I apologize in advance is this is not the correct “category” for this discussion. Please feel free to fix this.

Under the *realization functors* subsection of the entry on motivic homotopy theory, the last line reads:

“For a non-separably closed field k, there is a Gal(k^sep/k)-equivariant realization analogous to the Real realization.”

However no reference is given, and I do not believe such a reference exists. This is an extremely subtle point. See for example Wickelgren’s paper:

http://people.math.gatech.edu/~kwickelgren3/papers/Etale_realization.pdf

This does work in the unstable setting, but this is currently being written up by myself and Elden Elemanto, and we are working on the stable result. I’m not sure the best way to revise this entry. But for sure, if there is a reference, it should be added, and otherwise, I would suggest removing or rephrasing this sentence.

]]>Homotopy colimits of simplicial diagrams and homotopy limits of cosimplicial diagrams have their own special names: realization and totalization.

Is there a special name for homotopy limits of simplicial diagrams? In general, are there any examples in the literature where such homotopy limits are computed?

]]>(Homotopy) sifted colimits commute with finite (homotopy) products in the category of sets (respectively spaces).

Is it possible to point out a bigger class of categories for which this is true?

Jacob Lurie points out in a comment on MathOverflow (http://mathoverflow.net/questions/181188/commutation-of-simplicial-homotopy-colimits-and-homotopy-products-in-spaces) that this is false for arbitrary presentable ∞-categories.

On the other hand, it seems like this might be true for cartesian closed presentable ∞-categories, because the argument for sets seems to go through in this case.

Also, could it be true for algebras over a finitely accessible ∞-monad? The forgetful functor from algebras to spaces creates limits and sifted colimits, so commutativity should follow from commutativity in spaces.

In general, is it possible to describe a more general class of categories that covers the above examples?

]]>The nLab article retract has Corollary 1, which sketches a proof of the fact that retracts of homotopy (co)limit diagrams are again homotopy (co)limit diagrams.

What is the original reference (if any) for this statement?

]]>Does anyone of any connections that have been made between the notions of cybernetics, autopoeitic systems, or even George Spencer Browns Laws of Form, and homotopy theory or category theory?

I suppose right now such connections might be somewhat tenuous, as the latter is pretty philosophical, and the former is pretty formal, but it seems that there are perhaps connections to be made. Especially, with the notion of the distinctions in space that Spencer-Brown talks about, ideas of homology and homotopy occur to me. Also, possible connections with David Spivak’s ontology logs and various ideas about categorical database type things. I’m also wondering if there are any connections between the aforementioned more philosophical topics and internal logic of topoi, or even topological semantics.

Any references or ideas on this would be really appreciated, it’s just kind of a shot in the dark. Perhaps the question is not well formed.

Best, Jon Beardsley

]]>Created prime spectrum of a monoidal stable (∞,1)-category and cross-linked vigorously with related entries.

this needs to be further expand, clearly. More references etc.

]]>I am reading the article on Hurewicz connection

Theorem. A map $\pi:E \to B$ is a Hurewicz fibration iff there exists at least one Hurewicz connection for $\pi_!$.

I have two questions:

(1) How to formally construct the Hurewicz connection for $\pi_!$ ?

(2). It has scratched an idea of proof. May I know if anyone can reference a completed formal proof to me. (a paper or an article will be great !).

]]>I am reading your article on Dold fibration

I have two questions:

How to construct Dold fibration ? Just like Hurewicz fibration, Hurewicz fibration has a theorem: A map $\pi:E \to B$ is a Hurewicz fibration iff there exists at least one Hurewicz connection for $\pi_!$. Please reference some good readings to me on Dold fibration.

Thanks,

]]>I am reading your article on Dold fibration

I have two questions:

How to construct Dold fibration ? Just like Hurewicz fibration, Hurewicz fibration has a theorem: A map $\pi :E \to B$ is a Hurewicz fibration iff there exists at least one Hurewicz connection for $\pi_!$. Please reference some good readings to me on Dold fibration.

Thanks,

]]>I am reading your article on Dold fibration

I have two questions:

- How to construct Dold fibration ? Just like Hurewicz fibration, Hurewicz fibration has a theorem: A map $\pi:E \to B$ is a Hurewicz fibration iff there exists at least one Hurewicz connection for $\pi_!$.
- Please reference some good readings to me on Dold fibration.

Thanks,

Tom

]]>I am reading the article on Hurewicz connection

Theorem. A map $\pi:E \to B$ is a Hurewicz fibration iff there exists at least one Hurewicz connection for $\pi_!$.

I have two questions:

(1) How to formally construct the Hurewicz connection for $\pi_!$ ?

(2). It has scratched an idea of proof. May I know if anyone can reference a completed formal proof to me. (a paper or an article will be great !).

]]>1) Define a sort of complex for higher homotopy groupoids. Define the same thing for "forms" (for 1-forms: connections/holonomy functors are already there).

2) Prove that the two complexes above, defined in a certain way, are dual (for holonomy functors and 1-homotopy it's already true). "Cohomotopy".

3) Prove that the reduction to a point (monoid) of the complexes above is abelian and isomorphic to the homotopy groups, and that the central extension of whatever you find is homology.

4) "Fundamental theorem of topology": Let M and N be locally diffeomorphic. Then they are globally diffeomorphic IF AND ONLY IF their higher homotopy complexes (see above) are isomorphic.

The 4) is probably the dream of many. I think that a sufficient and necessary condition for diffeomorphisms, homology and homotopy is not there simply because the classic higher homotopy groups are "not very nice".

What are your "dream-results"? ]]>

Let $G$ be a Lie group and let $\mathbf{B}G$ be the smooth stack of principal $G$-bundles. Then we have a natural equivalence $G\simeq \Omega\mathbf{B}G$. If we apply the internal hom $[S^1,-]$ to the homotopy pullback diagram defining $\Omega\mathbf{B}G$ (and if internal homs preserve homotopy pullbacks, something I need be reassured about) then we get a homotopy pullback diagram realizing the free loop space object $\mathcal{L}G=[S^1,G]$ as the based loop space of $\mathcal{L}\mathbf{B}G=[S^1,\mathbf{B}G]$. On the other hand, since $G$ is a smooth group, then so is $\mathcal{L}G$ and we can consider the smooth stack $\mathbf{B}\mathcal{L}G$, whose based loop space is again $\mathcal{L}G$.

What is the relation between $\mathbf{B}\mathcal{L}G$ and $\mathcal{L}\mathbf{B}G$? (if any)

From what I read in the lines after equation (3.2) in Konrad Waldorf’s Spin structures on loop spaces that characterize string manifolds I guess that for a connected $G$ the two stacks $\mathbf{B}\mathcal{L}G$ and $\mathcal{L}\mathbf{B}G$ should actually be equivalent, but I don’t clearly see this yet.

]]>A stub for Cartan-Eilenberg categories.

]]>Created the page telescope conjecture since I noticed it was linked to by Morava K-theory but didn’t exist. Might add more later, specifically about how this is generalized to the setting of axiomatic stable homotopy categories and how it is true after localizing at $BP$, $E(n)$ and some other spectra, but believed to be false in general.

]]>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 ]]>

REMINISCENCES OF GROTHENDIECK AND HIS SCHOOL

LUC ILLUSIE, WITH SPENCER BLOCH, VLADIMIR DRINFELD, ET AL.

I was surprised to learn that derived cats date from 1964 and apparently prior to the cotangent complex

and Quillen's homotopical algebra @ 1967??

Does anyone know the history more accurately than that? and why the derived cat and

homotopical algebra communities grew apart? Anyone maintain a foot in both camps? ]]>

I am reading the paper

- Jean-Marc Cordier, Tim Porter,
*Vogt’s theorem on categories of homotopy coherent diagrams*, Math. Proc. Cambridge Philos. Soc., 100, (1986), 65 – 90.

Before introducing the coherent nerve, the paper motivates the business of homotopy coherences in low dimension. It goes on with saying that the combinatorics of these becomes increasingly hard to handle in higher dimensions and that the business with coherent nerve solves this in one step. However, it seems that the combinatorics is exactly the combinatorics of Street’s orientals, but in the coherent homotopy literature nobody does a clear comparison.

Thus, I would like to understand how to get the equivalent handling using the Street’s $\omega$-nerve, and seeingt the proof that such an approach to coherent homotopy is equivalent to the one via Vogt-Cordiert-Porter machinery. I mean I would like to have precise mathematical statements, not just intutitive understanding.

Tim ? Urs ?

]]>Created homotopy equivalence of toposes.

]]>New stub model structure on operator algebras with redirects model structures on operator algebras, homotopy theory for operator algebras, homotopy theory for C*-algebras etc.

]]>