Mike Stay kindly added the standard QM story to [[path integral]].

I changed the section titles a bit and added the reference to the Baer-Pfaeffle article on the QM path integral. Probably the best reference there is on this matter.

]]>As far as I can tell, Ehrhard’s definition of comprehension requires not just that the fibers have terminal objects but that these are preserved by the reindexing functors. This is automatic if the fibration is a bifibration, as in Lawvere’s version; it’s fairly explicit in Ehrhard’s formulation, and somewhat implicit in Jacobs’ but I believe still present (his “terminal object functor” must, I think, be a *fibered* terminal object).

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 ]

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brief `category:people`

-entry for hyperlinking references

Updated the linkref `weak initial algebras' to`

weak inital’ (a.k.a. ‘weakly initial’)

brief `cateory:people`

-entry for hyperlinking references

Recording the result from Triantafillou 82, characterizing injective/projective objects in diagrams of vector spaces over (the opposite of) the orbit category.

(The degreewise ingredients in the rational model for topological G-spaces)

]]>I added some first statements about projective resolutions also to *projective object*.

Added a reference to

- John W. Gray,
*Coherence for the Tensor Product of 2-Categories, and Braid Groups*, pp.62-76 in Heller, Tierney (eds.),*Algebra, Topology, and Category Theory*, Academic Press New York 1976.

Create a new page to keep record of PhD theses in category theory (with links to the documents where possible), particularly older ones that are harder to discover independently. At the moment, this is just a stub, but I plan to fill it out more when I have the chance.

]]>I corrected the ’historical’ comment. The Cartan seminar is December 1956 and E. H. Brown’s paper was submitted in 1958 so essentially the two theories were developed in tandem.

]]>starting page on hierarchy of universes in type theory

Anonymouse

]]>stub for jet bundle

]]>created page

]]>As there had been a change to the entry for Ross Street I gave it a glance. Is there a reason that the second reference is to a paper without Ross as an author?I hesitate to delete it as there may be a hidden reason. (I have edited this discussion entry to remedy the point that Todd and Urs have made below. I also edited the title of this discussion!)

]]>- Yuri Berest, Oleg Chalykh,
*Deformed Calogero–Moser operators and ideals of rational Cherednik algebras*, Commun. Math. Phys. 400, 133–178 (2023) doi arXiv:2002.08691

Created this entry, per discussion on topology.

]]>added pointer to:

- Alberto Cattaneo,
*Poisson Structures from Corners of Field Theories*, talk at CQTS (22 Nov 2023) [slides: pdf, video: YT]

starting article on Coquand universes, aka universes à la Coquand

Anonymouse

]]>created stub for [[symplectic groupoid]], effectively just regording my blog entries on Eli Hawkins' program of geometric quantization of Poisson manifolds

]]>Spurred by an MO discussion, I added the observation that coproduct inclusions are monic in a distributive category.

]]>gave this entry some formatting and added links

]]>it has annoyed me for a long time that bilinear form did not exist. Now it does. But not much there yet.

]]>started adding something (the example of the Hopf fibration and some references).

What’s a canonical reference on the Whitehead products corresponding to the Hopf fibrations? Like what is an original reference and what is a textbook account?

]]>brief `category:people`

-entry for hyperlinking references