There is a question about the education curriculum (the most advanced study) on algebraic topology

I am interested in what topics and to which year of study a person should know who wants to make a big contribution to AT, higher category theory

Example:

2-3 course - Spectrum, algebraic K theory, model categories, stable homotopy theory, homology theory, Stolz-Teichner programme

Thank you ]]>

I am a newcomer to non-commutative geometry and really enjoying the literature. I have read several of the classic papers, assisted by the excellent lecture notes *Homological Methods in Non-commutative Geometry* by Kaledin. I have also read the papers by Reyes, Heunen, and van den Berg on the obstructions to extending the Zariski site to $\mathsf{Ring}$ in various ways. Now I see on the nLab page for noncommutative algebraic geometry there is the added statement:

This is the reason why non-commutative algebraic geometry is phrased in other terms, mostly in terms of monoidal categories “of (quasicoherent) abelian sheaves” (“2-rings”).

Are there a few names people can drop for me, or even specific papers, which are most significant for following up on this hint? Before coming across this page I worked out some things for sheaf theory on bicategories (e.g. of noncommutative algebras), but feel I must be on well-trodden ground.

]]>It’s well known that the category of points of the presheaf topos over $Ring_{fp}^{op}$, the dual of the category of finitely presented rings, is the category of all rings (without a size or presentation restriction). In fact this holds for any algebraic theory, not only for the theory of commutative rings. One can learn about this in our entries on *Gabriel-Ulmer duality*, *flat functors*, and Moerdijk/Mac Lane.

But what if we don’t restrict the site to consist only of the compact objects? What are the points of the presheaf topos over the large category $Ring^{op}$, to the extent that the question is meaningful because of size-related issues? What are the points of the presheaf topos over $Ring_{\kappa}^{op}$, the dual of the category of rings admitting a presentation by $\lt \kappa$ many generators and relations, where $\kappa$ is a regular cardinal? (The category $Ring_{\kappa}^{op}$ is essentially small, so the question is definitely meaningful.)

The question can be rephrased in the following way: What is an explicit description of the category of *finite* limit preserving functors $F : Ring_{\kappa}^{op} \to Set$? Any such functor gives rise to a ring by considering $F(\mathbb{Z}[X])$, but unlike in the case $\kappa = \aleph_0$ such a functor is not determined by this ring.

This feels like an extremely basic question to me; it has surely been studied in the literature. I appreciate any pointers! Of course I’ll record any relevant thoughts in the lab.

]]>Created a stub at Milnor K-theory, which is now just an MO answer of Cisinski. To be expanded at some later point when I study this in more detail.

]]>There’s quite a bit on the $n$-lab about effective epimorphisms in $\infty$-toposes, so maybe I’m just not putting it together correctly, but I’m wondering if anyone here knows if there’s a characterization (in print somewhere, preferably) of effective epimorphisms for $A_\infty$-ring spectra. In particular, I suspect there should be some kind of statement like: For $f:A\to B$, a morphism of connected $A_\infty$-ring spectra, we can recover $A$ from the Amitsur complex (dually, the Cech nerve) of $f$ as long as $\pi_0(f)$ is an isomorphism and $\pi_1(f)$ is a surjection.

This sort of statement is proven by Gunnar Carlsson in the context of what he calls “derived completion” but he works with the $S$-algebra framework of Elmendorf, Kriz, May and Mandell, and it seems like it should be a much more general topos theoretic statement for an $\infty$-topos. Incidentally, has anyone written down anything about the topos of $A_\infty$-ring spectra at all? At first glance it seems like most of Lurie’s work on the subject is for $E_\infty$-rings.

Thanks! Jon

]]>Polarizations and Grothendieck’s Standard conjectures, Milne http://www.jmilne.org/math/articles/2002aS.pdf

Remarks on Grothendieck's standard conjectures, Beilinson http://arxiv.org/abs/1006.1116

The standard conjectures, Kleiman (Motives, Proceedings of Symposia in Pure Mathematics , Seattle)

arithmetics analogues of standard conjectures, H. Guillet and C. Soulé

Une introduction aux motifs, Yves André (Chapter V)

The idea is increase the references of standard conjectures nLab. ]]>

I made a new page called twisted form. Unfortunately, this stole the redirect from a sub-heading on differential form. The page is still pretty much a stub. I hope to enlarge it soon.

]]>Here’s a sketchy derivation of the fact that a String structure on a manifold induces a Spin structure on its loop space. As to be expected from the supposed naturality of the construction, everything happens at the level of stacks (or at least, it should, modulo a few details I’ve not checked).

To begin with, the (higher) stack $\mathbf{B}String$ is defined as the homotopy pullback

$\array{ \mathbf{B}String &\to& *\\ \downarrow && \downarrow \\ \mathbf{B}Spin &\stackrel{\frac{1}{2}p_1}{\to}& \mathbf{B}^3U(1) }$Applying the internal hom $[S^1,-]$ to the above diagram we get the homotopy commutative diagram (does internal hom preserve homotopy limits?)

$\array{ [S^1,\mathbf{B}String] &\to& *\\ \downarrow && \downarrow \\ [S^1,\mathbf{B}Spin] &\stackrel{[S^1,\frac{1}{2}p_1]}{\to}& [S^1,\mathbf{B}^3U(1)] }$Now we consider the fiber integration/transgression morphis $exp 2\pi i\int_{S^1}:[S^1,\mathbf{B}^3U(1)]\to \mathbf{B}^2U(1)$:

$\array{ [S^1,\mathbf{B}String] &\to& *\\ \downarrow && \downarrow \\ [S^1,\mathbf{B}Spin] &\stackrel{[S^1,\frac{1}{2}p_1]}{\to}& [S^1,\mathbf{B}^3U(1)]\\ & & & \searrow^{exp 2\pi\int_{S^1}}\\ & & & \phantom{mm} \mathbf{B}^2U(1) }$to get the homotopy commutative diagram

$\array{ [S^1,\mathbf{B}String] &\to& *\\ \downarrow && \downarrow \\ [S^1,\mathbf{B}Spin] &\stackrel{exp 2\pi i\int_{S^1}[S^1,\frac{1}{2}p_1]}{\to}& \mathbf{B}^2U(1) }$which, assuming $[S^1,-]$ commutes with $\mathbf{B}$ we can rewrite as

$\array{ [S^1,\mathbf{B}String] &\to& *\\ \downarrow && \downarrow \\ \mathbf{B}[S^1,Spin] &\stackrel{exp 2\pi i\int_{S^1}[S^1,\frac{1}{2}p_1]}{\to}& \mathbf{B}^2U(1) }$i.e., as

$\array{ \mathcal{L}\mathbf{B}String &\to& *\\ \downarrow && \downarrow \\ \mathbf{B}\mathcal{L}Spin &\stackrel{exp 2\pi i\int_{S^1}[S^1,\frac{1}{2}p_1]}{\to}& \mathbf{B}^2U(1) }$and the bottom horizontal arrow is the canonical 2-cocycle on the loop group $\mathcal{L}Spin$. By the universal property of the homotopy pullback, the above homotopy commutative diagram therefore factors as

$\array{ \mathcal{L}\mathbf{B}String\\ & \searrow\\ && \mathbf{B}\widetilde{\mathcal{L}Spin} &\to& *\\ && \downarrow && \downarrow \\ && \mathbf{B}\mathcal{L}Spin &\stackrel{exp 2\pi i\int_{S^1}[S^1,\frac{1}{2}p_1]}{\to}& \mathbf{B}^2U(1) }$where $\widetilde{\mathcal{L}Spin}$ is the canonical $U(1)$-central extension of the loop group $\mathcal{L}Spin$. We are done: the morphism $\mathcal{L}\mathbf{B}String\to \mathbf{B}\widetilde{\mathcal{L}Spin}$, factoring the natural projection $\mathcal{L}\mathbf{B}String\to \mathcal{L}\mathbf{B}Spin\cong \mathbf{B}\widetilde{\mathcal{L}Spin}$ is the universal morphism inducing the transgression from String structureson a Spin manifold to Spin structures on its loop space.

Namely, by definition, a String structur on $M$ is a lift of the Spin structure $M\to \mathbf{B}Spin$ to $M\to\mathbf{B}String$. Applying the internal hom $[S^1,-]$ to this lift we obtain a lift of the morphism $\mathcal{L}M\to \mathcal{L}\mathbf{B}Spin\cong \mathbf{B}\mathcal{L}Spin$ to a morphism $\mathcal{L}M\to \mathcal{L}\mathbf{B}String$. Since the projection $\mathcal{L}\mathbf{B}String\to \mathbf{B}\widetilde{\mathcal{L}Spin}$ factors through $\mathbf{B}\widetilde{\mathcal{L}Spin}$ we get a lift of the natural morphism $\mathcal{L}M\to \mathbf{B}\mathcal{L}Spin$ to a morphism $\mathcal{L}M\to \mathbf{B}\widetilde{\mathcal{L}Spin}$. But this is precisely the definition of a Spin structure on $\mathcal{L}M$.

]]>I expect I have not given the best code for all of this so someone may want to improve it in that respect.

Graham, also writes in his paper:

In view of the ease with which Whitehead's methods handle the

classifications of Olum and Jajodia, it is surprising that the

papers \cite{olum:1953} and \cite{jaj:1980} (both of which were

written after the publication of \cite{whjhc:1949}) make

respectively no use, and so little use, of \cite{whjhc:1949}.

We note here that B. Schellenberg, who was a student of Olum, has

rediscovered in \cite{sch:1973} the main classification theorems

of \cite{whjhc:1949}. The paper \cite{sch:1973} relies heavily on

earlier work of Olum. ]]>

R. Brown and P.J. Higgins, ``On the connection between the second

relative homotopy groups of some related spaces'', _Proc. London Math. Soc._ (3) 36 (1978) 193-212.

and the construction was generalised to filtered spaces in two paper with Higgins in JPAA 1981. The point of these constructions is not _about_ homotopy theory, but providing new algebraic structures to be used as algebraic tools for understanding and computation of homotopical invariants. In particular, our book "Nonabelian algebraic topology" brings together in Part I lots of nonabelian constructions and calculations in 2-dimensional homotopy theory, while the work with Loday extends these nonabelian constructions and calculations to higher dimensions, using strict $n$-fold groupoids. One particular construction which came out of the latter work, the nonabelian tensor product of groups, has a current bibliography of 120 items, largely because of the interest in the construction by group theorists.

One reason for these explicit results is the idea of computation in homotopy theory using strict colimits, enabled by Higher Homotopy Seifert-van Kampen Theorems. These work by dealing with structured spaces, i.e. filtered spaces or $n$-cubes of spaces. The philosophical implications need discussion, possibly in this forum.

Note Grothendieck's Section 5 in his "Esquisses d'un Programme" on the limitations for geometry of the notion of topological space. He advocates sophisticated notions of stratification.

For me, the fact that these strict structures can be defined for say filtered spaces is itself of significance, since the detailed proofs are not straightforward, and there is "just enough room" to make the proofs work.

Ronnie ]]>