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    • a category:reference-entry for the upcoming book

      v1, current

    • Created a stub with a definition and an example.

      v1, current

    • I added to biadjunction the statement and some references for the fact that any incoherent one can be improved to a coherent one.

    • 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 a change, I added some actual text to this category:people-entry, highlighting a little the content and relevance of (parts of) the research.

      diff, v5, current

    • Discussion of the formulas for the standard characteristic forms has been missing in various entries (e.g. at Chern class at characteristic form, etc.). Since there is little point in discussing the Chern forms independently from the Pontrjagin forms etc. I am now making it a stand-alone section to be !include-ed into relevant entries, to have it all in one place.

      Not done yet, though, but it’s a start.

      v1, current

    • I am going to polish the entry local system now.

      The following is long forgotten discussion that had been sitting in a query box there. Everybody involved should check what of that still needs further discussion and then have that discussion here on the forum.

      Urs: I am hoping that maybe David Speyer, whose expositional blog entry is linked to below, or maybe somebody else would enjoy filling in some material here.

      Bruce: Could it perhaps be “On a topological space (why do we need connected?) this is the same as a sheaf of flat sections of a finite-dimensional vector bundle equipped with flat connection;”. I guess by “flat connection” in this general topological context we would mean simply a functor from the homotopy groupoid to the category of vector spaces?

      Zoran Škoda: connected because otherwise you do not have even the same dimension of the typical stalk of teh lcoally constant sheaf. Maybe there is a fancy wording with groupoids avoiding this, but when you have a representation on a single space, you need connectedness.

      Ronnie Brown I do not have time to write more tonight but mention that there is a section of the paper

      • (with P.J.HIGGINS), “The classifying space of a crossed complex”, Math. Proc. Camb. Phil. Soc. 110 (1991) 95–120.

      on local systems, where a module over the fundamental groupoid of a space is regarded as a special case of a crossed complex. This seems convenient for the singular theories but has not been developed in a Cech setting. The homotopy classification theorem

      [X,ℬC]≅[ΠX *,C] [X, \mathcal{B}C] \cong [\Pi X_* ,C]

      where X *X_* is the skeletal filtration of the CW-complex XX, CC is a crossed complex, and ℬC\mathcal{B}C is the classifying space of CC, thus includes the local coefficient version of the classical Eilenberg-Mac Lane theory.

      Tim: Quoting an exercise in Spanier (1966) on page 58:

      A local system on a space XX is a covariant functor from the fundamental groupoid of XX to some category.

      A reference is given to a paper by Steenrod: Homology with local coefficients, Annals 44 (1943) pp. 610 - 627.

      Perhaps the entry could reflect the origins of the idea. The current one seems to me to be much too restrictive. There are other applications of the idea than the ones at present indicated, although of course those are important at the moment. Reference to vector bundles is not on the horizon in Spanier!!!!.

      Local systems with other codomains than vector spaces are used in rational homotopy theory.

      Urs: I am all in favor of emphasizing that “local system” is nothing but a functor from a fundamental groupoid. That’s of course right up my alley, compare the discussion with David Ben-Zvi at the “Journal Club”. Whoever finds the time to write something along these lines here should do so (and in clude in particular the reference Ronnie Brown gives above).

      BUT at the same time it seems to me that many practitioners will by defualt think of the explicitly sheaf-theoretic notion when hearing “local syetem” which the entry currently states. We should emphasize this explicitly, something like: “while in general a local system is to be thought of as a representation of a fundamental groupoid, often the term is understood exclusively in its realization within abelian sheaf theory as follows …”

      (to be continued in next comment)

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

      v1, current

    • just for completeness, to be able to link to it

      v1, current

    • Indicated where to find (in the nLab) a proof of the equivalence with the axiom of choice and with Zorn’s lemma (<– it’s there).

      diff, v21, current

    • A very old link to her webpage replaced by an up-to-date one!

      diff, v6, current

    • a minimum, just for completeness and to satisfy a link that had long been requetsed at Taylor series

      v1, current

    • a stub (nothing here yet), for the moment just to satisfy a link that had long been requested at determinant

      v1, current

    • added brief pointer to the derivation of SO(32)SO(32) gauge group via tadpole cancellation, and some references on type I phenomenology. Will add these also to string phenomenology and to GUT, as far as relevant there

      diff, v7, current

    • The relation between the homotopy theory of L ∞L_\infty-algebras and dg-Lie algebras is discussed, or at least mentioned, in several entries. But not all of them provide the same amount of information. So I am giving this its own page now, to provide a central resource.

      On the other hand, I had steam only for writing a kind of survey here, so far. But at least I’ll make all other relevant entries cross-point to/from here now.

      v1, current

    • concerning the discussion here: notice that an entry rig category had once been created, already.

    • 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

    • Moved the reference to the Lawvere commentary on Isbell to ’References’.

      diff, v16, current

    • added the statement of the Fubini theorem for ends to a new section Properties.

      (I wish this page would eventually give a good introduction to ends. I remember the long time when I banged my head against Kelly’s book and just didn’t get it. Then suddenly it all became obvious. It’s some weird effect with this enriched category theory that some of it is obvious once you understand it, but looks deeply mystifying to the newcomer. Kelly’s book for instance is a magnificently elegant resource for everyone who already understands the material, but hardly serves as an exposition of the ideas involved. I am hoping that eventually the nLab entries on enriched category theory can fill this gap. Currently they do not really. But I don’t have time for it either.)

    • Updated link of Barwick’s dag lectures, as the old link is dead

      diff, v9, current

    • included in large cardinal a jpg with a big diagram showing their relations.

    • am giving this its own entry, in order to have a place where to sort out the referencing of the “sh map”-terminology and its variants.

      Not done yet, but need to save.

      v1, current

    • started to add to internalization a list of links to examples. Probably we have much more.

    • One small question that has often occurred to me:

      • in the three usual axioms specifying how the unit interacts with parenthesizing in a monoidal bicategory, is there any known reason for drawing one of the three diagrams as a square (as opposed to a triangle, like the other two) even though one of the 1-cells is the identity id⊗\otimesid, except for the (certainly important) aesthetical/visual/psychological reason that otherwise (if using the conventional notation) the tip of the arrow giving the 2-cell would point from a 1-cell to a 0-cell?

      (Technical note: I chose the “Latest Changes” category, even though no change to monoidal bicategory was made yet, because monoidal bicategory appears to not have had a thread of its own yet, and it is not inconceivable that this page will evolve in the future and need a thread)

    • have added to (infinity,1)-operad the basics for the “(∞,1)(\infty,1)-category of operators”-style definition

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