Since about two weeks now, we have full-server backups – automatically built and quickly deployable.

Let’s stress-test the temporary test deployment to gauge availability of all features:

Occasionally you might be redirected to the true nLab or the true nForum, that it is to be expected, but feel free to report such issues here.

]]>Stub.

]]>have created full sub-2-category

also reworked full subcategory a little

]]>I brushed up the entry power a bit: wrote an Idea-section, created an Examples-section etc.

]]>I added to walking structure a 2-categorical theorem that implies that usually “the underlying X of the walking X is the initial X”. This fact seems like it should be well-known, but I don’t offhand know a reference for it, can anyone give a pointer?

]]>also created *axiom UIP*, just for completeness. But the entry still needs some reference or else some further details.

added a motivation from the pov of enriched category theory

]]>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 $F$ *equipped with* an isomorphism $F\cong hom_C(-,c)$ is not a represent**able** functor, it is a represent**ed** 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 $F : D \to C$ … is an object $lim F$ of $C$ *equipped with* morphisms to the objects $F(d)$ for all $d \in D$…” (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 $G$ be a group” rather than “let $(G,\cdot,e)$ 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 $F\cong hom_C(-,c)$” or “functor equipped with an isomorphism $F\cong hom_C(-,c)$.” (As long as it doesn’t mean something stupid like “functor equipped with an object $c$ such that there exists an isomorphism $F\cong hom_C(-,c)$.”) 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 ]

]]>

I created [[cone morphism]] and immediately realized I needed help.

I thought it was cool when I finally understood it after staring at it a few times at [[Understanding Constructions in Set]]. However, my understanding is in terms of objects and components of cones, but there is probably some slick way to define it all in one fell swoop.

]]>Added a diagram to [[cone]] and changed some notation to be compatible with [[cone morphism]] and [[Understanding Constructions in Set]]

]]>Added table of contents and section headers

Anonymouse

]]>I tried to polish and impove the idea-section at lax natural transformation after pointing to it from MO

]]>created [[Tannaka duality]]

with a short proof of the duality for the category of permutation representations of a group, using the Yoneda lemma three or four times in a row and nothing else.

either I am mixed up (in which case we'll roll back), or I guess this is the way that it's usually done in the literature? I haven't really checked. Sorry, I just needed that quickly as a lemma for my discussion at [[homotopy group of an infinity-stack]]

]]>starting discussion page

]]>stub on rational dagger categories

Anonymouse

]]>I edited the formatting of [[internal category]] a bit and added a link to [[internal infinity-groupoid]]

it looks like the first query box discussion there has been resolved. Maybe we can remove that box now?

]]>I added two recent examples of enriched categories: tangent bundle categories and Lawvere theories.

]]>Since this relationship is discussed on both enriched category and internal category, it seems useful to have a dedicated page, which also makes it easier to add references.

]]>I have edited the old entry *n-fold category* a little, brought the content into proper order, fixed the link to cat-n-groups and cross-linked with *n-fold complete Segal space*.

Created.

]]>This concept is a fundamental concept in double category theory, and is referenced on other pages, so it seemed worth creating a (for now, minimal) page.

]]>Started a page on this concept, which appears useful in 2-category theory, but is still not particularly well known.

]]>I added a section to filtered category about generalized filteredness relative to a class of small categories, as studied by Adamek-Borceux-Lack-Rosicky, and mentioned that it yields a better notion of $\kappa$-filteredness for the finite regular cardinal $2$, as pointed out by Zhen in another thread.

]]>oidification of magma

Anonymous

]]>starting a page on Poisson/commutator brackets of flux observables in (higher) gauge theory.

The title of the entry follows the title of Freed, Moore & Segal 2007a because that’s a good succinct description of the subject matter, but I don’t mean the entry to be restricted to their particular perspective (in fact, is their uncertainty relation not ultimately a definition – their Def. 1.29 – rather than a derivation from first principles?)

The most insightful discussion of the matter that I have seen so far is that in Cattaneo & Perez 2017, which is motivated by application to first-order formulation of gravity (where this has found a lot of attention), but I think the arguments apply verbatim to Yang-Mills theory, too (where however I haven’t seen it find any attention yet(?)).

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