Corrected a hyperlink.

]]>some minimum, in order to give a home to today’s

- Dong-Yang Wang, Ya-Dong Yang, Xing-Bo Yuan,
*$b \to c \tau \bar \nu$ decays in supersymmetry with R-parity violation*(arXiv:1905.08784)

a stub, as requested here

]]>in order to give a home to this article:

- Michael Atiyah, Robbert Dijkgraaf, Nigel Hitchin,
*Geometry and physics*, Phil. Trans. R. Soc. A 13 March 2010 vol. 368 no. 1914 913-926 (doi;10.1098/rsta.2009.0227)

started self-dual higher gauge theory. Just minimal idea and list of references so far.

]]>stub for homotopy type theory

]]>Page created, but author did not leave any comments.

]]>brief `category:people`

-entry for hyperlinking references at *collar neighbourhood theorem* (hope I identified the correct Wikipedia page for this author)

some minimum, for completeness and to record references

]]>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]

]]>Finally added to *fracture theorem* the basic statement of the “arithmetic fracture square”, hence the following discussion.

The number theoretic statement is the following:

+– {: .num_prop #ArithmeticFractureSquare}

The integers $\mathbb{Z}$ are the fiber product of all the p-adic integers $\underset{p\;prime}{\prod} \mathbb{Z}_p$ with the rational numbers $\mathbb{Q}$ over the rationalization of the former, hence there is a pullback diagram in CRing of the form

$\array{ && \mathbb{Q} \\ & \swarrow && \nwarrow \\ \mathbb{Q}\otimes_{\mathbb{Z}}\underset{p\;prime}{\prod} \mathbb{Z}_p && && \mathbb{Z} \\ & \nwarrow && \swarrow \\ && \underset{p\;prime}{\prod} \mathbb{Z}_p } \,.$Equivalently this is the fiber product of the rationals with the integral adeles $\mathbb{A}_{\mathbb{Z}}$ over the ring of adeles $\mathbb{A}_{\mathbb{Q}}$

$\array{ && \mathbb{Q} \\ & \swarrow && \nwarrow \\ \mathbb{A}_{\mathbb{Q}} && && \mathbb{Z} \\ & \nwarrow && \swarrow \\ && \mathbb{A}_{\mathbb{Z}} } \,.$=–

In the context of a modern account of categorical homotopy theory this appears for instance as (Riehl 14, lemma 14.4.2).

+– {: .num_remark}

Under the function field analogy we may think of

$Spec(\mathbb{Z})$ as an arithmetic curve over F1;

$\mathbb{A}_{\mathbb{Z}}$ as the ring of functions on the formal disks around all the points in this curve;

$\mathbb{Q}$ as the ring of functions on the complement of a finite number of points in the curve;

$\mathbb{A}_{\mathbb{Q}}$ is the ring of functions on punctured formal disks around all points, at most finitely many of which do not extend to the unpunctured disk.

Under this analogy the arithmetic fracture square of prop. \ref{ArithmeticFractureSquare} says that the curve $Spec(\mathbb{Z})$ has a cover whose patches are the complement of the curve by some points, and the formal disks around these points.

This kind of cover plays a central role in number theory, see for instance thr following discussions:

=–

]]>added pointer to

- Luis Scoccola,
*Nilpotent Types and Fracture Squares in Homotopy Type Theory*(arXiv:1903.03245)

I decided it would be a good idea to split off realizability topos into a separate entry (it had been tucked away under partial combinatory algebra). I’ve only just begun, mainly to get down the connection with COSHEP. A good (free, online) reference is Menni’s thesis.

]]>Updating reference to cubical type theory. This page need more work.

]]>Added an alternative description of the category of orbits via normalizers.

]]>Stub about blockchain platform EOS known for high performance.

]]>Page created, but author did not leave any comments.

]]>added a brief historical comment to *Higgs field* and added the historical references

started adding some basic technical details to topological K-theory

]]>Page created, but author did not leave any comments.

]]>added a cross reference to prorepresentable functor

The link to Marc Hoyois notes is dead. Perhaps his higher Galois theory has much the same content (as I do not know the notes refered to here and his webpage does not seem to list this pdf file.)

]]>I got tired of having to fight my way through Kelly’s monster yet again, and created transfinite construction of free algebras. I couldn’t really think of a good name for this page; suggestions are welcome.

]]>am splitting this off from *M5-brane*

Page created, but author did not leave any comments.

]]>**Edit to**: GUT by Urs Schreiber at 2018-04-01 01:21:13 UTC.

**Author comments**:

added pointer to textbook account

]]>