https://ncatlab.org/jamesdolan/published/Algebraic+Geometry ]]>

Somebody kindly pointed out by email to me two mistakes on the page Pr(infinity,1)Cat. I have fixed these now (I think).

The serious one was in the section Embedding into Cat where it said that $Pr(\infty,1)Cat \to (\infty,1)Cat$ preserves limits and colimits. But it only preserves limits. This is HTT, prop. 5.5.3.13. The wrong statement was induced from a stupid misreading of HTT, theorem. 5.5.3.18. Sorry.

The other mistake was that it said “full subcategory”. But of course by the very definition of $Pr(\infty,1)Cat$ if is not full in $(\infty,1)Cat$. I have fixed that, too, now.

]]>I was dissatisfied with the discussion at semisimple category because it only defined a semisimple *monoidal Vect-enriched* category, completely ignoring the more common notion of semsimple abelian category.

So, I stuck in the definition of semisimple abelian category.

However, I still think there is a lot that could be improved here: when is a semisimple abelian category which is also monoidal a semsimple monoidal category in some sense like that espoused here???

I think this article is currently a bit under the sway of Bruce Bartlett’s desire to avoid abelian categories. This could be good in some contexts, but not necessarily in all!

]]>Created sound doctrine as a stub to record relevant references.

]]>I came to think that the pattern of interrelations of notions in the context of locally presentable categories deserves to be drawn out explicitly. So I started:

Currently it contains the following table, to be further fine-tuned. Comments are welcome.

| | | inclusion of left exaxt localizations | generated under colimits from small objects | | localization of free cocompletion | | generated under filtered colimits from small objects |
|–|–|–|–|–|—-|–|–|
| **(0,1)-category theory** | (0,1)-toposes | $\hookrightarrow$ | algebraic lattices | $\simeq$ Porst’s theorem | subobject lattices in accessible reflective subcategories of presheaf categories | | |
| **category theory** | toposes | $\hookrightarrow$ | locally presentable categories | $\simeq$ Adámek-Rosický’s theorem | accessible reflective subcategories of presheaf categories |
$\hookrightarrow$ | accessible categories |
| **model category theory** | model toposes | $\hookrightarrow$ | combinatorial model categories | $\simeq$ Dugger’s theorem | left Bousfield localization of global model structures on simplicial presheaves | | |
| **(∞,1)-topos theory** | (∞,1)-toposes |$\hookrightarrow$ | locally presentable (∞,1)-categories | $\simeq$ <br/> Simpson’s theorem | accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories |
$\hookrightarrow$ |accessible (∞,1)-categories |

For a while, we've been blanking spam pages and putting them in their own category; then people are supposed to use these when they create a new page. This effectively removes the spam page from circulation without having to invoke deletion. Then we found some other junk pages that weren't spam but either mistakes or whatever is going on with those weird slideshow pages. Then we put some of these in category: empty since they weren't spam. But we remarked that they really could all go there.

So now I've put them all there and regularised the names.

Please use these when you make a new page! Only a few people have to remember to do this and they will all go away. (Zoran remembers, but I usually forget.)

]]>I added linear logic and type theory (homotopy type theory was already there) to true, which I renamed to truth to make it a noun (although something like true proposition, which I made a redirect, could also work). I then edited false (now falsehood) to include everything in truth.

]]>I wanted to understand Borel's Theorem better, so I wrote out a fairly explicit proof of the one-dimensional case.

]]>https://imgur.com/iioIWEx

Is this a known issue? ]]>

motivated by the blog discussion I added to [[rational homotopy theory]] a section Differential forms on topological spaces

]]>created [[Sullivan model]]

]]>I have expanded, streamlined and re-organized a little at *differential forms on simplices*.

made *curved L-infinity algebra* explicit

am in the process of adding some notes on how the D=5 super Yang-Mills theory on the worldvolume of the D4-brane is the double dimensional reduction of the 6d (2,0)-superconformal QFT in the M5-brane.

started a stubby *double dimensional reduction* in this context and added some first further pointers and references to *M5-brane*, to *D=5 super Yang-Mills theory* and maybe elsewhere.

But this still needs more details to be satisfactory, clearly.

]]>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.

I started a bare minimum at *adinkra* and cross-linked with *dessins d’enfants*.

Adinkras were introduced as a graphical tool for classifying super multiplets. Later they were realized to also classify super Riemann surfaces in a way related to dessins d’enfants.

I don’t really know much about this yet. Started the entry to collect some first references. Hope to expand on it later.

]]>Has anyone developed models for the homotopy theory of $H \mathbb{Q}$.module spectra over rational topological spaces a bit?

I expect there should be a model on the opposite category of dg-modules over rational dg-algebras. Restricted to the trivial modules it should reduce to the standard Sullivan/Quillen model of rational homotopy theory. Restricted to the dg-modules over $\mathbb{Q}$ it should reduce to the standard model for the homotopy theory of rational chain complexes, hence equivalently that of $H \mathbb{Q}$-module spectra.

Is there any work on this?

]]>The pages apartness relation and antisubalgebra disagree about the definition of an antiideal: do we assume $\neg(0\in A)$ or $\forall p\in A, p\# 0$? Presumably there is a similar question for antisubgroups, etc. In particular, the general universal-algebraic definition at antisubalgebra would give $\neg (0\in A)$ as the definition (since $0$ is a constant and $\bot$ is a nullary disjunction), contradicting the explicit definition of antiideal later on the same page.

Does this have something to do with whether $\#$-openness is assumed explicitly or not? The page apartness relation claims that, at least for antiideals, openness is automatic as long as the ring operations are strongly extensional. But antisubalgebra assumes openness explicitly, in addition to strong extensionality of the algebraic operations.

Finally, do we ever really need the apartness to be tight?

]]>Started work on syntopogenous space.

]]>I added a remark to inhabited set that one can regard writing $A\neq\emptyset$ to mean “$A$ is inhabited” as a reference to an inequality relation on sets other than denial.

]]>I gave *Adams operations* some details in the Definition section

I have begun an entry

meant to contain detailed notes, similar in nature to those at *Introduction to Stable homotopy theory* (but just point-set topology now).

There is a chunk of stuff already in the entry, but it’s just the beginning. I am announcing this here not because there is anything to read yet, but just in case you are watching the logs and are wondering what’s happening. In the course of editing this I am and will be creating plenty of auxiliary entries, such as *basic line bundle on the 2-sphere*, and others.

Just for procrastination purposes, yesterday I had started some minimum at *asymptotic safety*.