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• Created.

• Redirect: rationals

• Updated.

• Created.

• Created.

• 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)

• Created.

• Deleted a space from the name for consistency.

• Deleted a space from the name for consistency.

• Created.

• corrected a typo

Anonymous

• Corrected error in examples of nilpotent completions (claims only apply to connective spectra). Also, the parser complained about “Invalid LaTeX block: X^{\hat}_p” (which is a piece of TeX that was already present in the article, I didn’t add it), so I changed that too.

• added to equalizer statement and proof that a category has equalizers if it has pullbcks and products

• I have added to coequalizer basic statements about its relation to pushouts.

In the course of this I brought the whole entry into better shape.

• added to polynomial monad the article by Batanin-Berger on homotopy theory of algebras over polynomial monads.

• Following discussion in some other threads, I thought one should make it explicit and so I created an entry

Currently this contains some (hopefully) evident remarks of what “dependent linear type theory” reasonably should be at least, namely a hyperdoctrine with values in linear type theories.

The entry keeps saying “should”. I’d ask readers to please either point to previous proposals for what “linear dependent type theory” is/should be, or criticise or else further expand/refine what hopefully are the obvious definitions.

This is hopefully uncontroversial and should be regarded an obvious triviality. But it seems it might be one of those hidden trivialities which deserve to be highlighted a bit more. I am getting the impression that there is a big story hiding here.

Thanks for whatever input you might have.

• So far just the brief Idea and a few commented references.

• started to add to internalization a list of links to examples. Probably we have much more.

• I found the definition of a scheme to be slightly unclear/insufficiently precise at one point, so I have tweaked things slightly, and added more details. Indeed, it is quite common to find a formulation similar to ’every point has an open neighbourhood isomorphic to an affine scheme’, whereas I think it important to be clear that one does not have the freedom to choose the sheaf of rings on the local neighbourhood, it must be the restriction of the structure sheaf on $X$.

• I added the description of lax (co)limits of Cat-valued functors via (co)ends and ordinary (co)limits. I should probably flesh this out more.

I’ve adopted the convention on twisted arrows at twisted arrow category, which is opposite of that in GNN.

In the case of ordinary 2-category, when the diagram category is a 1-category, is the expression of lax (co)limits via ordinary weighted (co)limits really as simple as taking the weights $C_{\bullet/}$ or $C_{/\bullet}$? I can’t find a reference that spells that out clearly; if there really is such a simple description it should be put on the lax (co)limit page.

• I’m having trouble understanding the relationship between what is described in the page generalized algebraic theory and what is in the referenced article by Cartmell.

For example, the nLab page seems to contemplate three levels of symbols—“supersort”, “sort”, and “operation”—while if I understand correctly, Cartmell’s GATs only have symbols at two levels, for sorts and operations. Also, I would only expect a GAT’s sort symbols to be applied to terms, not types as the nLab page seems to contemplate. (The nLab page speaks of “derived operations in the theory of sorts” rather than types, but I believe the same concept is intended.) My intuition is that the world of GATs in Cartmell’s sense more or less corresponds to a certain sublanguage of LF (rather than $F_\omega$), so there shouldn’t be anything like a symbol that is applied to arguments that are types and yields a type.

• started some minimum at exceptional field theory (the formulation of 11d supergravity that makes the exceptional U-duality symmetry manifest)

• Page created, but author did not leave any comments.

The stable tangent bundle of a unit sphere bundle $S(\mathcal{V})$ in a real vector bundle $\mathcal{V} \overset{p}{\longrightarrow} M$ (Example \ref{UnitSphereBundles}) over a smooth manifold $M$ is isomorphic to the pullback of the direct sum of the stable tangent bundle of the base manifold with that vector bundle:

$T^{stab} S(\mathcal{V}) \; \simeq \; S(p)^\ast \big( T^{stab} M \oplus_M \mathcal{V} \big) \,.$

Still need to add a more canonical reference and/or a proof.

• for each pair $x,y \in D_0$ an object $D(x,y) \in C$ and not $D(x,y) \in C_0$ as $C_0$ is not defined

Anonymous

• I dropped a query box over at Hurewicz fibration about a small difference of definition between the Lab and May's Concise Topology (in May, one only needs to have the RLP with respect to inclusions of CGWH spaces (into their cylinder objects at 0), while on the Lab, it says it must hold for all topological spaces.

By the way, I like that page a lot, since it doesn't have an excessively long idea section (actually, there isn't any idea section, but I like that). It's written so it's easy to find the definition without wading through all of the clutter.

I was wondering if it would be possible to swap the idea section and definition section around in a few articles and see if it makes them more readable. There's nothing more irritating than skipping the idea section only to find that the definition references the concepts introduced there. A definition should be readable entirely without having to read the idea section.

• Created opetopic type theory with a bit of explanation based on what I understood based on what Eric Finster explained and demonstrated to me today.

This is the most remarkable thing.

I have added pointers to his talk slides and to his online opetopic type system, but I am afraid unguided exposition to either does not reveal at all the utmost profoundness of what Eric made me see when he explained and demonstrated OTT to me on his notebook. I hope he finds time and a way to communicate this insight.

• Page created, but author did not leave any comments.

• Added the original reference by Tyutin (the T in BRST).

• Gave this entry a little Idea-section and added mentioning that only numerable topological fiber bundles are guaranteed to be Serre fibrations.

Also I changed plain “bundle” to “fiber bundle” everywhere, including the title

• tried to polish a bit the matrial at Chern-Weil theory.

(not that there is much, yet, but still)

• have expanded the single sentence at differential geometry to something like a paragraph, indicating how differential geometry is the “higher geometry modeled on the pre-geometry $\mathcal{G} = CartSp$

• Added direct descriptions of the various universal fibrations.

• am splitting off this entry from exotic smooth strcuture, in order to facilitate linking to specifically the case of exotic 7-sphere.

Accordingly, so far the bulk of the entry is just copied over from the corresponding section at exotic smooth structure,

But I also added a new paragraph,

and that is what motivated me to split this off. Namely it occured to me that from the point of view of M-theory on 8-manifolds, Milnor’s classical construction of exotic smooth 7-spheres as boundaries of 8-manifolds is very particularly the construction of near horizon limits of black M2-brane spacetimes in the context of M-theory on 8-manifolds.

This must be known in the literature, and I’d like to collect what is known about it. So far I found a brief comment in this direction, in section 3.2 of

Will be adding more as I find more.

• added hyperlinks to some more of the keywords (such as Giraud theorem).

I see lots of room to clean up this old entry, but will leave it at that for the moment.

• Lukas Müller, Extended Functorial Field Theories and Anomalies in Quantum Field Theories (arXiv:2003.08217)
• added more complete publication data nd added DOI-s for Volumes 1 and 2. Couldn’t find DOI for vol 3 yet (?)

• I have added to M5-brane a fairly detailed discussion of the issue with the fractional quadratic form on differential cohomology for the dual 7d-Chern-Simons theory action (from Witten (1996) with help of Hopkins-Singer (2005)).

In the new section Conformal blocks and 7d Chern-Simons dual.

• I have started a category:reference page

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

• brief category:people-entry for hyperlinking references at M5-brane

• brief category:people-entry for hyperlinking references at M5-brane

• I have been further working on the entry higher category theory and physics. There is still a huge gap between the current state of the entry and the situation that I am hoping to eventually reach, but at least now I have a version that I no longer feel ashamed of.

Here is what i did:

• Partitioned the entry in two pieces: 1. “Survey”, and 2. “More details”.

• The survey bit is supposed to give a quick idea of what the set of the scene of fundamental physics is. It starts with a kind of creation story of physics from $\infty$-topos theory, which – I think – serves to provide a solid route from just the general abstract concept of space and process to the existence and nature of all $\sigma$-model quantum field theories of “$\infty$-Chern-Simons theory”-type (which includes quite a few) and moreover – by invoking the “holographic principle of higher category theory” – all their boundary theories, which includes all classical phase space physics.

The Survey-bit continues with indicating the formalization of the result of quantizing all these to full extended quantum field theories. It ends with a section meant to indicate what is and what is not yet known about the quantization step itself. This is currently the largest gap in the mathematical (and necessarily higher categorical) formalization of physics: we have a fairly good idea of the mathematics that describes geometric background structure for physics and a fairly good idea of the axioms satisfied by the quantum theories obtained from these, but the step which takes the former to the latter is not yet well understood.

• The “More details”-bit is stubby. I mainly added one fairly long subsection on the topic of “Gauge theory”, where I roughly follow the historical route that eventually led to the understanding that gauge fields are modeled by cocycles in higher (nonabelian) differential cohomology.