperlinking references at*twistor string theory*

- Discussion Type
- discussion topicDavid Skinner
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 19th 2021

brief

`category:people`

-entry for hyperlinking references at*twistor string theory*

- Discussion Type
- discussion topicAnastasia Volovich
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 19th 2021

brief

`category:people`

-entry for hyperlinking references at*twistor string theory*

- Discussion Type
- discussion topicevent
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 19th 2021

- Discussion Type
- discussion topicJohn Fogarty
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 19th 2021

brief

`category:people`

-entry for hyperlinking references at*geometric invariant theory*

- Discussion Type
- discussion topicFrances Kirwan
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 19th 2021

brief

`category:people`

-entry for hyperlinking references at*geometric invariant theory*and*homotopy of rational maps*

- Discussion Type
- discussion topicgeometric invariant theory
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 19th 2021

added author- and source-links for

- David Mumford, John Fogarty, Frances Clare Kirwan,
*Geometric invariant theory*, Ergebnisse der Mathematik und ihrer Grenzgebiete (2)**34**, Springer-Verlag (1965) (ISBN:978-3-540-56963-3, pdf)

- David Mumford, John Fogarty, Frances Clare Kirwan,

- Discussion Type
- discussion topicR. J. Milgram > history
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jul 19th 2021

brief

`category:people`

-entry for hyperlinking references at*homotopy of rational maps*

- Discussion Type
- discussion topicC. P. Boyer > history
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jul 19th 2021

brief

`category:people`

-entry for hyperlinking references at*homotopy of rational maps*

- Discussion Type
- discussion topicJ. C. Hurtubise
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 19th 2021

brief

`category:people`

-entry for hyerlinking references at*homotopy of rational maps*

- Discussion Type
- discussion topicB. M. Mann
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 19th 2021

brief

`category:people`

-entry for hyperlinking references at*homotopy of rational maps*

- Discussion Type
- discussion topicmoduli space of monopoles
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active Jul 19th 2021

starting something, but nothing here yet. For the moment this is just a home for

- Michael Atiyah, Nigel Hitchin,
*The geometry and dynamics of magnetic monopoles*M. B. Porter Lectures. Princeton University Press, Princeton, NJ, 1988 (jstor:j.ctt7zv206)

- Michael Atiyah, Nigel Hitchin,

- Discussion Type
- discussion topicJoachim Wehler
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 19th 2021

brief

`category:people`

-entry for hyperlinking references at*Riemann surface*and*Stein manifold*

- Discussion Type
- discussion topicHeinrich Behnke
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 19th 2021

brief

`category:people`

-entry for hyperlinking references at*Stein manifold*

- Discussion Type
- discussion topicStein manifold
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active Jul 19th 2021

have added a tad more content to

*Stein manifold*and cross-linked a bit more

- Discussion Type
- discussion topicAntonio Alarcón
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 19th 2021

brief

`category:people`

-entry for hyperlinking references at*minimal surface*and at*Oka principle*

- Discussion Type
- discussion topicminimal surface
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 19th 2021

- Discussion Type
- discussion topicOka principle
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Jul 19th 2021

added publication data to:

- Finnur Lárusson,
*Model structures and the Oka principle*, Journal of Pure and Applied Algebra Volume 192, Issues 1–3, 1 September 2004, Pages 203-223 (math.CV/0303355, doi:10.1016/j.jpaa.2004.02.005)

- Finnur Lárusson,

- Discussion Type
- discussion topicKantaro Ohmori
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by David_Corfield
- Last Active Jul 19th 2021

- Discussion Type
- discussion topicMichele Del Zotto
- Category Latest Changes
- Started by David_Corfield
- Comments 1
- Last comment by David_Corfield
- Last Active Jul 19th 2021

- Discussion Type
- discussion topicactegory
- Category Latest Changes
- Started by maxsnew
- Comments 9
- Last comment by mattecapu
- Last Active Jul 19th 2021

- Discussion Type
- discussion topicYuta Kusakabe
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 19th 2021

brief

`category:people`

-entry for hyperlinking references at*Oka manifold*

- Discussion Type
- discussion topicOka manifold
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Jul 19th 2021

I have added hyperlinks to

Franc Forstnerič,

*Oka manifolds*, Comptes Rendus Mathematique, Acad. Sci. Paris 347 (2009), 1017–20 (arXiv:0906.2421, doi:10.1016/j.crma.2009.07.005)Franc Forstnerič,

*Oka maps*, Comptes Rendus Mathematique, Acad. Sci. Paris, Ser. I 348 (2010) 145-148 (arxiv/0911.3439, doi:10.1016/j.crma.2009.12.004)

and added pointer to

Franc Forstnerič, Finnur Lárusson,

*Survey of Oka theory*, New York J. Math., 17a (2011), 1-28 (arXiv:1009.1934, eudml:232963)Franc Forstnerič (appendix by Finnur Lárusson),

*Oka manifolds: From Oka to Stein and back*, Annales de la Faculté des sciences de Toulouse, Mathématiques, Série 6, Tome 22 (2013) no. 4, pp. 747-809 (numdam:AFST_2013_6_22_4_747_0)

From the latter I quoted the fact (now this Prop.) that complex projective space is Oka.

On another note, I am suspecting that there is little material that wouldn’t want to go both to

*Oka manifold*and*Oka principle*and am thinking the two might need to be merged.

- Discussion Type
- discussion topicgeometry of physics -- categories and toposes
- Category Latest Changes
- Started by Urs
- Comments 21
- Last comment by Urs
- Last Active Jul 19th 2021

I’ll be preparing here notes for my lectures

*Categories and Toposes (schreiber)*, later this month.

- Discussion Type
- discussion topicGaeta topos
- Category Latest Changes
- Started by David_Corfield
- Comments 3
- Last comment by David_Corfield
- Last Active Jul 18th 2021

Added bornological topos as an example.

- Discussion Type
- discussion topicCohesive Toposes -- Combinatorial and Infinitesimal Cases
- Category Latest Changes
- Started by David_Corfield
- Comments 1
- Last comment by David_Corfield
- Last Active Jul 18th 2021

- Discussion Type
- discussion topicfree cocompletion
- Category Latest Changes
- Started by Urs
- Comments 30
- Last comment by Joshua Meyers
- Last Active Jul 18th 2021

I was looking again at this entry, while preparing my category theory notes elsewhere, and I find that this entry is really bad.

With the co-Yoneda lemma in hand (every presheaf is a colimit of representables, and that is dealt with well on its page), the statement of free cocompletion fits as an easy clear Idea into 2 lines, and as a full proof in maybe 10.

The entry should just say that!

Currently the section “technical details” starts out right, but somehow forgets along the way what it means to write a proof in mathematics.

On the other hand, the section “Gentle introduction” seems to be beating about the bush forever. Does this really help newbies?

- Discussion Type
- discussion topicfinitary functor
- Category Latest Changes
- Started by David_Corfield
- Comments 1
- Last comment by David_Corfield
- Last Active Jul 18th 2021

- Discussion Type
- discussion topicKiyoshi Oka
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 17th 2021

brief

`category:people`

-entry for hyperlinking references at*Oka principle*

- Discussion Type
- discussion topiccomplex projective space
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Jul 17th 2021

I have split off

*complex projective space*from*projective space*and added some basic facts about its cohomology.

- Discussion Type
- discussion topiccore of a ring
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Urs
- Last Active Jul 17th 2021

gave

*core of a ring*some minimum content

- Discussion Type
- discussion topicinverse
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 17th 2021

For what it’s worth, I have removed the list of “Remarks” and instead turned its items into numbered Examples or Propositions (with proofs).

Some of the items – which were not so much about inverses as about isomorphisms (such as two-out-of-3) I moved to

*isomorphism*. That entry, too, is lacking professional fromatting, but I’ll leave it as is for the time being.

- Discussion Type
- discussion topicrational fiber lemma
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Jul 16th 2021

- Discussion Type
- discussion topicEhrenfeucht-Fraïssé comonad
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by Mike Shulman
- Last Active Jul 15th 2021