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    • starting an entry, for the moment mainly in order to record the fact that “crossed homomorphisms” are equivalently homomorphic sections of the corresponding semidirect product group projection. This is obvious, but is there a reference that makes it explicit?

      v1, current

    • Added to T-duality a section with the discussion of the usual path-integral heuristics for why the two sigma-models on T-dual backgrounds yield equivalent quantum field theories.

    • moved the discussion of the Lie algebra 𝔰𝔲(2)\mathfrak{su}(2) out of the middle of “Properties – General” into its own subsection “Properties – Lie algebra”.

      Will copy this also over to su(2)

      diff, v18, current

    • I corrected an apparent typo:

      A 2-monad TT as above is lax-idempotent if and only if for any TT-algebra a:TA→Aa \colon T A \to A there is a 2-cell θ a:1⇒η∘a\theta_a \colon 1 \Rightarrow \eta \circ a

      to

      A 2-monad TT as above is lax-idempotent if and only if for any TT-algebra a:TA→Aa \colon T A \to A there is a 2-cell θ a:1⇒η A∘a\theta_a \colon 1 \Rightarrow \eta_A \circ a

      It might be nice to say η A\eta_A is the unit of the algebra….

      diff, v22, current

    • For the purposes of negative thinking, it may be useful to recognise that every ∞\infty-category has a (−1)(-1)-morphism, which is the source and target of every object. (In the geometric picture, this comes as the (−1)(-1)-simplex of an augmented simplicial set.)

      Jonathan Arnoult has pointed out on CT Zulip that this is misleading: it sounds like it implies that every ∞\infty-category is monoidal! And John Baez pointed out that the analogy to augmented simplicial sets fails because in an augmented simplicial set each 0-simplex has only one face, rather than a separate “source” and “target” that are both the same (−1)(-1)-simplex.

      I suggest we just remove this paragraph and the query box following it, since I can’t think of a way to rephrase it that would be more helpful than unhelpful. But I’m open to other suggestions.

      diff, v16, current

    • Added section on irreflexive comparisons, which generalises linear orders in constructive mathematics

      Anonymous

      diff, v15, current

    • Stub. For the moment just for providing a place to record this reference:

      • Jean Thierry-Mieg, Connections between physics, mathematics and deep learning, Letters in High Energy Physics, vol 2 no 3 (2019) (doi:10.31526/lhep.3.2019.110)

      v1, current

    • This is a brief description of the construction that started appearing in category-theoretic accounts of deep learning and game theory. It appeared first in Backprop As Functor (https://arxiv.org/abs/1711.10455) in a specialised form, but has slowly been generalised and became a cornerstone of approaches unifying deep learning and game theory (Towards Foundations of categorical Cybernetics, https://arxiv.org/abs/2105.06332), (Categorical Foundations of Gradient-based Learning, https://arxiv.org/abs/2103.01931).

      Our group here in Glasgow is using this quite heavily, so since I couldn’t find any related constructions on the nLab I decided to add it. This is also my first submission. I’ve read the “HowTo” page, followed the instructions, and I hope everything looks okay.

      There’s quite a few interesting properties of Para, and eventually I hope to add them (most notably, it’s an Para is an oplax colimit of a functor BM -> Cat, where B is the delooping of a monoidal category M).

      A notable thing to mention is that I’ve added some animated GIF’s of this construction. Animating categorical concepts is something I’ve been using as a pedagogical tool quite a bit (more here https://www.brunogavranovic.com/posts/2021-03-03-Towards-Categorical-Foundations-Of-Neural-Networks.html) and it seems to be a useful tool getting the idea across with less friction. If it renders well (it seems to) and is okay with you, I might add more to the Optics section, and to the neural networks section (I’m hoping to get some time to add our results there).

      Bruno Gavranović

      v1, current

    • For now creating page, it needs to be (much) further expanded.

      v1, current

    • Added a reference to Tall–Wraith. Changed ∘:P⊗ kP→P\circ: P \otimes_k P \to P to ∘:P⊙ kP→P\circ: P \odot_k P \to P. Added redirects.

      diff, v3, current

    • Several references and a link to “writings” page

      diff, v2, current

    • Added definition of Tambara-Yamagami (TY) fusion category, the simplest example of non-invertible symmetries in the 2d defect language.

      v1, current

    • created a “category: reference”-page The Stacks Project

      I have only now had a closer look at this and am impressed by the scope this has. Currently a total of 2288 pages. It starts with all the basics, category theory, commutative algebra and works its way through all the details to arrive at algebraic stacks.

      So besides my usual complaint (Why behave as if there are not sites besides the usual suspects on CRing opCRing^{op} and either give a general account or call this The Algebraic Stacks Project ? ) I am enjoying seeing this. We should have lots of occasion to link to this. Too bad that this did not start out as a wiki.

    • A class of structures similar to quantum groups.

      v1, current

    • I needed an entry to be able to point to which collects pointers to the various entries on “dualities” in string theory. So I created one: duality in string theory.

    • Starting page on Tr(ϕ 3)\mathrm{Tr}(\phi^3)

      Anonymouse

      v1, current

    • Canonical bases for the (representations of) quantum groups and Lie groups.

      v1, current

    • I reverted back to

      This polynomial is an invariant of the knot, the Alexander polynomial of the knot.

      I see no reason for a strange comma before “is”. The second comma is before the appositional phrase.

      diff, v23, current

    • I have edited at Tychonoff theorem:

      1. tidied up the Idea-section. (Previously there was a long paragraph on the spelling of the theorem before the content of the theorem was even mentioned)

      2. moved the proofs into a subsection “Proofs”, and added a pointer to an elementary proof of the finitary version, here

      Notice that there is an ancient query box in the entry, with discussion between Todd and Toby. It would be good to remove this box and turn whatever conclusion was reached into a proper part of the entry.

      At then end of the entry there is a line:

      More details to appear at Tychonoff theorem for locales

      which however has not “appeared” yet.

      But since the page is not called “Tychonoff theorem for topological spaces”, and since it already talks about locales a fair bit in the Idea section, I suggest to remove that line and to simply add all discussion of localic Tychonoff to this same entry.

    • Started lift.

      weak factorization system has redirects from: lifting property, right lifting property, left lifting property, lifting problem, lifting problems.

      Would it be better to have these redirect to lift?

    • created quick stub for framed bicategory

      but my machine's battery will die any second now...

    • I added a reference on Gabriel filters on quantales.

      Very strange: version one on show had a reference on Etendues (entered by T. Holder), but when I clicked edit there was none in the edit window, just in show window. I edited and one can not see it in any history, nowhere. So there was something in show cache from 2014 which is not recorded in any history edit.

      diff, v2, current

    • Add a strengthening of the coherence result, also due to Power.

      diff, v7, current

    • starting page on surfaceology

      Anonymouse

      v1, current

    • Explained how to prove the commutativity of addition in EFA

      diff, v11, current

    • I gave Fourier-Mukai transform a bit of an Idea-section. It overlaps substantially with the Definition section now, but I thought one needs to say the simple basic idea clearly in words first. Also added a few more pointers to literature.

    • Anonymous “helpfully” changed the statement

      A matrix is a list of lists.

      to

      A matrix is a function M:[n]×[m]→XM:[n]\times[m]\rightarrow X from the Cartesian product [n]×[m][n]\times[m] to a set XX.

      which I have reverted back.

      diff, v12, current

    • a minimum of an Idea-section, but mainly to record some references

      v1, current

    • a stub entry, for the moment just to make the link work

      v1, current

    • Pointed out that the ring of big Witt vectors is represented by the free lambda-ring on one generator (no coincidence; this follows by easy abstract nonsense).

      diff, v32, current

    • have now spelled out at Tor in simple terms how Tor 1 Ab(A,B)Tor_1^{Ab}(A,B) is a torsion group, so far for the case that AA is finite.

    • added to equalizer statement and proof that a category has equalizers if it has pullbcks and products

    • brief category:people-entry for hyperlinking references

      v1, current

    • Rename the page to reflect the fact that coherence and strictification are distinct, but related, theorems.

      diff, v7, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • I have added

      • Paolo Facchi, Giovanni Gramegna, Arturo Konderak, Entropy of quantum states (arXiv:2104.12611)

      and added publication details to

      • A. P. Balachandran, T. R. Govindarajan, Amilcar R. de Queiroz, A. F. Reyes-Lega, Algebraic approach to entanglement and entropy, Phys. Rev. A 88, 022301 (2013) (arXiv:1301.1300)

      and grouped together more discernibly the references on operator-algebraic entropy

      diff, v50, current

    • added at core the remark that the core is right adjoint to the forgetful functor Grpd→CatGrpd \to Cat.

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

    • Adding reference

      • Peter Freyd, Algebraic real analysis, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 (tac:20-10)

      Anonymous

      diff, v8, current

    • This is a page for Sridhar Ramesh’s notion of introspective theory. Precise definitions to be added.

      Aaron David Fairbanks

      v1, current

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