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• brief category:people-entry for hyperlinking references

• a category:reference-entry

• Page created. Idempotent monoids should be to monoids as idempotent monads are to monads.

I’ve added the examples of idempotent elements in (ordinary) monoids (1), idempotent morphisms in categories (2), solid rings (3), idempotent monads (4), idempotent $1$-morphisms in bicategories (5), and “solid ring spectra” (6) ―What are other examples?

Also, should idempotent monoids have a unit? The examples 1 and 2 I mentioned above don’t, but 3, 4, and 6 do, while whether 5 does or doesn’t seems to vary a bit among the literature (AFAIU).

• Maximilian Doré, Samson Abramsky, Towards Simplicial Complexes in Homotopy Type Theory (pdf)

• starting something

• I have added pointers to Mikhail Kapranov’s talks on the sphere spectrum in relation to super-algebra, and added some words at the beginning that this was the original motivation for the proposed definition of spectral supergeometry in the entry.

Also I fixed the link to the video recording of Krapranov’s 2013 talk. The previous link no longer worked but there is a YouTube copy of the video. Fixed this also at superalgebra, see there at Kaprananov 13

• Baez and Dolan mainly did the periodic table of k-tuply monoidal n-categories; this article was written like all we did was “slightly distort” some existing table.

• A couple of properties.

• Explain the connection with enriched monads

• brief category:people-entry for hyperlinking references

• brief category:people-entry for hyperlinking references

• added a brief section (here) on the original “Conner-Floyd orientation”

$\array{ M SU &\longrightarrow& K \mathrm{O} \\ \big\downarrow && \big\downarrow \\ M \mathrm{U} &\longrightarrow& K \mathrm{U} }$
• started an entry on the Borel construction, indicating its relation to the nerve of the action groupoid.

• I added some content in protomodular category. It’s mostly drawn from Bourn’s papers. It will need brushing up, cross-linking, etc.

There are many further related concepts. Don’t know how important they are, e.g., Bourn says

The dual of a topos is arithmetical.

Is that a standard concept? And ’affine categories’?

• Added link to Bourn’s most helpful 2017 textbook From Groups to Categorial Algebra : Introduction to Protomodular and Mal’tsev Categories. Revised reference to the Borceux-Bourn 2004 monograph.

• brief category:people-entry for hyperlinking references

• starting something – not done yet but need to save

• I was involved in some discussion about where the word “intensional” as in “intensional equality” comes from and how it really differs from “intenTional” and what the point is of having such a trap of terms.

Somebody dug out Martin-Löf’s lecture notes “Intuitionistic type theory” from 1980 to check. Having it in front of me and so before I forget, I have now briefly made a note on some aspects at equality in the section Different kinds of equalits (below the first paragraph which was there before I arrived.)

Anyway, on p. 31 Martin-Löf has

intensional (sameness of meaning)

I have to say that the difference between “sameness of meaning” and “sameness of intenTion”, if that really is the difference one wants to make, is at best subtle.

• brief category:people-entry for hyperlinking references

• brief category:people-entry for hyperlinking references

• brief category:people-entry for hyperlinking references at homotopy theory

• the book Monoidal Functors, Species and Hopf Algebras is very good, but still being written. Clearly the current link under which it is found on the web is not going to be the permanent link. So I thought it is a bad idea to link to it directly. Instead I created that page now which we can reference then from nLab entries. When the pdf link changes, we only need to adapt it at that single page.

• Where the page has

The index theorem is supposed to have an interpretation in terms of the quantum field theory of the superparticle on the given space,

is the “is supposed to” necessary? Why not “has an interpretation”? Is it just the general issue of any translation from mathematics to physics?

• Added how small categories can be thought of as semigroups.

• Did anyone ever write out on the $n$Lab the proof that for $X$ locally compact and Hausdorff, then $Map(X,Y)$ with the compact-open topology is an exponential object? (Many entries mention this, but I don’t find any that gets into details.)

I have tried to at least add a pointer in the entry to places where the proof is given. There is prop. 1.3.1 in

• Marcelo Aguilar, Samuel Gitler, Carlos Prieto, sections 1.2, 1.3 of Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)

but of course there are more canonical references. I also added pointer to

• Eva Lowen-Colebunders, Günther Richter, An Elementary Approach to Exponential Spaces, Applied Categorical Structures May 2001, Volume 9, Issue 3, pp 303-310 (publisher)
• brief category:people-entry for hyperlinking references

• brief category:people-entry for hyperlinking references

• Chris Nagele, Oliver Janssen, Matthew Kleban, Decoherence: A Numerical Study (arXiv:2010.04803)
• Is the Strøm model category left proper? I know that pushout along cofibrations of homotopy equivalences of the form $A \to \ast$ are again homotopy equivalences. (e.g. Hatcher 0.17) Maybe the proof directly generalizes, haven’t checked.

• following a suggestion by Zoran, I have created a stub (nothing more) for Kuiper’s theorem

• am finally giving this its own entry. Nothing much here yet, though, still busy fixing some legacy cross-linking…

• started some remarks at physical unit. But I really need to stop with that now and do more urgent things…

• The entry used to start out with the line “not to be confused with neutral element”. This was rather suboptimal. I have removed that sentence and instead expanded the Idea-section to read now as follows:

Considering a ring $R$, then by the unit element one usually means the neutral element $1 \in R$ with respect to multiplication. This is the sense of “unit” in terms such as nonunital ring.

But more generally a unit element in a unital (!) ring is any element that has an inverse element under multiplication.

This concept generalizes beyond rings, and this is what is discussed in the following.

• brief category:people-entry for hyperlinking references