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- Discussion Type
- discussion topicfunction extensionality
- Category Latest Changes
- Started by Urs
- Comments 47
- Last comment by nLab edit announcer
- Last Active 3 days ago

created function extensionality

- Discussion Type
- discussion topicinterval type
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by nLab edit announcer
- Last Active 4 days ago

created interval type

- Discussion Type
- discussion topicperfectoid space
- Category Latest Changes
- Started by Urs
- Comments 21
- Last comment by J-B Vienney
- Last Active 4 days ago

added some minimum content to

*perfectoid space*. Also added a corresponding paragraph to*function field analogy*.

- Discussion Type
- discussion topicreconstruction theorem
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by varkor
- Last Active 4 days ago

added hyperlinks to some more of the keywords (such as

*Giraud theorem*).I see lots of room to clean up this old entry, but will leave it at that for the moment.

- Discussion Type
- discussion topicPing Gao
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 4 days ago

- Discussion Type
- discussion topictwo-variable adjunction
- Category Latest Changes
- Started by Mike Shulman
- Comments 5
- Last comment by varkor
- Last Active 4 days ago

I have added a reference to Cheng-Gurski-Riehl to two-variable adjunction, and some comments about the cyclic action.

- Discussion Type
- discussion topicwormhole
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 4 days ago

added pointer to the original article

- Albert Einstein, Nathan Rosen,
*The Particle Problem in the General Theory of Relativity*, Phys. Rev.**48**73 (1935) (doi:10.1103/PhysRev.48.73)

and made

*Einstein-Rosen bridge*a redirect- Albert Einstein, Nathan Rosen,

- Discussion Type
- discussion topictype of propositions
- Category Latest Changes
- Started by Mike Shulman
- Comments 7
- Last comment by nLab edit announcer
- Last Active 4 days ago

Is there any point to having both type of propositions and Prop? The analogous page-name Type is a redirect to type of types.

- Discussion Type
- discussion topicelliptic cohomology -- references
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active 4 days ago

a bare list of references, to be

`!include`

-ed into relevant entries, such as at*elliptic cohomology*, but also at*equivariant elliptic cohomology*,*elliptic genus*,*Witten genus*etc.(in an attempt to clean up and harmonize the referencing across all these entries – still some way to go towards that goal, but it should be a start)

- Discussion Type
- discussion topicKiran Luecke
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 4 days ago

- Discussion Type
- discussion topicTate K-theory
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active 4 days ago

am finally splitting this off from

*Tate curve*

- Discussion Type
- discussion topicEmerton-Gee stack
- Category Latest Changes
- Started by Anton Hilado
- Comments 4
- Last comment by Anton Hilado
- Last Active 4 days ago

- Discussion Type
- discussion topicDavid Savitt
- Category Latest Changes
- Started by Anton Hilado
- Comments 1
- Last comment by Anton Hilado
- Last Active 4 days ago

- Discussion Type
- discussion topicGalois representation
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Anton Hilado
- Last Active 4 days ago

added pointer to

- Tom Lovering,
*Etale cohomology and Galois Representations*, 2012 (pdf)

for review of how Galois representations are arithmetic incarnations of local systems/flat connections. Added the same also to

*local system*and maybe elsewhere.- Tom Lovering,

- Discussion Type
- discussion topicjudgmental equality
- Category Latest Changes
- Started by David_Corfield
- Comments 3
- Last comment by nLab edit announcer
- Last Active 4 days ago

Added a comment linking it to definitional equality.

- Discussion Type
- discussion topicprismatic cohomology
- Category Latest Changes
- Started by DavidRoberts
- Comments 36
- Last comment by Anton Hilado
- Last Active 5 days ago

Stub to record today’s reference

- Bhargav Bhatt, Peter Scholze,
*Prisms and Prismatic Cohomology*, preprint (2019) arXiv:1905.08229

- Bhargav Bhatt, Peter Scholze,

- Discussion Type
- discussion topicmodularity theorem
- Category Latest Changes
- Started by nLab edit announcer
- Comments 8
- Last comment by Anton Hilado
- Last Active 5 days ago

- Discussion Type
- discussion topicMünchhausen trilemma
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active 5 days ago

- Discussion Type
- discussion topicepistemology
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Guest
- Last Active 5 days ago

- Discussion Type
- discussion topicSabine Hossenfelder
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by nLab edit announcer
- Last Active 5 days ago

- Discussion Type
- discussion topicempiricism
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by zskoda
- Last Active 5 days ago

- Discussion Type
- discussion topiccontext
- Category Latest Changes
- Started by Urs
- Comments 30
- Last comment by nLab edit announcer
- Last Active 5 days ago

[ forwarding old discussion that used to be at context ]

The following discussion was initiated by a previous version of the above entry which referred to “cartesian multicategories” rather than finitely complete categories.

+–{: .query} Mike: What is a cartesian multicategory, and how do I interpret the theory of groups in one? I can guess what it would mean for a multicategory to have finite products. But if I interpret the multiplication as a morphism $G\times G\to G$, then I’m not using the multicategory structure, so we might as well just be in a category with finite products. And if I interpret the multiplication as a multimap $(G;G)\to G$, then I don’t know how to interpret the axiom of inverses, since there is no ’diagonal’ $G\to (G;G)$ or ’projection’ $G\to ()$.

*Toby*: I'm not sure why I generalised to cartesian multicategories, but it is a nontrivial generalisation. (Perhaps I was planning to show, as an example, how the category of contexts of the canonical language of a multicategory becomes a monoidal category or something, but that doesn't seem very useful. Maybe I was just doing unnecessary generality, but of course it's not the absolutely most general situation either.)Anyway … you make a multicategory cartesian much as you might make a monoidal category cartesian by equipping it with appropriate diagonal and projection maps. The problem is that, while $G \to G \otimes G$ and $G \to 1$ make sense in a monoidal category, they don't make sense in a multicategory. But you fix this by filtering through Yoneda.

So a

**cartesian multicategory**is a multicategory equipped with, for each object $G$ and object $X$, a function $\check{G}^*_X\colon hom(G;X) \leftarrow hom(G,G;X)$ and a function $\hat{G}^*_X\colon hom(G;X) \leftarrow hom(;X)$. (H'm, my commas and semicolons are the opposite of yours; no matter.) Then these are subject to various coherence requirements that should be obvious.Mike: Okay, I see. Though I’m guessing you wanted those natural transformations to go the other way. Are there any naturally occurring examples of cartesian multicategories that are not cartesian monoidal categories? Even if there are, I’m inclined to regard the concept as esoteric enough that it would be clearer to just say “category with finite products” in this introductory article.

*Toby*: Ah, the curse of contravariance! Going over the whole introduction again, I think that I understand why I mentioned multicategories, which is that a context like $a\colon G, b\colon G$ is more naturally interpreted as a list $(G, G)$ of objects than as a single object $G \times G$. But if we were really to go in that direction, then we'd also want the context $a\colon G, b\colon G, (a b)^2 = a^2 b^2$ to be interpreted as a list in its own right rather than an actual subobject of $G \times G$, and that's going a bit far … farther than I understand clearly, in any case. So in fact I let the category be finitely complete so that we could form that subobject (referred to only via the link to internal logic, of course).Mike: True. Is a one-object cartesian multicategory the same as a Lawvere theory, aka an operad relative to the theory of categories with finite products? If so, then perhaps the relevant place to work is a multicategory relative to the theory of lex categories? Can that be generalized to stronger logics?

*Toby*: Yes, that seems to be right, that Lawvere theories are equivalent to one-object cartesian multicategories (cartesian multimonoids? cartesian operads?). So this should work.Of course, one thing that contexts do is to form an honest category even if you start with a multicategory. So here we're trying to go backwards and see what bare-bones starting point could lead to the same category of contexts of the equational theory of a group. =–

- Discussion Type
- discussion topicskyrmion
- Category Latest Changes
- Started by Urs
- Comments 50
- Last comment by Urs
- Last Active 5 days ago

- Discussion Type
- discussion topicRoger Howe
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 5 days ago

- Discussion Type
- discussion topicterminal coalgebra for an endofunctor
- Category Latest Changes
- Started by varkor
- Comments 4
- Last comment by nLab edit announcer
- Last Active 5 days ago

- Discussion Type
- discussion topicreductive dual pair
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 5 days ago

- Discussion Type
- discussion topicAllan Merino
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 5 days ago

- Discussion Type
- discussion topicspin representation
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by nLab edit announcer
- Last Active 5 days ago

have noted down the basic properties of the irreducible representations of the Lorentzian spin group, at

*spin representation – Properties*.

- Discussion Type
- discussion topicEilenberg-Moore category
- Category Latest Changes
- Started by FinnLawler
- Comments 33
- Last comment by varkor
- Last Active 5 days ago

I’ve added to Eilenberg-Moore category an explicit definition of EM objects in a 2-category and some other universal properties of EM categories, including Linton’s construction of the EM category as a subcategory of the presheaves on the Kleisli category.

Question: can anyone tell me what Street–Walters mean when they say that this construction (and their generalised one, in a 2-category with a Yoneda structure) exhibits the EM category as the ‘category of sheaves for a certain generalised topology on’ the Kleisli category?

- Discussion Type
- discussion topiclimit
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by varkor
- Last Active 5 days ago

I toiuched the formatting and the hyperlinking of the paragraphs on compatibility of limits with other universal constructions.

Merged the previous tiny subsections on this to a single one, now

*Compatibility with universal constructions*.added the hyperlink to the stand-alone entry

*adjoints preserve (co-)limits*.Will create an analogous stand-alone entry for

*limits commute with limits*.

- Discussion Type
- discussion topicfree loop space of classifying space
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by Urs
- Last Active 5 days ago

am recording an actual proof that

$\mathcal{L} \big( \overline{W}\mathcal{G} \big) \;\; \simeq \;\; \mathcal{G}_{ad} \sslash \mathcal{G}$I expected that a proof for this folklore theorem would be citeable from the literature, but maybe not quite. This MO reply points to Lemma 9.1 in arXiv:0811.0771, which has the idea (in topological spaces), but doesn’t explicitly verify all ingredients. I have tried to make it fully explicit (in simplicial sets).

- Discussion Type
- discussion topicSullivan model of loop space
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Urs
- Last Active 5 days ago

I have created

*Sullivan model of free loop space*with the formula and pointers to the literature.

- Discussion Type
- discussion topicAchim Krause
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 5 days ago

- Discussion Type
- discussion topicJean-Marie Maranda
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active 5 days ago

brief

`category:people`

-entry for hyperlinking references at*enriched category*.Does anyone know a reflection of this author on the web? Is this maybe the Jean Maranda here?

- Discussion Type
- discussion topicnuclear magnetic resonance
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 5 days ago

- Discussion Type
- discussion topicAsif Equbal
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 5 days ago

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

- Discussion Type
- discussion topicrelative Langlands program
- Category Latest Changes
- Started by Anton Hilado
- Comments 2
- Last comment by Anton Hilado
- Last Active 6 days ago

- Discussion Type
- discussion topicquantum logic gate
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active 6 days ago

a stub, to make links work

(This used to be a stub “quantum circuit” which I just quasi-duplicated at a more extensive entry

*quantum circuit diagram*. But since*quantum gate*was already redirecting here – which is how I discovered/remembered that this entry exists – no harm is done by making that it’s new title.)

- Discussion Type
- discussion topictopological cyclic homology
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Anton Hilado
- Last Active 6 days ago

- Discussion Type
- discussion topicHochschild cohomology
- Category Latest Changes
- Started by Urs
- Comments 101
- Last comment by Anton Hilado
- Last Active 6 days ago

I am in the process of preparing a piece on Hochschild cohomology

while the Lab is down, I'll abuse the forum here for posting my source. Probably not well suited for reading it, but just so the effort is not wasted should my notebook get run over by a bus. That could happen, as the buses here in Sheffield go on the wrong side of the road.

See followup comment...

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

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

- Discussion Type
- discussion topicreductionism
- Category Latest Changes
- Started by nLab edit announcer
- Comments 4
- Last comment by Urs
- Last Active 6 days ago

- Discussion Type
- discussion topicpluralism
- Category Latest Changes
- Started by nLab edit announcer
- Comments 8
- Last comment by Urs
- Last Active 6 days ago

- Discussion Type
- discussion topicskeleton
- Category Latest Changes
- Started by Alec Rhea
- Comments 18
- Last comment by Urs
- Last Active 6 days ago

- Discussion Type
- discussion topicneutral non-well-founded set theory
- Category Latest Changes
- Started by nLab edit announcer
- Comments 5
- Last comment by Urs
- Last Active 6 days ago

starting page on the most general notion of set theories without the axiom of foundation and Mostowski’s principle

Anonymous

- Discussion Type
- discussion topicUrs Schreiber
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active 6 days ago

- Discussion Type
- discussion topicdecategorification
- Category Latest Changes
- Started by bwebster
- Comments 7
- Last comment by nLab edit announcer
- Last Active 6 days ago

I thought it might be good if somebody explained the relationship between decategorification and extended TQFT. My understanding from talking to physicists is that you should multiply your space by $S^1$; is this right in a mathematical sense? I've added a query box asking roughly the same thing.

Also, I attempted to add a sidebar, mostly just to try it out, and somehow it's not rendering right. Anyone want to explain what I did wrong?

- Discussion Type
- discussion topicepistemic modal logic
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 6 days ago

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

starting article on epistemology of mathematics which was previously linked on philosophy of mathematics but went to no article.

Anonymous

- Discussion Type
- discussion topicphilosophy of physics
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 7 days ago

Adding reference

- Valia Allori (2017).
*A New Argument for the Nomological Interpretation of the Wave Function: The Galilean Group and the Classical Limit of Nonrelativistic Quantum Mechanics.*(philsci-archive:14023)

Anonymous

- Valia Allori (2017).

- Discussion Type
- discussion topicGeorg Hegel
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active 7 days ago

- Discussion Type
- discussion topicphilosophy of mathematics
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 7 days ago

- Discussion Type
- discussion topichigher-level foundations
- Category Latest Changes
- Started by Mike Shulman
- Comments 2
- Last comment by nLab edit announcer
- Last Active 7 days ago

- Discussion Type
- discussion topicHomePage
- Category Latest Changes
- Started by Urs
- Comments 78
- Last comment by Guest
- Last Active 7 days ago

I have added at

*HomePage*in the section*Discussion*a new sentence with a new link:If you do contribute to the nLab, you are strongly encouraged to similarly drop a short note there about what you have done – or maybe just about what you plan to do or even what you would like others to do. See Welcome to the nForum (nlabmeta) for more information.

I had completly forgotton about that page

*Welcome to the nForum (nlabmeta)*. I re-doscivered it only after my recent related comment here.

- Discussion Type
- discussion topiccyclotomic spectrum
- Category Latest Changes
- Started by Urs
- Comments 25
- Last comment by Urs
- Last Active 7 days ago

Started a bare minimum at

*cyclotomic spectrum*. So far it’s essentially just a pointer to the canonical reference by Blumberg-Mandell. (Thomas Nikolaus and Peter Scholze have a new foundation of the theory in preparation for which notes however are not public yet, also Clark Barwick has something in preparation, for which you may find notes by looking at his website and being clever in deducing hidden URLs, he says.)For the moment the only fact that I have actually recorded in the entry is a fact that is trivial for anyone familiar with the theory,but which looks interesting from the point of view of the story at

*Generalized cohomology of M2/M5-branes (schreiber)*: the global equivariant sphere spectrum for all the cyclic groups (all the A-type finite groups in the ADE classification…) carries canonical cyclotomic structure and as such is the tensor unit among cyclotomic spectra.Apart from mentioning this, I have added brief cross-links with

*topological cyclic homology*,*equivariant sphere spectrum*,*cyclic group*and maybe other entries.

- Discussion Type
- discussion topiccyclic loop space
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active 7 days ago

I need a word for the homotopy quotient $(\mathcal{L}X)/S^1$ of free loop spaces $\mathcal{L}X$ by their canonical circle action. It seems that the only term in use with respect to this is “twisted loop space”, which however usually refers just to the constant loops $(\mathcal{L}_{const}X)//S^1$. Since under nice conditions the derived functions on the $\mathcal{L}Spec(A)/S^1$ is the cyclic homology complex of $A$, I suggest that a good name is “cyclic loop space”. I made a quick note at

*cyclic loop space*, just to fix and disambiguate terminology.

- Discussion Type
- discussion topicPontryagin-Thom construction -- references
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by nLab edit announcer
- Last Active 7 days ago

this is a bare list of references, to be

`!include`

-ed into relevant entries(various parts of this list have been contained at

*Cohomotopy*,*Cohomotopy charge map*,*Thom’s theorem*,*Pontryagin-Thom construction*and elsewhere; this here now to ease harmonizing/completing these lists)

- Discussion Type
- discussion topicset theory
- Category Latest Changes
- Started by TobyBartels
- Comments 12
- Last comment by nLab edit announcer
- Last Active Nov 28th 2022

There is discussion about material and structural set theories at set theory, spilt over from the Café.