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    • Added material on diagonal maps and the product functor, mentioning for instance the fact that the product functor is right adjoint to a diagonal functor.

      diff, v22, current

    • A stub, for the moment just to have a place for recording a couple of references (which were previously at fusion category.

      v1, current

    • Added the contents of the canonical isomorphism induced by some non-canonical isomorphism as coming from Lack’s proof.

      diff, v32, current

    • in analogy to what I just did at classical mechanics, I have now added some basic but central content to quantum mechanics:

      • Quantum mechanical systems

      • States and observables

      • Spaces of states

      • Flows and time evolution

      Still incomplete and rough. But I have to quit now.

    • brief category:people-entry for hyperlinking references

      v1, current

    • Asked to clarify, made a definition.


      v1, current

    • I noticed that the entry classifying space is in bad shape. I have added a table of contents and tried to structure it slightly, but much more needs to be done here.

      I have added a paragraph on standard classifying spaces for topological principal bundles via the geometric realization of the simplicial space associated to the given topological group.

      In the section “For crossed complexes” there is material that had been provided by Ronnie Brown which needs to be harmonized with the existing Idea-section. It proposes something like a general axiomatics on the notion of “classifying space” more than giving details on the geometric realization of crossed complexes

    • The Idea-section at quasi-Hopf algebra had been confused and wrong. I have removed it and written a new one.

    • Following discussion in some other threads, I thought one should make it explicit and so I created an entry

      Currently this contains some (hopefully) evident remarks of what “dependent linear type theory” reasonably should be at least, namely a hyperdoctrine with values in linear type theories.

      The entry keeps saying “should”. I’d ask readers to please either point to previous proposals for what “linear dependent type theory” is/should be, or criticise or else further expand/refine what hopefully are the obvious definitions.

      This is hopefully uncontroversial and should be regarded an obvious triviality. But it seems it might be one of those hidden trivialities which deserve to be highlighted a bit more. I am getting the impression that there is a big story hiding here.

      Thanks for whatever input you might have.

    • brief category:people entry for hyperlinking reretences

      v1, current

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

      v1, current

    • added an Examples-section (here) “In 2d gravity on String worldsheets”

      diff, v3, current

    • I am changing the page title – this used to be “A first idea of quantum field theory”, which of course still redirects. The “A first idea…” seemed a good title for when this was an ongoing lecture that was being posted to PhysicsForums. I enjoyed the double meaning one could read into it, but it’s a bad idea to carve such jokes into stone. And now that the material takes its place among the other chapters of geometry of physics, with the web of cross-links becoming thicker, the canonical page name clearly is “perturbative quantum field theory”.

      diff, v187, current

    • added pointer to these two recent references, identifying further L L_\infty-algebra structure in Feynman amplitudes/S-matrices of perturbative quantum field theory:

      diff, v11, current

    • Adding reference

      • Hee-Joong Chung, Dohyeong Kim, Minhyong Kim, Jeehoon Park, Hwajong Yoo, Entanglement entropies in the abelian arithmetic Chern-Simons theory [2312.17138]


      diff, v8, current

    • this is a bare list of references which used to be (and still is) at entanglement entropy. But since the same references are now also needed at long-range entanglement, I am putting them in a separate page here, to be !include-ed into both these entries

      v1, current

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

      v1, current

    • Balancing doesn’t mention duals anywhere, and makes sense even without duals. I removed an incorrect statement and replaced it with the correct one. Not sure if it needs a reference, but the correct result appears as Lemma 4.20 in (where it’s attributed to Deligne, but the citation is to Yetter).

      diff, v7, current

    • 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:ABf: A\to B. The canonical map coker(kerf)ker(cokerf)\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(cokerf)B\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 ff: 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 ff a monic and you get a decomposition into iso iso kernel and for ff 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 2:2:\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 tfAbtfAb are the maps whose cokernel (in AbAb) is torsion. Thus 2:2:\mathbb{Z}\to\mathbb{Z} is monic and epic, so tfAbtfAb is not balanced. And since 2:2:\mathbb{Z}\to\mathbb{Z} is its own canonical map, that canonical map is monic and epic in tfAbtfAb, so this isn’t a counterexample.

      Zoran: says at one place that Palamodov’s version of semi-abelian category is preabelian + canonical morphism is epi and mono. =–

      [end forwarded discussion]

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • I have added to M5-brane a fairly detailed discussion of the issue with the fractional quadratic form on differential cohomology for the dual 7d-Chern-Simons theory action (from Witten (1996) with help of Hopkins-Singer (2005)).

      In the new section Conformal blocks and 7d Chern-Simons dual.

    • brief category:people-entry for hyperlinking references

      v1, current

    • Mike Stay kindly added the standard QM story to path integral.

      I changed the section titles a bit and added the reference to the Baer-Pfaeffle article on the QM path integral. Probably the best reference there is on this matter.