I am trying to compile references with results on configuration-space models for the iterated loop spaces $\Omega^n S^n$, $n \geq 2$.

From Segal 1973 thm 1 this is the topological group completion of the configuration space of points in $\mathbb{R}^n$ under “compress the configurations and form their disjoint union”.

The evident guess that this, in turn, should just be the configuration space of points labeled by a sign (charge) $\in \{\pm 1\}$, where oppositely charged points in the configuration are allowed to undergo pair creation/annihilation, was famously shown to be *wrong* (“close” but wrong) by McDuff 1975 p. 6.

A fix to this issue was claimed by Caruso & Waner 1981, who say that it works when signed points are replaced by oriented little cubes, suitably construed.

However, their definition is at least not easy to read, which made Okuyama 2004, 05 claim a “simpler” model that uses configurations of just line segments with signed endpoints subject to a clever annihilation relation.

Here is a **question** I have:

Okuyama’s Def. 3.3 (p. 7) specialized to $n=2$ looks to me like movements (paths) in this configuration space of line segments (intervals) in the plane can never make the intervals move around each other for the path to form a braid — contrary to what one should expect (?) in this dimension. (Namely the definition is such that the intervals stretch along the $x$-direction, say, while never being allowed to coincide in their $y$-coordinate — unless I am misreading something?)

Now this model is picked up and generalized to the equivariant setting by Okuyama & Shimakawa 2007 — however they no longer claim that it works for finite $n$ (cf. above their Th. 1.1, p. 2).

This makes me wonder if the claim in Okuyama 2004, 05 is wrong for finite $n$, or else what I am missing in my question above.

]]>G-d-operads are equivariant d-operads

Natalie Stewart

]]>https://ncatlab.org/nlab/search?query=*

500 Internal Server Error

target of repeat operator is not specified: /*/i

And here is the exception backtrace:

/home/nlab/www/nlab/app/controllers/wiki_controller.rb:154:in `block in search'

⋮ ]]>

Added the property that final functors and discrete fibrations form an orthogonal factorisation system.

]]>Fiberwise cores of G-∞-categories appear all over the place

Natalie Stewart

]]>have added to (infinity,1)-operad the basics for the “$(\infty,1)$-category of operators”-style definition

]]>Adding a page on d-operads

Natalie Stewart

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

]]>I am giving this bare list of references its own entry, so that it may be `!include`

-ed into related entries (such as *topological quantum computation*, *anyon* and *Chern-Simons theory* but maybe also elsewhere) for ease of updating and synchronizing

Changed $Sch/S$ to $Sch/X$.

]]>In my master thesis I have stumbled upon some issue which makes me go crazy. I want to state the problem but I should discuss it so that it would not mislead me.

There is a model at hand, a field theory which interacts with gravity non minimally via metric in the lagrangian. This field theory is nontrivial topologically and prequantizes. Nonetheless, it has a prequantization when we fix the metric but it seems that it has no canonical prequantization as a field theory on the bundle of fields + metrics (satisfying certain conditions). Moreover, there is an obstruction to prequantization of such a field theory, since if we compute metric energy-momentum tensor locally it shall not glue properly to a global one.

Initially I intended to find out how gravity interacts with nontrivial topologically field theories so as to conclude inconsistencies with gravity even at the level of prequantization.

It seems that given a prequantum field theory which has interaction with gravity (via energy-momentum tensor) we must deform it (canonically some way, so that it could be functorial, may be additive…) so as to extend the prequantization to the bundle of fields + metrics.

Why this idea? I guess the answer is that we already do this with even locally-defined field theories. An obstruction to a proper nonminimal interaction (via Energy-Momentum tensor) of a field theory with gravity is that the Energy-Momentum tensor must be divergence-free. When we deform a given field theory we usually fall to such field configurations that satisfy divergence-freeness or even fall on-shell of the theory.

So we had:

obstruction to nonminimal interaction - divergence-freeness of EMT (analytical condition). We deform a bundle of a theory but not lagrangians. This deformation is “canonical”.

Now we have in addition:

obstruction to nonminimal interaction - EMT should be a globally defined tensor on a manifold (topological condition). We deform a field theory but I guess here we change the Lagrangians so that they could prequantize on a bundle of fields + metrics.

What do you think I could do in this case? ]]>

added to action groupoid a section on action oo-groupoids

]]>Created a separate page, in order to talk about their monad in more detail.

]]>Creating page for now, adding more content soon.

]]>For now creating the page. I’m not an expert on the topic, so if someone wants to chime in, please do!

]]>starting something, to go along with *third stable homotopy group of spheres*

I added to Tim’s stub on cellular homology. Still a bit rough around the edges perhaps. An example (say real projective space) would also be nice.

]]>prompted by this MO question I saw that our entry *parameterized spectrum* was a bit thin on information. I have now at least briefly added a section *Yoga of six functors* with mentioning of and pointers to the Wirthmüller context property, the Beck-Chevalley condition and the interpretation as linear homotopy type theory.

added pointer to:

- Thomas Lamiaux:
*Computing Cohomology Rings in Cubical Agda*, talk at*Running HoTT 2024*, CQTS@NYUAD (April 2024) [video:kt]

The equivariant version of symmetric sequences

Natalie Stewart

]]>removed link to old philosophy paper

steveawodey

]]>Fix mistaken uniqueness claims

]]>I added a little bit to maximal ideal (first, a first-order definition good for commutative rings, and second a remark on the notion of scheme, adding to what Urs wrote about closed points).

The second bit is almost a question to myself: I don’t feel I really grok the notion of scheme (why it’s this and not something slightly different that’s the natural definition, the Tao if you like). In particular, it’s where *fields* – simple objects in the category of commutative rings – make their entrance in the notion of covering by affine opens that I don’t feel I really understand.

For now creating the page. More information to be added in the future.

]]>Added redirect coaction compatible localization and split the entry into sections.

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