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

• 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.

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since it seems to be outdated, and instead added a pointer to the personal webpage.

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

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

• 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]

• Page created, but author did not leave any comments.

Anonymous

• 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.
• 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)?

• Add Kanamori reference (from David Roberts's blog)

• Fixed a LaTeX typo in Remark 3.12. I’m not an expert so could someone double-check?

• Added link to 2014 Awodey-Butz-Simpson-Streicher APAL paper “Relating first-order set theories, toposes and categories of classes” on BIST

• am giving this a little entry, to be able to link to the term

• Change title to use simple ’ rather than some unicode symbol.

• 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)

• Written definition and basic properties. Locale case still missing, I’m not an expert.

• (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).

• 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)
• added disambiguation with the notion in probability theory

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

• brief category:people-entry for hyperlinking references at Stein manifold

• a stub – for the moment just so as to have a place where to record references

• brief category:people-entry for hyperlinking references

• Add page structure, idea section.

I preserved most of the original page under a definition section.

• brief category:people-entry for hyperlinking references at Oka manifold

• The links to Bob Walters’ archive don’t work now. Has it gone elsewhere?

I’ve added in another source for the lectures.

• 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?

• Did a little editing. Added redirect to ’finitary category’.

• brief category:people-entry for hyperlinking references at Oka principle

• 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.

• The statement from Bousfield-Kan 72, Chapter II

• A very brief description of the Ehrenfeucht-Fraisse comonad, as defined by Abramsky-Shah, which described the Ehrenfeucht-Fraisse game.

Eigil Rischel