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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• for completeness

• Removed an incorrect historical claim (Dwyer and Kan did throughly investigate relative categories already in 1980s, way before 2000s).

• need to record some results on equivariant tubular neighbourhoods etc. Didn’t know where to put these, so I thought we’d need a dedicated entry on equivariant differential topology.

• created at internal logic an Examples-subsection and spelled out at Internal logic in Set how by turning the abstract-nonsense crank on the topos Set, one does reproduce the standard logic.

• Created:

## Definition

More precisely, an operad $O$ in Set induced a monad $T$ on Set:

$T(S)=\coprod_{n\ge0} O_n \times_{\Sigma_n} S^n.$

Such a monad $T$ is equipped with a canonical weakly cartesian natural transformation to the moand $Sym$ arising from the commutative operad.

## Properties

A theorem of Joyal \cite{Joyal} states that there is a monoidal equivalence between the monoidal category of endofunctors $Set\to Set$ that admits a weakly cartesian natural transformation to $Sym$ and the monoidal category of species, i.e., symmetric sequences in Set with the substitution product.

In particular, the category of analytic monads on Set is equivalent to the category of operads in Set.

## The colored case

The correspondence carries over to colored operads (with a set of colors $C$) if we use the slice category $Set/C$ instead of Set.

## The nonsymmetric case

A similar correspondence can be established for nonsymmetric case, except that we must include the data of a transformation to $Sym$, which is no longer unique.

## The homotopical case

The correspondence generalizes to (∞,1)-categories, with some statements becoming more elegant. See Gepner–Haugseng–Kock \cite{GHK}.

## References

• André Joyal, Foncteurs analytiques et espèces de structures, Combinatoire énumérative (Montréal/Québec, 1985), Lecture Notes in Mathematics 1234 (1986), 126-159. doi.

• Mark Weber, Generic morphisms, parametric representations and weakly Cartesian monads, Theory Appl. Categ. 13 (2004), 191–234.

• David Gepner, Rune Haugseng, Joachim Kock, ∞-Operads as Analytic Monads, arXiv:1712.06469.

• the entry braid group said what a braid is, but forgot to say what the braid group is; I added in a sentence, right at the beginning (and fixed some other minor things).

• I’ve added Peter May’s Galois theory example to M-category in a section “Applications”.

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A book by Tom Leinster.

London Mathematical Society Lecture Note Series 298 (2004).

Cambridge University Press.

ISBN 0 521 53215 9.

• am giving this book a category:reference-entry for ease of referencing

• am finally giving this an entry

• starting something

• started Lie algebra cohomology,

(for the moment mainly to record that reference on super Lie algebra cocycles)

• following Zoran’s suggestion I added to the beginning of the Idea-section at monad a few sentences on the general idea, leading then over to the Idea with respect to algebraic theories that used to be the only idea given there.

Also added a brief stub-subsection on monads in arbitrary 2-categories. This entry deserves a bit more atention.

• started a Properties-section at Lawvere theory with some basic propositions.

Would be thankful if some experts looked over this.

Also added the example of the theory of sets. (A longer list of examples would be good!) And added the canonical reference.

• I’ve added a definition to locally cartesian closed model category, although I’m open to debate about whether this is the right definition. This definition is more or less exactly what one needs to interpret dependent products in type theory with function extensionality (I plan to add a proof of this). But it’s certainly less obviously correct from a pure model-categorical viewpoint. For one thing, it doesn’t imply that we have a cartesian closed model category, which one would naively expect a notion of “locally cartesian closed model category” to do.

Anonymous

• stub article on Mochizuki’s corollary 3.1.2

Anonymous

• Am making a start on trying to understand something of Mochizuki’s IUTT papers. I do not hold out any promises on how far I am going to get, or how long it is going to take me! Even this very first definition is going to me a long time, I think, as I intend to try to fill out all details. All help will be appreciated!

• I have changed the title of this article, as well as references to the object within it. Use of the term “Hawaiian Earring” is objected to by Hawaiian mathematicians. Please see these two threads, one by native Hawaiian and math PhD Dr. Marissa Loving, and the other by an expert on the Hawaiian Earring, Dr. Jeremy Brazas.

I have retitled the article “Shrinking wedge of circles”, which is the name used for this space in Hatcher’s “Algebraic Topology”. I have a retained a note in the body of the article that the space is sometimes referred to as the “Hawaiian earring space”.

This small change in name helps to make mathematics a more inclusive and just field, especially in consideration of the historical marginalization and exclusion of indigenous mathematicians. By taking this action, the nLab site can help to spread a change in language more widely, including on other math reference sites.

I hope that this change is readily accepted and approved by the nLab community. Thank you!

Justin Lanier

• Stub. For the moment just for providing a place to record this reference:

• Jean Thierry-Mieg, Connections between physics, mathematics and deep learning, Letters in High Energy Physics, vol 2 no 3 (2019) (doi:10.31526/lhep.3.2019.110)
• I understood that the old terminology was ’projective system’, and ’projective limit’ refereed to the limit of a projective system. Can anyone confirm that? if I am right the present entry is slightly incorrect, but this needs checking first before changing it.

• brief category:people-entry for hyperlinking references (at FGA)

• brief category:people-entry for hyperlinking references at FGA explained and elsewhere

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• brief category:people-entry for hyperlinking references at FGA explained, and elsewhere

• a category:reference-entry (“FGA” used to redirect to EGA)

• I updated a link on FGA explained to one of the chapters, to point to the arXiv version (the one on constructing Hilbert and Quot schemes). I also added a link to the conference page itself, which has links to scans of lecture notes, as the direct lecture notes links seem to be broken.

• Added a link to the retyped version of SGA 4 1/2.

• Wrote “Initial Object” section I mostly just copy-pasted things from the Understanding limits in Set page and modified them. I might have made a mistake in my definition of a colimit, so that should be checked.

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Anonymous

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• Wrote a minimum of substance to this entry. My interest is prompted by current thinking on $n$-dimensional analogues of Euler angles. There are several articles (mainly in quantum chemistry and mathematical physics literature) around 1969-1974, which introduce some analogues via calculational procedures. In my taste an elementary geometrical introduction using both intrinsic and extrinsic approach, in a spirit of Euler, will be more suggestive.

• added to simplicial object a section on the canonical simplicial enrichment and tensoring of $D^{\Delta^{op}}$ for $D$ having colimits and limits.

• Some tidying up and additions at simplex category, in particular a section on its 2-categorical structure, and more on universal properties.

I’ve edited the definition to focus more on the augmented simplex category $\Delta_a$ instead of the ’topologists’ $\Delta$’, but I haven’t changed their names, because it seemed to me that that was the best way to keep everyone involved in the discussion at that page happy. (I also changed the ordinal sum functor from $+$ to $\oplus$, after Tim’s suggestion.)

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Anonymous

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Anonymous

• Added a reference of Robert Furber, Bart Jacobs at Giry monad.

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Anonymous

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