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- Discussion Type
- discussion topicStable categories and structured ring spectra
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active 6 days ago

- Discussion Type
- discussion topicelliptic genera as partition functions -- references
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active 6 days ago

a bare list of references, to be

`!include`

-ed into relevant entries (such as*Witten genus*,*M5-brane elliptic genus*but also inside*elliptic cohomology – references*) – for ease of harmonizing lists of references

- Discussion Type
- discussion topicadjoint string
- Category Latest Changes
- Started by varkor
- Comments 1
- Last comment by varkor
- Last Active 6 days ago

- Discussion Type
- discussion topicambidextrous adjunction
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by varkor
- Last Active 6 days ago

started a stub for

*ambidextrous adjunction*, but not much there yet

- Discussion Type
- discussion topicbiadjoint pair
- Category Latest Changes
- Started by varkor
- Comments 3
- Last comment by varkor
- Last Active 6 days ago

- Discussion Type
- discussion topicbiadjunction
- Category Latest Changes
- Started by Mike Shulman
- Comments 3
- Last comment by varkor
- Last Active 6 days ago

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

- Discussion Type
- discussion topicsmashing localization
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 7 days ago

am expanding out this reference item at several places:

- Tilman Bauer,
*Bousfield localization and the Hasse square*(2011) (pdf), chapter 6 in: Christopher Douglas, John Francis, André Henriques, Michael Hill (eds.),*Topological Modular Forms*, Mathematical Surveys and Monographs Volume 201, AMS 2014 (ISBN:978-1-4704-1884-7)

- Tilman Bauer,

- Discussion Type
- discussion topiccomonad
- Category Latest Changes
- Started by David_Corfield
- Comments 1
- Last comment by David_Corfield
- Last Active 7 days ago

- Discussion Type
- discussion topicpara construction
- Category Latest Changes
- Started by nLab edit announcer
- Comments 14
- Last comment by rongmin
- Last Active 7 days ago

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

- Discussion Type
- discussion topicAdrian Norbert Schellekens
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 7 days ago

- Discussion Type
- discussion topicChern- and Pontrjagin forms -- section
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 7 days ago

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.

- Discussion Type
- discussion topictwisted de Rham cohomology
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active 7 days ago

added pointer to the original article:

- Ryan Rohm, Edward Witten, around (23) and appendix of:
*The antisymmetric tensor field in superstring theory*, Annals of Physics Volume 170, Issue 2, September 1986, Pages 454-489 (doi:10.1016/0003-4916(86)90099-0)

- Ryan Rohm, Edward Witten, around (23) and appendix of:

- Discussion Type
- discussion topicGauss-Manin connection
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by Urs
- Last Active Jul 25th 2021

New stub, Gauss-Manin connection.

- Discussion Type
- discussion topiclocal system
- Category Latest Changes
- Started by Urs
- Comments 62
- Last comment by Urs
- Last Active Jul 25th 2021

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

$[X, \mathcal{B}C] \cong [\Pi X_* ,C]$where $X_*$ is the skeletal filtration of the CW-complex $X$, $C$ is a crossed complex, and $\mathcal{B}C$ is the classifying space of $C$, 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 $X$ is a covariant functor from the fundamental groupoid of $X$ 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)

- Discussion Type
- discussion topicHodge theory and Complex algebraic geometry
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jul 25th 2021

started a category:reference entry

in the course of this I added some stuff here and there, for instance at

*Abel-Jacobi map*. But very stubby for the moment.

- Discussion Type
- discussion topicGerardus Mercator
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 25th 2021

- Discussion Type
- discussion topicNicholas Mercator
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 25th 2021

brief

`category:people`

-entry for hyperlinking references at*logarithm*and*Mercator series*

- Discussion Type
- discussion topicMercator series
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jul 25th 2021

- Discussion Type
- discussion topicwell-ordering theorem
- Category Latest Changes
- Started by Todd_Trimble
- Comments 28
- Last comment by martinescardo
- Last Active Jul 24th 2021

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).

- Discussion Type
- discussion topiccontinuous logic
- Category Latest Changes
- Started by David_Corfield
- Comments 5
- Last comment by David_Corfield
- Last Active Jul 24th 2021

Added a reference

- Simon Cho,
*Categorical semantics of metric spaces and continuous logic*, (arXiv:1901.09077)

- Simon Cho,

- Discussion Type
- discussion topicImma Gálvez-Carrillo
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Jul 24th 2021

- Discussion Type
- discussion topicpartial sum
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 24th 2021

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

*Taylor series*

- Discussion Type
- discussion topiclogarithm
- Category Latest Changes
- Started by TobyBartels
- Comments 2
- Last comment by Urs
- Last Active Jul 24th 2021

- Discussion Type
- discussion topicrelation between determinant and trace
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 24th 2021

I am hereby giving the material at

*determinant – Properties – As a polynomial in traces of powers*its own entry, for ease of cross-linking to it (notably at*trace*, but also at*characteristic polynomial*and elsewhere)

- Discussion Type
- discussion topicNewton identities
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 24th 2021

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

*determinant*

- Discussion Type
- discussion topicL-infinity-algebra
- Category Latest Changes
- Started by Urs
- Comments 21
- Last comment by DavidRoberts
- Last Active Jul 24th 2021

expanded L-infinity-algebra as indicated on the nCafe, here

- Discussion Type
- discussion topictype I string theory
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by David_Corfield
- Last Active Jul 24th 2021

added brief pointer to the derivation of $SO(32)$ 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

- Discussion Type
- discussion topiccofree coalgebra
- Category Latest Changes
- Started by DavidRoberts
- Comments 1
- Last comment by DavidRoberts
- Last Active Jul 24th 2021

- Discussion Type
- discussion topicspin network
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Grant_Bradley
- Last Active Jul 23rd 2021

a stub, just to satisfy a requested link at

*string diagram*

- Discussion Type
- discussion topicrelation between L-infinity algebras and dg-Lie algebras
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 23rd 2021

The relation between the homotopy theory of $L_\infty$-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.

- Discussion Type
- discussion topicrig category
- Category Latest Changes
- Started by Urs
- Comments 21
- Last comment by David_Corfield
- Last Active Jul 23rd 2021

concerning the discussion here: notice that an entry

*rig category*had once been created, already.

- Discussion Type
- discussion topictensor product of functors
- Category Latest Changes
- Started by IngoBlechschmidt
- Comments 3
- Last comment by Hurkyl
- Last Active Jul 23rd 2021

I noticed we didn’t have a standalone entry on the tensor product of functors, so I created a stub and linked it with

*tensor product*,*tensor product of modules*,*co-Yoneda lemma*,*weighted colimit*,*free cocompletion*, and*geometric realization*. More links and more content are still missing though, for instance I did not discuss the enriched setting.

- Discussion Type
- discussion topicL-infinity algebras in physics
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Jul 23rd 2021

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

Markus B. Fröb,

*Anomalies in time-ordered products and applications to the BV-BRST formulation of quantum gauge theories*(arXiv:1803.10235)Alex Arvanitakis,

*The $L_\infty$-algebra of the S-matrix*(arXiv:1903.05643)

- Discussion Type
- discussion topicdense functor
- Category Latest Changes
- Started by David_Corfield
- Comments 13
- Last comment by Guest
- Last Active Jul 23rd 2021

- Discussion Type
- discussion topicend
- Category Latest Changes
- Started by Urs
- Comments 21
- Last comment by Urs
- Last Active Jul 23rd 2021

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.)

- Discussion Type
- discussion topicDonald Yau
- Category Latest Changes
- Started by David_Corfield
- Comments 1
- Last comment by David_Corfield
- Last Active Jul 23rd 2021

- Discussion Type
- discussion topicNiles Johnson
- Category Latest Changes
- Started by David_Corfield
- Comments 1
- Last comment by David_Corfield
- Last Active Jul 23rd 2021

- Discussion Type
- discussion topiccodensity monad
- Category Latest Changes
- Started by Thomas Holder
- Comments 12
- Last comment by David_Corfield
- Last Active Jul 23rd 2021

- Discussion Type
- discussion topicdifferential forms on simplices
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jul 23rd 2021

I have expanded, streamlined and re-organized a little at

*differential forms on simplices*.

- Discussion Type
- discussion topicmodel structure on spectra
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jul 23rd 2021

stub for model structure on spectra

- Discussion Type
- discussion topicspectral algebraic geometry
- Category Latest Changes
- Started by Théo de Oliveira S.
- Comments 2
- Last comment by Urs
- Last Active Jul 23rd 2021

- Discussion Type
- discussion topiclarge cardinal
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by nLab edit announcer
- Last Active Jul 22nd 2021

included in

*large cardinal*a jpg with a big diagram showing their relations.

- Discussion Type
- discussion topicSullivan model
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Jul 22nd 2021

created Sullivan model

- Discussion Type
- discussion topicnilpotent L-infinity algebra
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 22nd 2021

added this remark:

Under the formal duality between $L_\infty$-algebras and their Chevalley-Eilenberg dgc-algebras, connective nilpotent $L_\infty$-algebras correspond bijectively to the connected Sullivan models (Berglund 2015, Thm. 2.3).

and pointer to

- Alexander Berglund,
*Rational homotopy theory of mapping spaces via Lie theory for $L_\infty$ algebras*, Homology, Homotopy and Applications, Volume 17 (2015) Number 2 (arXiv:1110.6145, doi:10.4310/HHA.2015.v17.n2.a16)

- Alexander Berglund,

- Discussion Type
- discussion topicMasahiro Sugawara
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 22nd 2021

brief

`category:people`

-entry for hyperlinking references at*H-space*and*strong homotopy map*

- Discussion Type
- discussion topichomomorphism of L-∞ algebras
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 22nd 2021

- Discussion Type
- discussion topicinternalization
- Category Latest Changes
- Started by Urs
- Comments 43
- Last comment by Urs
- Last Active Jul 22nd 2021

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

- Discussion Type
- discussion topicred-shift conjecture
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Jul 22nd 2021

created

*red-shift conjecture*

- Discussion Type
- discussion topiciterated algebraic K-theory
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Urs
- Last Active Jul 22nd 2021

- Discussion Type
- discussion topicmonoidal bicategory
- Category Latest Changes
- Started by Peter Heinig
- Comments 22
- Last comment by Urs
- Last Active Jul 21st 2021

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$\otimes$id, 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)

- 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

- Discussion Type
- discussion topicsylleptic ∞-group
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 21st 2021

added pointer to:

- Nick Gurski, Angélica M. Osorno, Section 2.2 of
*Infinite loop spaces, and coherence for symmetric monoidal bicategories*, Adv. Math. 246 (2013) 1-32 (arXiv:1210.1174)

- Nick Gurski, Angélica M. Osorno, Section 2.2 of

- Discussion Type
- discussion topicAngélica M. Osorno
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 21st 2021

brief

`category:people`

-entry for hyperlinking references at*monoidal 2-category*and elsewhere

- Discussion Type
- discussion topicsylleptic 3-group
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jul 21st 2021

… whereby the periodic table is fianlly un-grayed

- Discussion Type
- discussion topicMartin Neuchl
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 21st 2021

brief

`category:people`

-entry for hyperlinking references at*monoidal 2-category*and*braided monoidal 2-category*

- Discussion Type
- discussion topicsymmetric monoidal 2-category
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 21st 2021

added pointer to:

- Brian Day, Ross Street, Section 5 of:
*Monoidal Bicategories and Hopf Algebroids*, Advances in Mathematics Volume 129, Issue 1, 15 July 1997, Pages 99-157 (doi:10.1006/aima.1997.1649)

- Brian Day, Ross Street, Section 5 of:

- Discussion Type
- discussion topicsylleptic monoidal 2-category
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Jul 21st 2021

*sylleptic monoidal 2-category*,*symmetric monoidal 2-category*… but now I am running out of steam…

- Discussion Type
- discussion topicLior Yanovski
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 21st 2021

brief

`category:people`

-entry for hyperlinking references at*Eckmann-Hilton argument*and*(∞,1)-operads*

- Discussion Type
- discussion topic(infinity,1)-operad
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Jul 21st 2021

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

- Discussion Type
- discussion topicbraided monoidal 2-category
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jul 21st 2021

at

*braided monoidal 2-categiry*the following query box was sitting, which I hereby move from there to here

+–{: .query} Ben Webster: I would very much like to know: what structure on a (triangulated/dg-/stable infinity/whatever you like) monoidal category would make its 2-category of module categories (give that phrase any sensible construal you like) is braided monoidal.

If one decategorifies this question, one gets the question “what structure on a ring makes its category of representations braided monoidal” and the answer to this question is well-known: a quasi-triangular quasi-Hopf structure.

I asked a MathOverflow question on the same topic. No interesting answers yet. =–

- Discussion Type
- discussion topicbraided 2-group
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Jul 21st 2021