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this is a bare list of references, meant to be !include-ed into the relevant References-sections at black hole information paradox and Bekenstein-Hawkind entropy
I had accidentally duplicated this page, but merging it now into the new one: Robert B. Griffiths
added pointer to today’s
added table of contents, headers, redirects, and a link to family of sets and union
Anonymous
at foundation of mathematics I have tried to start an Idea-section.
Also, I am hereby moving a bunch of old discussion boxes from there to here:
[ begin forwarded discussion ]
+– {: .query} Urs asks: Concerning the last parenthetical remark: I suppose in this manner one could imagine -categories as a foundation for -categories? What happens when we let ?
Toby answers: That goes in the last, as yet unwritten, section. =–
+– {: .query} Urs asks: Can you say what the problem is?
Toby answers: I'd say that it proved to be overkill; ETCS is simpler and no less conceptual. In ETCC (or whatever you call it), you can neatly define a group (for example) as a category with certain properties rather than as a set with certain structure. But then you still have to define a topological space (for example) as a set with certain structure (where a set is defined to be a discrete category, of course). I think that Lawvere himself still wants an ETCC, but everybody else seems to have decided to stick with ETCS.
Roger Witte asks: Surely in ETCC, you define complete Heyting algebras as particular kinds of category and then work with Frames and Locales (ie follow Paul Taylor’s leaf and apply Stone Duality). You should be able to get to Top by examining relationships between Loc and Set. I thought Top might be the the comma category of forgetful functor from loc to set op and the contravariant powerset functor. Thus a Topological space would consist of a triple S, L, f where S is a set, L is a locale and f is a function from the objects of the locale to the powerset of S. A continuous function from S, L, f to S’, L’, f’ is a pair g, h where g is a function from the powerset of S’ to the powerset of S and g is a frame homomorphism from L’ to L and (I don’t know how to draw the commutation square). However I think this has too many spaces since lattice structures other than the inclusion lattice can be used to define open sets.
Toby: It's straightforward to define a topological space as a set equipped with a subframe of its power set. So you can define it as a set , a frame , and a frame monomorphism , or equivalently as a set , a locale , and an epimorphism of locales, where is the discrete space on as a locale. (Your ’However, […]’ sentence is because you didn't specify epimorphism/monomorphism.) This is a good perspective, but I don't think that it's any cleaner in ETCC than in ETCS.
Roger Witte says Thanks, Toby. I agree with your last sentence but my point is that this approach is equally clean and easy in both systems. The clean thing about ETCC is the uniformity of meta theory and model theory as category theory. The clean thing about ETCS is that we have just been studying sets for 150 years, so we have a good intuition for them.
I was responding to your point ’ETCC is less clean because you have to define some things (eg topological spaces) as sets with a structure’. But you can define and study the structure without referring to the sets and then ’bolt on’ the sets (almost like an afterthought).
Mike Shulman: In particular cases, yes. I thought the point Toby was trying to make is that only some kinds of structure lend themselves to this naturally. Groups obviously do. Perhaps topological spaces were a poorly chosen example of something that doesn’t, since as you point out they can naturally be defined via frames. But consider, for instance, a metric space. Or a graph. Or a uniform space. Or a semigroup. All of these structures can be easily defined in terms of sets, but I don’t see a natural way to define them in terms of categories without going through discrete categories = sets.
Toby: Roger, I don't understand how you intend to bolt on sets at the end. If I define a topological space as a set , a frame , and a frame monomorphism from to the power frame of , how do I remove the set from this to get something that I can bolt the set onto afterwards? With semigroups, I can see how, from a certain perspective, it's just as well to study the Lawvere theory of semigroups as a cartesian category, but I don't see what to do with topological spaces.
Roger Witte says If we want to found mathematics in ETCC we want to work on nice categories rather than nice objects. Nice objects in not nice categories are hard work (and probably ’evil’ to somke extent). Thus the answer to Toby is that to do topology in ETCC you do as much as possible in Locale theory (ie pointless topology) and then when you finally need to do stuff with points, you create Top as a comma like construction (ie you never take away the points but you avoid introducing them as long as possible). Is it not true that the only reason you want to introduce points is so that you can test them for equality/inequality (as opposed to topological separation)?
Mike, I spent about two weeks trying to figure out how to get around Toby’s objection ’topology’ and now you chuck four more examples at me. My gut feeling is that the category of directed graphs is found by taking the skeleton of CAT, that metric locales are regular locales with some extra condition to ensure a finite basis, that Toby can mak
[ to be continued in next comment ]
moved material about category theory and foundations from foundations of mathematics to its own page
Anonymous
added to symmetric monoidal category a new Properties-section As models for connective spectra with remarks on the theorems by Thomason and Mandell.
I noticed that there was a neglected stub entry universe that failed to link to the fairly detailed (though left in an unpolished state full of forgotten discussions) Grothendieck universe.
I renamed the former to universe > history and made “universe” redirect to “Grothendieck universe”
Added:
A definitive source (by one of the authors of the theory) is
a bare list of references, to be !includ-ed into the list of references of relevant entries, such as at quantum computing and quantum programming, for ease of updating and syncing
Added reference to the paper
Paul Blain Levy, Formulating Categorical Concepts using Classes, arXiv:1801.08528
created small set
Adding reference
as a construction of the locale of real numbers can be found in section 5.3 of that article
Anonymous
The entry used to be a plain disambiguation between reflective subcategory and an empty list of other meaning of reflection.
But since the only evident point that entries like rotation, translation, boost, etc. could point to for related concepts is an entry called reflection. So maybe that should be the main meaning here, and alternative meaning be secondary.
I slightly edited accordingly. But besides the different link structure now, there is still no substantial content here.
brief category:people-entry for hyperlinking references at associative n-category and homotopy.io
started page on a class of set theories, which include unsorted set theory, two-sorted set theory, and three-sorted set theory, and where membership is a relation. Usually contrasted with dependently sorted set theory, where membership is a typing judgment.
Anonymous
starting article on set theories with three sorts, an example of which is structural ZFC.
Anonymous
added to locale a section relation to toposes stating localic reflection
adding reference
Anonymous
expanded the discussion at equivariant homotopy theory
expanded the statement of the classical Elmendorf theorem
added the statement of the general Elmendorf theorem in general model categories
added remarks on G-equivariant oo-stacks, as special cases of this
have added under “Selected writings” the Kervaire-saga (here and at he other author’s pages):
Michael Hill, Michael Hopkins, Douglas Ravenel, Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem, New Mathematical Monographs, Cambridge University Press (2021) [doi:10.1017/9781108917278]
Michael Hill, Michael Hopkins, Douglas Ravenel, On the non-existence of elements of Kervaire invariant one, Annals of Mathematics 184 1 (2016)[doi:10.4007/annals.2016.184.1.1, arXiv:0908.3724, talk slides]
Michael Hill, Michael Hopkins, Douglas Ravenel, The Arf-Kervaire problem in algebraic topology: Sketch of the proof, Current Developments in Mathematics, 2010: 1-44 (2011) (HHRKervaire.pdf:file, doi:10.4310/CDM.2010.v2010.n1.a1)
Michael Hill, Michael Hopkins, Douglas Ravenel, The Arf-Kervaire invariant problem in algebraic topology: introduction (2016) [pdf]
I made “constructive logic” redirect to here (“constructive mathematics”) instead of to “intuitionistic mathematics”, as it used to
Added reference
Anonymous
Urs has added Euler integration prompted by Tom’s post at nCafe; I wanted to do that and will contribute soon. I noticed there is no entry integral in Lab, but it redirects to integration. I personally think that integral as a mathematical object is a slightly more canonical name for a mathematical entry than integration, if the two are not kept separated. Second, the entry is written as an (incomplete) disambiguation entry and with a subdivision into measure approach versus few odd entries. I was taught long time ago by a couple of experts in probability and measure theory that a complete subordination to the concept of integral to a concept of measure is pedagogically harmful, and lacks some important insights. This has also to do with the choice of the title: integration points to a process, and the underlying process may involve measure. Integral is about an object which is usually some sort of functional, or operator, on distributions which are to be acted upon.
Thus I would like to rename the entry into integral (or to create a separate entry from integration) and make it into a real entry, the list of variants being just a section, unlike in the disambiguation only version. What do you think. Then I would add some real ideas about it.
adding accents to the page name, to make Dmitri’s requested links work at localic group
moving section on precompact spaces under properties section instead of definition section
Anonymous
Created weak excluded middle, which is equivalent to the one of de Morgan’s laws that is not intuitionistically valid.
these are to De Morgan Heyting algebras what Heyting categories are to Heyting algebras and what Boolean categories are to Boolean algebras.
Anonymous