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- Discussion Type
- discussion topicAdded link to Street's handwritten 1988 note on the Gray tensor product of 2cats
- Category Latest Changes
- Started by Keith Harbaugh
- Comments 5
- Last comment by DavidRoberts
- Last Active 17 hours ago

Added the following to the page on the Gray tensor product:

The Gray tensor product as the left Kan extension of a tensor product on the full subcategory Cu of 2Cat is on page 16 of

- Ross Street, “Gray’s tensor product of 2-categories”, 22-page handwritten note, (1988) (PDF at Macquarie)

Since I’m really new at this, feel very free to give advice or corrections, thanks, Keith

- Discussion Type
- discussion topicmonadic decomposition
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active 1 day ago

New page: monadic decomposition.

- Discussion Type
- discussion topichomotopy Kan extension
- Category Latest Changes
- Started by Urs
- Comments 21
- Last comment by Mike Shulman
- Last Active 1 day ago

created homotopy Kan extension

this is to go along with the discussion at limit in a quasi-category that I am currently working on.

- Discussion Type
- discussion topicminimal dg-model of 4-sphere over 3-sphere
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active 1 day ago

I have corrected and expanded my note (at

$\array{ S^4 \\ \downarrow \\ S^3 } \,,\phantom{AA} \array{ S^4//S^1 \\ \downarrow \\ S^3 } \,,\phantom{AA} \array{ S^0 \\ \downarrow \\ S^3 }$*4-sphere*: here) of the result of Roig-Saralegi 00, p. 2 on minimal rational dg-models of the following maps over $S^3$induced from the “suspended Hopf action” of $S^1$ on $S^4$.

My aim in extracting this is to rename the generators given in Roig-Saralegi 00, p. 2 such as to make their degrees and their pattern more manifest. I hope I got it right now:

$\array{ \text{fibration} & \array{\text{vector space underlying} \\ \text{minimal dg-model}} & \array{ \text{differential on} \\ \text{minimal dg-model} } \\ \array{ S^4 \\ \downarrow \\ S^3 } & Sym^\bullet \langle h_3\rangle \otimes \left\langle \underset{deg = 2p}{ \underbrace{ \omega_{2p} }}, \underset{deg = 2p + 4}{ \underbrace{ f_{2p + 4} }} \,\vert\, p \in \mathbb{N} \right\rangle & d \colon \left\{ \begin{aligned} \omega_0 & \mapsto 0 \\ \omega_{2p+2} &\mapsto h_3 \wedge \omega_{2p} \\ f_4 & \mapsto 0 \\ f_{2p+6} & \mapsto h_3 \wedge f_{2p + 4} \end{aligned} \right. \\ \array{ S^0 \\ \downarrow \\ S^3 } & Sym^\bullet \langle h_3\rangle \otimes \left\langle \underset{deg = 2p}{ \underbrace{ \omega_{2p} }}, \underset{ deg = 2p }{ \underbrace{ f_{2p} }} \,\vert\, p \in \mathbb{N} \right\rangle & d \colon \left\{ \begin{aligned} \omega_0 & \mapsto 0 \\ \omega_{2p+2} &\mapsto h_3 \wedge \omega_{2p} \\ f_0 & \mapsto 0 \\ f_{2p+2} &\mapsto h_3 \wedge f_{2p} \end{aligned} \right. \\ \array{ S^4//S^1 \\ \downarrow \\ S^3 } & Sym^\bullet \langle h_3 , f_2 \rangle \otimes \left\langle \underset{deg = 2p}{ \underbrace{ \omega_{2p} }}, \underset{ deg =2p + 4 }{ \underbrace{ f_{2p + 4} }} \,\vert\, p \in \mathbb{N} \right\rangle & d \colon \left\{ \begin{aligned} \omega_0 & \mapsto 0 \\ \omega_{2p+2} &\mapsto h_3 \wedge \omega_{2p} \\ f_2 & \mapsto 0 \\ f_{2p+4} & \mapsto h_3 \wedge f_{2p + 2} \end{aligned} \right. }$

- Discussion Type
- discussion topicGrothendieck construction for model categories
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 2 days ago

I have given

*Grothendieck construction for model categories*its own entry, in order to have a place for recording references. In particular I added pointer to the original references (Roig 94, Stanculescu 12)(There used to be two places in the entry

*Grothendieck construction*where an attempt was made to list the literature on the model category version, but they didn’t coincide and were both inclomplete. So I have replaced them with pointers to the new entry.)

- Discussion Type
- discussion topicPROP
- Category Latest Changes
- Started by fpaugam
- Comments 25
- Last comment by Mike Shulman
- Last Active 3 days ago

- I would like to include something on wheeled properads (or wheeled PROPs) in the nlab. It seems to me that a wheeled prop is something like a symmetric monoidal category with duals for every object generated by one object. Is this right? Is there a place in the litterature where i can find the relation between wheeled properads used by Merkulov and some kinds of symmetric monoidal categories with duality?

Before changing the PROP entry to add this variant, i would like to have a nice reference on this.

- Discussion Type
- discussion topicE8
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active 3 days ago

added statement of and references for some of the homotopy groups of $E_8$ to E8

- Discussion Type
- discussion topiccohomotopy
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active 4 days ago

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 topicPoisson-Lie T-duality
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 6 days ago

started a bare minimum at

*Poisson-Lie T-duality*, for the moment just so as to have a place to record the two original references

- Discussion Type
- discussion topicMassey product
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active 6 days ago

added to

*Massey product*a paragraph on their relation to $A_\infty$-algebras and a bunch of references on that relation

- Discussion Type
- discussion topicCahiers topos
- Category Latest Changes
- Started by Urs
- Comments 39
- Last comment by Sam Staton
- Last Active 6 days ago

created Cahiers topos.

Do I understand correctly that this gadget is named after the

*journal*that Dubuc’s original article appeared in? What a strange idea.

- Discussion Type
- discussion topicmonoidal category
- Category Latest Changes
- Started by Urs
- Comments 46
- Last comment by Richard Williamson
- Last Active Mar 9th 2018

Todd,

when you see this here and have a minute, would you mind having a look at

*monoidal category*to see if you can remove the query-box discussion there and maybe replace it by some crisp statement?Thanks!

- Discussion Type
- discussion topicLimit spaces
- Category Latest Changes
- Started by spitters
- Comments 7
- Last comment by DavidRoberts
- Last Active Mar 7th 2018

Created the bare minimum at limit spaces. I was surprised it was missing.

- Discussion Type
- discussion topicTools for the advancement of objective logic
- Category Latest Changes
- Started by jonsterling
- Comments 10
- Last comment by Mike Shulman
- Last Active Mar 7th 2018

Currently, the page Tools for the advancement of objective logic says that the described paper discusses the “concrete particular”, but this paper does not discuss “concrete particular”.

[For context, Lawvere regards “concrete particular” as a category error, as he has stated elsewhere—basically to Lawvere, the abstract and the concrete are two aspects of a universal/general concept, and not to be discussed at the level of particulars. The passage from the particular to the universal/general is marked by choosing a subclass of observables about the particular as definitive—this generates the abstract general, and thence the concrete generals. The connection between the particulars and the concrete general comes from the fact that each particular can be observed in the ways specified by the abstract general, and that the abstract generals embed as representables into the concrete generals.]

To be precise, the 1994 paper talks about the particular, and then the abstract and concrete general.

However, I haven’t edited the page yet because I believe that in order to keep the nlab internally consistent, one may need to also update the page abstract general, concrete general and concrete particular; however, I am aware of a significant discussion elsewhere in the nforum, and I see that maybe some consensus was already reached there in favor of the notion of the “concrete particular”.

The latter page contains the nlab’s synthesis of these ideas which does not quite match Lawvere’s—which is generally fine, but I think we should be careful about attributing this interpretation to Lawvere himself (which we are in danger of doing in the page on the 1994 paper).

Do you all have any thoughts on how to proceed?

- Discussion Type
- discussion topicprinciple of equivalence
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by David_Corfield
- Last Active Mar 7th 2018

at

*principle of equivalence*I have restructured the Examples-section: added new subsections in “In physics” on gauge transformations and on general covariance (just pointers so far, no text), and then I moved the section that used to be called “In quantum mechanics” to “Examples-In category theory” and renamed it to “In the definition of $\dagger$-categories” (for that is really what these paragraphs discuss, not any notion of equivalence in quantum mechanics, the application of $\dagger$-categories in that context notwithstanding)

- Discussion Type
- discussion topicHenselian pair
- Category Latest Changes
- Started by DavidRoberts
- Comments 7
- Last comment by Tim Campion
- Last Active Mar 6th 2018

Created Henselian pair and linked to it from Henselian ring and Hensel’s lemma. Probably the definition could be given in more generality for topological rings. The existing definition seems to implicitly assume the discrete topology, since in that case, restricted formal power series are just polynomials, and that’s what appears in the definition I copied over from the Stacks Project.

- Discussion Type
- discussion topicpure morphism
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by zskoda
- Last Active Mar 5th 2018

pure morphism (much more to be said, and more references, but no time now)

- Discussion Type
- discussion topicadditive analytic geometry
- Category Latest Changes
- Started by David_Corfield
- Comments 2
- Last comment by David_Corfield
- Last Active Mar 5th 2018

Someone started additive analytic geometry.

- Discussion Type
- discussion topiclax F-adjunction
- Category Latest Changes
- Started by Mike Shulman
- Comments 2
- Last comment by DavidRoberts
- Last Active Mar 4th 2018

Created lax F-natural transformation and lax F-adjunction in order to record the fact that F-categories give an abstract context in which to formulate Johnstone’s definition of fibration in a 2-category.

- Discussion Type
- discussion topicStore Comonad
- Category Latest Changes
- Started by KeithEPeterson
- Comments 3
- Last comment by KeithEPeterson
- Last Active Mar 4th 2018

Added page store+comonad

I mostly just copied and pasted the state+monad page and added a reference.

- Discussion Type
- discussion topicmere proposition
- Category Latest Changes
- Started by Urs
- Comments 15
- Last comment by Mike Shulman
- Last Active Mar 4th 2018

an entry for

*mere proposition*had been missing. Created a minimum, just so as to satisfy links.

- Discussion Type
- discussion topicn-fold category
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by Mike Shulman
- Last Active Mar 3rd 2018

I have edited the old entry

*n-fold category*a little, brought the content into proper order, fixed the link to cat-n-groups and cross-linked with*n-fold complete Segal space*.

- Discussion Type
- discussion topichydrogen line
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Mar 3rd 2018

Detection of the 21cm hydrogen absorption line expected in the CMB has been claimed now. Such a detection is thought to have implications for observational cosmology comparable in relevance to those of the recent gravitational wave detection. I have collected some original articles and reviews here in an otherwise empty entry

*hydrogen line*

- Discussion Type
- discussion topiccompact element
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Todd_Trimble
- Last Active Mar 2nd 2018

created

*compact element*

- Discussion Type
- discussion topicNew Spaces for Mathematics and Physics
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active Mar 2nd 2018

I have started a category:reference page

such as to be able to point to it for reference, e.g. from Kontsevich 15 etc.

- Discussion Type
- discussion topicsober space
- Category Latest Changes
- Started by Urs
- Comments 38
- Last comment by martinescardo
- Last Active Mar 1st 2018

at

*sober space*the only class of examples mentioned are Hausdorff spaces. What’s a good class of non-Hausdorff sober spaces to add to the list?

- Discussion Type
- discussion topicquasitopos
- Category Latest Changes
- Started by Mike Shulman
- Comments 7
- Last comment by Todd_Trimble
- Last Active Mar 1st 2018

Added some remarks, mostly about extensivity and exactness, to quasitopos.

- Discussion Type
- discussion topicComma Double Category
- Category Latest Changes
- Started by maxsnew
- Comments 3
- Last comment by Mike Shulman
- Last Active Feb 27th 2018

I started comma double category. Since I care about equipments more than double categories in general, and because it actually is an instance of a comma object, I made the article mostly about virtual double categories. I wrote down a couple of conjectures about when the comma has units and composites, but haven’t verified them yet and not sure when I will.

- Discussion Type
- discussion topiccolax/lax comma object
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Feb 27th 2018

New page: colax/lax comma object

- Discussion Type
- discussion topicdouble category of algebras
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Feb 27th 2018

New page: double category of algebras.