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poset of subobjects did not yet have a remark on the frequently occurring question whether to define subobjects as morphisms or as isomorphism classes thereof, nor did it yet (seem) to have a thread. Added both. In particular, added the phrase “groupoid of subobjects”. Added two reference where the morphism-versus-class-of-morphisms issue is discussed. Refrained from making any recommendations or comparisions, only writing “alternative definition”.
Whenever I see this topic coming up, I am reminded of the “Isomorphic types are equal?!” debate.
Since I do not have much experience in this debate, I shy away from trying to connect isomorphic-types-are-equal with poset of subobjects, but it seems to me that something relevant could be said in this regard in poset of subobjects by someone more experienced.
brief category:people-entry for hyperlinking references at Hermite polynomial, countable set and pi (if we mention him at pi, somebody should reall mention him at Euler number)
This is a bare sub-section, to be !include-ed into relevant entries (at D=5 Maxwell theory, at self-dual higher gauge theory and at Perry-Schwarz Lagrangian).
a stub, for the moment just so as to complete a pattern of entries, but I added pointer to
Paul Wesson, Space-Time-Matter: Modern Kaluza-Klein Theory, World Scientific 1989 (doi:10.1142/3889)
Paul Wesson, James M. Overduin, Principles of Space-Time-Matter: Cosmology, Particles and Waves in Five Dimensions, World Scientific 2018 (doi:10.1142/10871)
brief category:people-entry for hyperlinking references at Kaluza-Klein theory and 5d electromagnetism
recently there were some questions about it here on the nForum: now there is an entry on the Kaluza-Klein mechanism
Added to BF-theory the reference that right now I am believing is the earliest one:
Gary Horowitz, Exactly soluable diffeomorphism invariant theories Commun. Math. Phys. 125, 417-437 (1989)
But maybe I am wrong. Does anyone have an earlier one? I saw pointers to A. Schwarz articles from the late 70s, but I am not sure if he really considered BF as such.
bief category:people-entry, for hyperlinking the reference
that has been asking for this link since January 2020 (here)
I’m making some edits to locally finitely presentable category, and removing some old query boxes. A punchline was extracted, I believe, from the first query box. The second I don’t think is too important (it looks like John misunderstood).
+–{: .query} Mike: Do people really call finitely presentable objects “finitary”? I’ve only seen that word applied to functors (those that preserve filtered colimits). Toby: I have heard ’finite’; see compact object. Mike: Yes, I’ve heard ’finite’ too. =–
+– {: .query} Toby: In the list of equivalent conditions above, does this essentially algebraic theory also have to be finitary?; that is, if it's an algebraic theory, then it's a Lawvere theory?
Mike: Yes, it certainly has to be finitary. Possibly the standard meaning of “essentially algebraic” implies finitarity, though, I don’t know. Toby: I wouldn't use ’algebraic’ that way; see algebraic theory. John Baez: How come the first sentence of this paper seems to suggest that the category of models of any essentially algebraic theory is locally finitely presentable? The characterization below, which I did not write, seems to agree. Here there is no restriction that the theory be finitary. Does this contradict what Mike is saying, or am I just confused?
Mike: The syntactic category of a non-finitary essentially algebraic theory is not a category with finite limits but a category with -limits where is the arity of the theory. A finitary theory can have infinitely many sorts and operations; what makes it finitary is that each operation only takes finitely many inputs, hence can be characterized by an arrow whose domain is a finite limit. I think this makes the first sentence of that paper completely consistent with what I’m saying. =–
created a bare minimum at light-cone gauge quantization, just so as to be able to sensibly link to it from elsewhere
This is a page with fully explicit component computations of properties of the Hodge star operator on Minkowski spacetime. It is a bare sub-section, to be !include-ed inside sections of relevant entries (under “Examples” at Hodge star operator and under “Properties” at Minkowski spacetime)
Thomas Holder is busy creating Hegelian taco as I write.
brief category:people-entry for hyperlinking references at perturbative quantum field theory
copied over, from overcategory, statement and proof of computing limits in undercategories
brief category:people-entry for hyperlinking references at vector meson dominance
John Baez added to graphic category some facts noticed at the cafe; I made left shelf a link and added the appropriate redirect to rack.
added pointer to
and publication data to
I’m not entirely happy with the introduction (“Statement”) to the page axiom of choice. On the one hand, it implies that the axiom of choice is something to be considered relative to a given category (which is reasonable), but it then proceeds to give the external formulation of AC for such a , which I think is usually not the best meaning of “AC relative to ”. I would prefer to give the Statement as “every surjection in the category of sets splits” and then discuss later that analogous statements for other categories (including both internal and external ones) can also be called “axioms of choice” — but with emphasis on the internal ones, since they are what correspond to the original axiom of choice (for sets) in the internal logic.
(I would also prefer to change “epimorphism” for “surjection” or “regular/effective epimorphism”, especially when generalizing away from sets.)
Added alternative terminology “local right adjoint” and “strongly cartesian monad” from Berger-Mellies-Weber. They claim the former “has become the more accepted terminology” than “parametric right adjoint”; does anyone know other references to support this? (I think it’s certainly more logical, in that it fits with the general principle of “local” meaning “on slice categories” — not to be confused with the different general principle of “local” meaning “in hom-objects”.)
I felt there should be an entry category of presheaves. So I started one.
I have split off ordinal sum from the entry on joins as it is needed in other entries as well. I have not revised the entry just doing a cut and paste, so it needs more work!