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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

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- Discussion Type
- discussion topicdual number
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 14 hours ago

- Discussion Type
- discussion topiccompact closed category
- Category Latest Changes
- Started by Noam_Zeilberger
- Comments 15
- Last comment by John Baez
- Last Active 16 hours ago

I added a note to compact closed category on the fact that the inclusion from compact closed categories into SMCCs has a left adjoint, pointing to an article by Day where he describes the free compact closed category over a closed symmetric monoidal category as a localization. Question: this left adjoint is not full, but I believe it is faithful – does anyone know how to prove that?

- Discussion Type
- discussion topicSeverin Bunk
- Category Latest Changes
- Started by nLab edit announcer
- Comments 5
- Last comment by nLab edit announcer
- Last Active 17 hours ago

- Discussion Type
- discussion topicsupergeometry of fermion fields -- references
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 17 hours ago

a bare list of references, to be

`!include`

-ed into the References-sections of relevant entries (such as at*supergeometry*and*fermion*), for ease of synchronization

- Discussion Type
- discussion topicsupergeometry
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active 17 hours ago

added to supergeometry a link to the recent talk

- Mikhail Kapranov,
*Categorification of supersymmetry and stable homotopy groups of spheres*(video)

- Mikhail Kapranov,

- Discussion Type
- discussion topicdualizable object
- Category Latest Changes
- Started by Urs
- Comments 35
- Last comment by John Baez
- Last Active 18 hours ago

edited dualizable object a little, added a brief paragraph on dualizable objects in symmetric monoidal $(\infty,n)$-categories

- Discussion Type
- discussion topicgroup object
- Category Latest Changes
- Started by Urs
- Comments 16
- Last comment by John Baez
- Last Active 1 day ago

Added to group object the Yoneda-embedding-style definition and added supergroup to the list of examples.

- Discussion Type
- discussion topicdouble negation
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Urs
- Last Active 1 day ago

I have added to the References at

*double negation*pointer to Andrej’s exposition:which is really good. I have also added this to

*double negation transformation*, but clearly that entry needs some real references, too.

- Discussion Type
- discussion topicextension (double category theory)
- Category Latest Changes
- Started by mattecapu
- Comments 1
- Last comment by mattecapu
- Last Active 1 day ago

- Discussion Type
- discussion topicrestriction (double category theory)
- Category Latest Changes
- Started by mattecapu
- Comments 1
- Last comment by mattecapu
- Last Active 1 day ago

- Discussion Type
- discussion topicframed bicategory
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by mattecapu
- Last Active 1 day ago

created quick stub for framed bicategory

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

- Discussion Type
- discussion topictight and loose morphisms
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by mattecapu
- Last Active 1 day ago

- Discussion Type
- discussion topiccompanion pair
- Category Latest Changes
- Started by mattecapu
- Comments 7
- Last comment by mattecapu
- Last Active 1 day ago

- Discussion Type
- discussion topicconjunction
- Category Latest Changes
- Started by mattecapu
- Comments 2
- Last comment by mattecapu
- Last Active 1 day ago

same as in companion pair

- Discussion Type
- discussion topicdescent morphism
- Category Latest Changes
- Started by Mike Shulman
- Comments 33
- Last comment by DavidRoberts
- Last Active 1 day ago

Created descent morphism.

In adding links, I discovered that Euclidean-topological infinity-groupoid and separated (infinity,1)-presheaf use the phrase “descent morphism” to refer to the

*comparison functor*mapping into the category of descent data. If no one has any objections, I would like to change this to avoid confusion, but I’m not sure what to change it to: would “comparison functor” be good enough?

- Discussion Type
- discussion topicManuela Sobral
- Category Latest Changes
- Started by DavidRoberts
- Comments 1
- Last comment by DavidRoberts
- Last Active 1 day ago

- Discussion Type
- discussion topicsemantics
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 5 days ago

- Discussion Type
- discussion topicmodel
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by nLab edit announcer
- Last Active 5 days ago

I felt we needed a dedicated entry on model/category of models. So I started one. But just a puny stub so far.

- Discussion Type
- discussion topicisomorphism in a Segal type
- Category Latest Changes
- Started by Guest
- Comments 3
- Last comment by nLab edit announcer
- Last Active 6 days ago

- Discussion Type
- discussion topiccohesive homotopy type theory
- Category Latest Changes
- Started by Urs
- Comments 38
- Last comment by nLab edit announcer
- Last Active 6 days ago

I am working on an entry cohesive homotopy type theory.

This started out as material split off from cohesive (infinity,1)-topos, but is expanding now.

- Discussion Type
- discussion topicRezk type
- Category Latest Changes
- Started by Guest
- Comments 3
- Last comment by nLab edit announcer
- Last Active 6 days ago

- Discussion Type
- discussion topicbounded total order
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 6 days ago

- Discussion Type
- discussion topicop modality
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active 6 days ago

- Discussion Type
- discussion topicSegal type
- Category Latest Changes
- Started by Guest
- Comments 5
- Last comment by nLab edit announcer
- Last Active 6 days ago

- Discussion Type
- discussion topichom type
- Category Latest Changes
- Started by Guest
- Comments 2
- Last comment by nLab edit announcer
- Last Active 6 days ago

- Discussion Type
- discussion topicparametric dependent type theory
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by nLab edit announcer
- Last Active 6 days ago

starting page on type theories with parametricity, as a separate article from the article on polymorphism.

Anonymouse

- Discussion Type
- discussion topicUlrik Buchholtz
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active 6 days ago

Adding reference

- Ulrik Buchholtz,
*Higher Structures in Homotopy Type Theory*, in:*Reflections on the Foundations of Mathematics*, Synthese Library**407**Springer (2019) [doi:10.1007/978-3-030-15655-8_7, arXiv:1807.02177]

Anonymouse

- Ulrik Buchholtz,

- Discussion Type
- discussion topicJonathan Weinberger
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by nLab edit announcer
- Last Active 6 days ago

- Discussion Type
- discussion topicDaniel Gratzer
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by nLab edit announcer
- Last Active 6 days ago

- Discussion Type
- discussion topicdirected univalence axiom
- Category Latest Changes
- Started by Guest
- Comments 2
- Last comment by nLab edit announcer
- Last Active 6 days ago

- Discussion Type
- discussion topicsimplicial type theory
- Category Latest Changes
- Started by nLab edit announcer
- Comments 5
- Last comment by nLab edit announcer
- Last Active 6 days ago

- Discussion Type
- discussion topiccoCartesian fibration of (∞,1)-operads
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 6 days ago

- Discussion Type
- discussion topicspin-statistics theorem
- Category Latest Changes
- Started by Urs
- Comments 21
- Last comment by Urs
- Last Active 6 days ago

stub for spin-statistics theorem. Just recording a first few references so far.

- Discussion Type
- discussion topichigher parallel transport
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active 6 days ago

added pointer to

- Arthur Parzygnat,
*Two-dimensional algebra in lattice gauge theory*(arXiv:1802.01139)

which provides a wealth of computational details and illustrative graphics.

- Arthur Parzygnat,

- Discussion Type
- discussion topicGiry monad
- Category Latest Changes
- Started by zskoda
- Comments 74
- Last comment by nLab edit announcer
- Last Active 6 days ago

Added a reference of Robert Furber, Bart Jacobs at Giry monad.

- Discussion Type
- discussion topicexponentiable functor
- Category Latest Changes
- Started by John Baez
- Comments 12
- Last comment by SiyaM
- Last Active 7 days ago

I added the word “strict” here:

The theorem is then that the following are equivalent:

- $p$ is a
**strict**Conduché functor. - $p$ is exponentiable in the 1-category $Cat$.
- $p$ is exponentiable in the strict 2-category $Cat$.

because strict and weak Conduché functors are being distinguished in this article.

- $p$ is a

- Discussion Type
- discussion topic2-category
- Category Latest Changes
- Started by Urs
- Comments 34
- Last comment by varkor
- Last Active 7 days ago

promted by demand from my Basic-Course-On-Category-Theory-Students I expanded the entry 2-category:

mentioned more relations to other concepts in the Idea-section;

added an Examples-section with a bunch of (classes of) examples;

added a list of references. Please add more if you can think of more!

- Discussion Type
- discussion topicdouble category
- Category Latest Changes
- Started by John Baez
- Comments 27
- Last comment by varkor
- Last Active 7 days ago

- I added more info on pseudo double categories and double bicategories to double category. I also simplified the picture of a square, which had been bristling with scary unnecessary detail. There's a slight blemish in the left vertical arrow, which I can't see how to fix.

- Discussion Type
- discussion topicsynthetic differential geometry
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Dmitri Pavlov
- Last Active 7 days ago

I worked on synthetic differential geometry:

I rearranged slightly and then expanded the "Idea" section, trying to give a more comprehensive discussion and more links to related entries. Also added more (and briefly commented) references. Much more about references can probably be said, I have only a vague idea of the "prehistory" of the subject, before it became enshrined in the textbooks by Kock, Lavendhomme and Moerdijk-Reyes.

Also, does anyone have an electronic copy of that famous 1967 lecture by Lawvere on "categorical dynamics"? It would be nice to have an entry on that, as it seems to be a most visionary and influential text. If I understand right it gave birth to topos theory, to synthetic differential geometry and all that just as a spin-off of a more ambitious program to formalize physics. If I am not mistaken, we are currently at a point where finally also that last bit is finding a full implmenetation as a research program.

- Discussion Type
- discussion topicrelative adjoint functor
- Category Latest Changes
- Started by Peter Heinig
- Comments 5
- Last comment by RodMcGuire
- Last Active Aug 6th 2024

Made some some small improvements (ordering of sections, note on how the definition defaults to the usual definition of adjoints, fixing broken link in the references, etc) in relative adjoint functor.

- Discussion Type
- discussion topicdouble category of algebras
- Category Latest Changes
- Started by Mike Shulman
- Comments 4
- Last comment by varkor
- Last Active Aug 6th 2024

New page: double category of algebras.

- Discussion Type
- discussion topiccohomotopy
- Category Latest Changes
- Started by Urs
- Comments 45
- Last comment by perezl.alonso
- Last Active Aug 6th 2024

At the old entry

*cohomotopy*used to be a section on how it may be thought of as a special case of non-abelian cohomology. While I (still) think this is an excellent point to highlight, re-reading this old paragraph now made me feel that it was rather clumsily expressed. Therefore I have rewritten (and shortened) it, now the third paragraph of the Idea-section.(We had had long discussion about this entry back in the days, but it must have been before we switched to nForum discussion, because on the nForum there seems to be no trace of it.)

- Discussion Type
- discussion topicfibration of quasi-categories
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Hurkyl
- Last Active Aug 6th 2024

am starting fibrations of quasi-categories

- Discussion Type
- discussion topicspherical function
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Aug 6th 2024

- Discussion Type
- discussion topicmapping class group
- Category Latest Changes
- Started by perezl.alonso
- Comments 1
- Last comment by perezl.alonso
- Last Active Aug 6th 2024

Is there a particular reference where these (or rather, their super analogs) are computed for super Riemann surfaces?

- Discussion Type
- discussion topicspatial locale
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 17
- Last comment by Dmitri Pavlov
- Last Active Aug 5th 2024

- Discussion Type
- discussion topicsemidirect product group
- Category Latest Changes
- Started by nLab edit announcer
- Comments 11
- Last comment by nLab edit announcer
- Last Active Aug 5th 2024

- Discussion Type
- discussion topicinequality space
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by nLab edit announcer
- Last Active Aug 5th 2024

- Discussion Type
- discussion topicmodular theory
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Tom Mainiero
- Last Active Aug 5th 2024

- Discussion Type
- discussion topicenveloping von Neumann algebra
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by Tom Mainiero
- Last Active Aug 4th 2024

- Discussion Type
- discussion topicprealgebra real number
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active Aug 4th 2024

- Discussion Type
- discussion topicepimorphism
- Category Latest Changes
- Started by Urs
- Comments 16
- Last comment by nLab edit announcer
- Last Active Aug 4th 2024

felt like adding a handful of basic properties to epimorphism

- Discussion Type
- discussion topicSurj
- Category Latest Changes
- Started by David_Corfield
- Comments 8
- Last comment by varkor
- Last Active Aug 4th 2024

- Discussion Type
- discussion topicDirac manifold
- Category Latest Changes
- Started by perezl.alonso
- Comments 1
- Last comment by perezl.alonso
- Last Active Aug 4th 2024

pointer

- Saskia Demulder, Thomas Raml.
*Poisson-Lie T-duality defects and target space fusion*(2022). (arXiv:2208.04662).

- Saskia Demulder, Thomas Raml.

- Discussion Type
- discussion topicconformal invariance and scale invariance
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by zskoda
- Last Active Aug 3rd 2024

added pointer to:

- Martin Lüscher, Gerhard Mack:
*Global Conformal Invariance in Quantum Field Theory*, Comm. Math. Phys.**41**3 (1975) 203-234 [doi:10.1007/BF01608988, jstor:cmp/1103898909, inspire:90687]

- Martin Lüscher, Gerhard Mack:

- Discussion Type
- discussion topicdistributivity pullback
- Category Latest Changes
- Started by Mike Shulman
- Comments 7
- Last comment by varkor
- Last Active Aug 3rd 2024

- Discussion Type
- discussion topiclocally cartesian closed category
- Category Latest Changes
- Started by Todd_Trimble
- Comments 5
- Last comment by nLab edit announcer
- Last Active Aug 3rd 2024

In locally cartesian closed category, I wrote out an explicit proof that pullback functors $f^\ast \colon C/Y \to C/X$ between slices preserve exponentials (so that Frobenius reciprocity is satisfied).

- Discussion Type
- discussion topicD-brane
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by perezl.alonso
- Last Active Aug 3rd 2024

- Discussion Type
- discussion topicfactorization algebra
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by zskoda
- Last Active Aug 2nd 2024

added the full definition to factorization algebra

- Discussion Type
- discussion topicfull image
- Category Latest Changes
- Started by varkor
- Comments 3
- Last comment by J-B Vienney
- Last Active Aug 2nd 2024