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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality education elliptic-cohomology enriched fibration foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity 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 infinity integration integration-theory k-theory lie-theory limit limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monads monoid monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics planar 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 science set set-theory sheaf 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 type type-theory universal variational-calculus

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- Discussion Type
- discussion topicpro-object
- Category Latest Changes
- Started by Tim_Porter
- Comments 13
- Last comment by Jem Lord
- Last Active Nov 5th 2020

I understood that the old terminology was ’projective system’, and ’projective limit’ refereed to the limit of a projective system. Can anyone confirm that? if I am right the present entry is slightly incorrect, but this needs checking first before changing it.

- Discussion Type
- discussion topicL-complete module
- Category Latest Changes
- Started by nLab edit announcer
- Comments 4
- Last comment by DavidRoberts
- Last Active Nov 5th 2020

- Discussion Type
- discussion topicAndrew Salch
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 4th 2020

brief

`category:people`

-entry for hyperlinking references at*L-complete module*

- Discussion Type
- discussion topicMcKay correspondence
- Category Latest Changes
- Started by David_Corfield
- Comments 10
- Last comment by Urs
- Last Active Nov 4th 2020

I added references to John Baez’s two blog posts on The Geometric McKay Correspondence, Part I, Part II.

I hadn’t realised the length of legs in the Dynkin diagrams corresponds to the stabilizer order on vertices, edges, faces in the corresponding Platonic solid. So 2,3,5 for $E_8$ and the icosahedron.

- Discussion Type
- discussion topicLocal Quantum Physics -- Fields, Particles, Algebras
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 4th 2020

- Discussion Type
- discussion topiclocally presentable category
- Category Latest Changes
- Started by Tobias Fritz
- Comments 9
- Last comment by John Baez
- Last Active Nov 3rd 2020

- In the definition, the article states "every object in C is a small object (which follows from 2 and 3)". The bracketed remark doesn't seem quite right to me, since neither 2 nor 3 talk about smallness of objects. Presumably this should better be phrased as in A.1.1 of HTT, "assuming 3, this is equivalent to the assertion that every object in S is small".

Am I right? I don't (yet) feel confident enough with my category theory to change this single-handedly.

- Discussion Type
- discussion topicsalamander lemma
- Category Latest Changes
- Started by Todd_Trimble
- Comments 35
- Last comment by nLab edit announcer
- Last Active Nov 3rd 2020

Under definition 1 of salamander lemma, I fixed a mistake in the definition of $A_\Box$ where there was a direct sum of two submodules, where there needed to be a sum (i.e., join) instead.

- Discussion Type
- discussion topicpreimage
- Category Latest Changes
- Started by Peter Heinig
- Comments 7
- Last comment by nLab edit announcer
- Last Active Nov 3rd 2020

Made a few additions to preimage. Added missing word; added a brief mention of the widely-known general reason for the good preservation-properties of this endofunctor.

The mention of these properties had already been there in preimage, but a reason was still missing. My parenthetical remark should perhaps be expanded and harmonized with existing relevant material on the nLab ($\forall_f$ and $\exists_f$ are already well-documented on some pages), but this requires more care than I can apply to it today. Intend to return to the remark before long.

- Discussion Type
- discussion topicYimin Yang
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 3rd 2020

brief

`category:people`

-entry for hyperlinking references at*equivariant K-theory*

- Discussion Type
- discussion topicEuler's constant
- Category Latest Changes
- Started by nLab edit announcer
- Comments 5
- Last comment by Richard Williamson
- Last Active Nov 3rd 2020

- Discussion Type
- discussion topicEuler number
- Category Latest Changes
- Started by Richard Williamson
- Comments 1
- Last comment by Richard Williamson
- Last Active Nov 3rd 2020

Adding redirect for Euler’s constant.

- Discussion Type
- discussion topicBTZ black hole
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Urs
- Last Active Nov 3rd 2020

- Discussion Type
- discussion topicD1-D3 intersections in AdS2-CFT1 -- references
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Nov 3rd 2020

- Discussion Type
- discussion topicorbifold cohomology
- Category Latest Changes
- Started by Urs
- Comments 91
- Last comment by Urs
- Last Active Nov 2nd 2020

added references by Pronk-Scull and by Schwede, and wrote an Idea-section that tries to highlight the expected relation to global equivariant homotopy theory. Right now it reads like so:

On general grounds, since orbifolds $\mathcal{G}$ are special cases of stacks, there is an evident definition of cohomology of orbifolds, given by forming (stable) homotopy groups of derived hom-spaces

$H^\bullet(\mathcal{G}, E) \;\coloneqq\; \pi_\bullet \mathbf{H}( \mathcal{G}, E )$into any desired coefficient ∞-stack (or sheaf of spectra) $E$.

More specifically, often one is interested in viewing orbifold cohomology as a variant of Bredon equivariant cohomology, based on the idea that the cohomology of a global homotopy quotient orbifold

$\mathcal{G} \;\simeq\; X \sslash G \phantom{AAAA} (1)$for a given $G$-action on some manifold $X$, should coincide with the $G$-equivariant cohomology of $X$. However, such an identification (1) is not unique: For $G \subset K$ any closed subgroup, we have

$X \sslash G \;\simeq\; \big( X \times_G K\big) \sslash K \,.$This means that if one is to regard orbifold cohomology as a variant of equivariant cohomology, then one needs to work “globally” in terms of

*global equivariant homotopy theory*, where one considers equivariance with respect to “all compact Lie groups at once”, in a suitable sense.Concretely, in global equivariant homotopy theory the plain orbit category $Orb_G$ of $G$-equivariant Bredon cohomology is replaced by the global orbit category $Orb_{glb}$ whose objects are the delooping stacks $\mathbf{B}G \coloneqq \ast\sslash G$, and then any orbifold $\mathcal{G}$ becomes an (∞,1)-presheaf $y \mathcal{G}$ over $Orb_{glb}$ by the evident “external Yoneda embedding”

$y \mathcal{G} \;\coloneqq\; \mathbf{H}( \mathbf{B}G, \mathcal{G} ) \,.$More generally, this makes sense for $\mathcal{G}$ any orbispace. In fact, as a construction of an (∞,1)-presheaf on $Orb_{glb}$ it makes sense for $\mathcal{G}$ any ∞-stack, but supposedly precisely if $\mathcal{G}$ is an orbispace among all ∞-stacks does the cohomology of $y \mathcal{G}$ in the sense of global equivariant homotopy theory coincide the cohomology of $\mathcal{G}$ in the intended sense of ∞-stacks, in particular reproducing the intended sense of orbifold cohomology.

At least for topological orbifolds this is indicated in (Schwede 17, Introduction, Schwede 18, p. ix-x, see also Pronk-Scull 07)

- Discussion Type
- discussion topicprobability theory
- Category Latest Changes
- Started by David_Corfield
- Comments 4
- Last comment by Urs
- Last Active Nov 2nd 2020

Strangely, we don’t seem to have an nForum discussion for probability theory.

I added a reference there to

- Chris Heunen, Ohad Kammar, Sam Staton, Hongseok Yang,
*A Convenient Category for Higher-Order Probability Theory*, (arXiv:1701.02547)

It replaces the category of measurable spaces, which isn’t cartesian closed, with the category of quasi-Borel spaces, which is. As they point out in section IX, what they’re doing is working with concrete sheaves on an established category of spaces, rather like the move to diffeological spaces.

[Given the interest in topology around these parts at the moment, we hear of ’C-spaces’ as generalized topological spaces arising from a similar sheaf construction in C. Xu and M. Escardo, “A constructive model of uniform continuity,” in Proc. TLCA, 2013.]

- Chris Heunen, Ohad Kammar, Sam Staton, Hongseok Yang,

- Discussion Type
- discussion topicmeasure theory
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Nov 2nd 2020

added publication data to:

- Simon Henry,
*Measure theory over boolean toposes*, Mathematical Proceedings of the Cambridge Philosophical Society Volume 163 Issue 1, 2016 (arXiv:1411.1605, doi:10.1017/S0305004116000700)

- Simon Henry,

- Discussion Type
- discussion topicA Survey of Elliptic Cohomology
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 1st 2020

- Discussion Type
- discussion topicfive lemma
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by nLab edit announcer
- Last Active Nov 1st 2020

touched

*five lemma*

- Discussion Type
- discussion topichomotopy coherent diagram
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Tim_Porter
- Last Active Oct 31st 2020

I have added to homotopy coherent diagram and to model structure on algebras over an operad the fact that one can describe homotopy coherent diagrams as algebras over the Boardman-Vogt resolution of some operad. Vogt’s rectification theorem is then a special case of the general Berger-Moerdijk result.

In the course of this I reorganized and expanded homtopy coherent diagram a bit. It still needs to be polished a bit.

- Discussion Type
- discussion topicLoop Groups and Twisted K-Theory
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Oct 31st 2020

touched the first paragraph, making hyperlinked pointers to

*compact Lie groups*,*positive energy representations*and the*Verlinde ring*.Then I replaced “twisted euqivariant K-theoyry” with

*twisted ad-equivariant K-theory*and will give this its own entry now…

- Discussion Type
- discussion topictwisted ad-equivariant K-theory
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Oct 31st 2020

- Discussion Type
- discussion topicpositive energy representation
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Oct 31st 2020

have split-off a stub

*positive energy representation*from*loop group*

- Discussion Type
- discussion topicVerlinde ring
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 31st 2020

added complete referencing of FHT and cross-link with

*twisted ad-equivariant K-theory*

- Discussion Type
- discussion topicModels for Smooth Infinitesimal Analysis
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Oct 31st 2020

I am filling in more details (definitions, properties) about the various toposes at Models for Smooth Infinitesimal Analysis

- Discussion Type
- discussion topicsynthetic differential geometry
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Oct 31st 2020

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 topicsheaf and topos theory
- Category Latest Changes
- Started by DavidRoberts
- Comments 7
- Last comment by Urs
- Last Active Oct 31st 2020

Moved talk by McLarty to the History section of the refernces from functorial geometry

- Discussion Type
- discussion topicStructured Spaces
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 31st 2020

added DOI to:

- Monique Hakim,
*Topos annelés et schémas relatifs*, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64, Springer, Berlin, New York (1972) (doi:10.1007/978-3-662-59155-0)

- Monique Hakim,

- Discussion Type
- discussion topicPursuing Stacks
- Category Latest Changes
- Started by Tim_Porter
- Comments 11
- Last comment by Tim_Porter
- Last Active Oct 31st 2020

@Todd. Thanks for correcting my atrocious English!

Does anyone have any ideas as to how we could provide a bit more for this entry?

- Discussion Type
- discussion topicAwodey's proposal
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Urs
- Last Active Oct 31st 2020

- Discussion Type
- discussion topicstring diagram
- Category Latest Changes
- Started by nLab edit announcer
- Comments 33
- Last comment by nLab edit announcer
- Last Active Oct 30th 2020