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    • reformatted the entry group a little, expanded the Examples-section a little and then pasted in the group-related “counterexamples” from counterexamples in algebra. Mainly to indicate how I think this latter entry should eventually be used to improve the entries that it refers to.

    • maded explicit the identification of equivariant stable homotopy groups with equivariant generalized cohomology groups of the point: here

      diff, v12, current

    • Making a particular page for C_2 equivariant homotopy groups because real Betti realization is fundamental there and doesn’t exist over other groups.

      Natalie Stewart

      v1, current

    • Changing for consistency with the borromean rings

      Natalie Stewart

      diff, v4, current

    • I switched the colors so that they match the logo :)

      Natalie Stewart

      diff, v6, current

    • Page for Bastiaan

      Natalie Stewart

      v1, current

    • Added the new C_2-equivariant stable stems paper

      Natalie Stewart

      diff, v12, current

    • Added the new C_2-equivariant stable stems paper

      Natalie Stewart

      diff, v11, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • a bare minimum entry, for the moment just to record some references

      v1, current

    • I am starting an entry spontaneously broken symmetry. But so far no conceptualization or anything, just the most basic example for sponatenously broken global symmetry.

    • Updating reference to cubical type theory. This page need more work.

      diff, v55, current

    • Added characterization of κ\kappa-compact objects in λ\lambda-accessible categories.

      diff, v63, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • I am moving the following old query box exchange from orbifold to here.

      old query box discussion:

      I am confused by this page. It starts out by boldly declaring that “An orbifold is a differentiable stack which may be presented by a proper étale Lie groupoid” but then it goes on to talk about the “traditional” definition. The traditional definition definitely does not view orbifolds as stacks. Neither does Moerdijk’s paper referenced below — there orbifolds form a 1-category.

      Personally I am not completely convinced that orbifolds are differentiable stacks. Would it not be better to start out by saying that there is no consensus on what orbifolds “really are” and lay out three points of view: traditional, Moerdijk’s “orbifolds as groupoids” (called “modern” by Adem and Ruan in their book) and orbifolds as stacks?

      Urs Schreiber: please, go ahead. It would be appreciated.

      end of old query box discussion

    • added to supergeometry a link to the recent talk

      • Mikhail Kapranov, Categorification of supersymmetry and stable homotopy groups of spheres (video)

    • finally a stub for Segal condition. Just for completeness (and to have a sensible place to put the references about Segal conditions in terms of sheaf conditions).

    • added a few more references with brief comments to QFT with defects

      (this entry is still just a stub)

    • brief category:people-entry for hyperlinking references

      v1, current

    • while adding to representable functor a pointer to representable morphism of stacks I noticed a leftover discussion box that had still be sitting there. So hereby I am moving that from there to here:

      [ begin forwarded discussion ]

      +–{+ .query} I am pretty unhappy that all entries related to limits, colimits and representable things at nlab say that the limit, colimit and representing functors are what normally in strict treatment are just the vertices of the corresponding universal construction. A representable functor is not a functor which is naturally isomorphic to Hom(-,c) but a pair of an object and such isomorphism! Similarly limit is the synonym for limiting cone (= universal cone), not just its vertex. Because if it were most of usages and theorems would not be true. For example, the notion and usage of creating limits under a functor, includes the words about the behaviour of the arrow under the functor, not only of the vertex. Definitions should be the collections of the data and one has to distinguish if the existence is really existence or in fact a part of the structure.–Zoran

      Mike: I disagree (partly). First of all, a functor FF equipped with an isomorphism F≅hom C(−,c)F\cong hom_C(-,c) is not a representable functor, it is a represented functor, or a functor equipped with a representation. A representable functor is one that is “able” to be represented, or admits a representation.

      Second, the page limit says “a limit of a diagram F:D→CF : D \to C … is an object limFlim F of CC equipped with morphisms to the objects F(d)F(d) for all d∈Dd \in D…” (emphasis added). It doesn’t say “such that there exist” morphisms. (Prior to today, it defined a limit to be a universal cone.) It is true that one frequently speaks of “the limit” as being the vertex, but this is an abuse of language no worse than other abuses that are common and convenient throughout mathematics (e.g. “let GG be a group” rather than “let (G,⋅,e)(G,\cdot,e) be a group”). If there are any definitions you find that are wrong (e.g. that say “such that there exists” rather than “equipped with”), please correct them! (Thanks to your post, I just discovered that Kan extension was wrong, and corrected it.)

      Zoran Skoda I fully agree, Mike that “equipped with” is just a synonym of a “pair”. But look at entry for limit for example, and it is clear there that the limiting cone/universal cone and limit are clearly distinguished there and the term limit is used just for the vertex there. Unlike for limits where up to economy nobody doubt that it is a pair, you are right that many including the very MacLane representable take as existence, but then they really use term “representation” for the whole pair. Practical mathematicians are either sloppy in writing or really mean a pair for representable. Australians and MacLane use indeed word representation for the whole thing, but practical mathematicians (example: algebraic geometers) are not even aware of term “representation” in that sense, and I would side with them. Let us leave as it is for representable, but I do not believe I will ever use term “representation” in such a sense. For limit, colimit let us talk about pairs: I am perfectly happy with word “equipped” as you suggest.

      Mike: I’m not sure what your point is about limits. The definition at the beginning very clearly uses the words “equipped with.” Later on in the page, the word “limit” is used to refer to the vertex, but this is just the common abuse of language.

      Regarding representable functors, since representations are unique up to unique isomorphism when they exist, it really doesn’t matter whether “representable functor” means “functor such that there exists an isomorphism F≅hom C(−,c)F\cong hom_C(-,c)” or “functor equipped with an isomorphism F≅hom C(−,c)F\cong hom_C(-,c).” (As long as it doesn’t mean something stupid like “functor equipped with an object cc such that there exists an isomorphism F≅hom C(−,c)F\cong hom_C(-,c).”) In the language of stuff, structure, property, we can say that the Yoneda embedding is fully faithful, so that “being representable” is really a property, rather than structure, on a functor.

      [ continued in next comment ]

    • starting something. Not done yet but need to save

      v1, current

    • The equivariant version of commutative operads

      Natalie Stewart

      v1, current

    • The notion of G-categories is fundamental to the modern approach in equivariant homotopy theory

      Natalie Stewart

      v1, current

    • Adding a page for orbital cats

      Natalie Stewart

      v1, current

    • Add some explanation why Kan condition explains composition and inverse from the groupoid point of view.

      Chenchang Zhu

      diff, v4, current

    • Am trying to get some historical citations straight at linear type theory, maybe somebody can help me:

      what are the original sources of the idea that linear logic/type theory should generally be that of symmetric monoidal categories (“multiplicative intuitionistic LL”)?

      In order of appearance, I am aware of

      • de Paiva 89 gives one particular example of a non-star-autonomous SMC that deserves to be said to interpret “linear logic” and clearly identifies the general perspective.

      • Bierman 95 discusses semantics in general SMCs more generally

      • Barber 97 reviews this and explores a bit more.

      What (other) articles would be central to cite for this idea/perspective?

      I am aware of more recent reviews such as

      but I am looking for the correct “original sources”.

    • added to canonical form references (talk notes) on canonicity or not in the presence of univalence

    • Started this page to record various facts about large cocompletions as I survey some of the literature. It’s possible that some of these concepts should have their own pages eventually, but at the moment this seems awkward as many don’t already have names in the literature (or have names that are not ideal for various reasons).

      v1, current

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

    • Created a stub for this concept.

      v1, current

    • I added a note to compact closed category on the fact that the inclusion from compact closed categories into SMCCs has a left adjoint, pointing to an article by Day where he describes the free compact closed category over a closed symmetric monoidal category as a localization. Question: this left adjoint is not full, but I believe it is faithful – does anyone know how to prove that?

    • Added actual definition for pseudofunctor and modified notions, moved discussion from idea section to new discussion section at bottom of page.

      diff, v13, current

    • Robert Burkland

      Natalie Stewart

      v1, current

    • He’s a mathematician

      Natalie Stewart

      v1, current

    • creating article on tensor product of commutative monoids

      Anonymous

      v1, current

    • Making a page for Inbar

      Natalie Stewart

      v1, current

    • Modes are fundamental to multiplicative infinite loop space theory and multiplicativity in algebraic K theory

      Natalie Stewart

      v1, current

    • Recursively adding references for semiadditivity and ambidexterity

      Natalie Stewart

      v1, current

    • adding recursive references for m-semiadditivity and ambidexterity

      Natalie Stewart

      v1, current

    • Started page on generalized symmetries, with brief description of main Idea.

      v1, current

    • This entry defines Grothendieck topologies using sieves.

      However, in the original definition (Michael Artin’s seminar notes “Grothendieck topologies”), a Grothendieck topology on a category CC is defined as a set TT of coverings.

      More precisely (to cite from Artin’s notes), a Grothendieck topology is defined as families of maps {ϕ i:U i→U} i∈I\{\phi_i\colon U_i\to U\}_{i\in I} such that

      • for any isomorphism ϕ\phi we have {ϕ}∈T\{\phi\}\in T;

      • if {U i→U}∈T\{U_i\to U\}\in T and {V i,j→U i}∈T\{V_{i,j}\to U_i\}\in T for each ii, then {V i,j→U}∈T\{V_{i,j}\to U\}\in T;

      • if {U i→U}∈T\{U_i\to U\}\in T and V→UV\to U is a morphism, then U i× UVU_i\times_U V exist and {U i× UV→V}∈T\{U_i\times_U V\to V\}\in T.

      This is almost identical to the current definition of Grothendieck pretopology, except that in Artin’s definition only the relevant pullbacks are required to exist.

      It seems to me that the original definition by Artin is the one used most often in algebraic geometry.

      diff, v43, current

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