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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• 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.

• brief category:people-entry for hyperlinking references

• brief category:people-entry for hyperlinking references

• Created page.

• Created page.

• 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.

Anonymous

• starting article on the Munchhaüsen trilemma and its relevance to the foundations of mathematics

Anonymous

Anonymous

• person stub

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

• [ 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. =–

• just a stub for the moment, in order to make links work

• brief category:people-entry for hyperlinking references

• Fix duplicate redirect.

• starting something…

• brief category:people-entry for hyperlinking references

• 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?

• 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).

• brief category:people-entry for hyperlinking references

• 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?

• a stub entry, for the moment just to make links work

• brief category:people-entry for hyperlinking references

• Added an earlier reference for semifunctors.

• Created page. Will fill out more later.

• 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.)

• added at TC some references on computing THH for cases like $ko$ and $tmf$, here

• 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...

• starting page on reductionism to contrast with pluralism

Anonymous

• starting article on pluralism in philosophy and the foundations of mathematics

Anonymous

• Test edit, I can’t seem to get the page to accept the larger edit I’ve made.

• Just to say that I will be on family vacation until $\sim$ Aug 7.

It looks like we’ll have no WiFi on that house boat, so that I’ll not be able to reply here as usual. But I may try to hack myself into the Matrix via phone, from time to time.

• 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?

• starting stub article on epistemic modal logic. Not an expert on modal logics in general, but I think David Corfield might know more about this topic.

Anonymous

• 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

• typo

i11e

• Created with name “higher-level foundations” as suggested by Ulrik, but with “higher foundations” as a redirect.

• 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.

• 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.

• 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.

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