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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• 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.

• have touched this old entry for formatting, hyperlinking, grammar and spelling. The two comment paragraphs below the definition should probably either go up into the Idea-section and else be moved to their own Outloook-section or similar. Not sure.

• a stub, for the moment just so as to complete a pattern of entries, but I added pointer to

• 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.

• 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 $\kappa$-limits where $\kappa$ 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. =–

• Added actual definition and a brief idea.

• 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)

• Will start something here.

• starting something – not done yet but need to save

• In codomain fibration one calls the function

C \ (-) : C --> Cat

mapping c to the slice category (C \ c) a pseudofunctor. However I fail to see how this is not functorial.

A morphism f : a --> b is sent to the functor (C \ f) : (C \ a) --> (C \ b) defined by (g : c --> a) |--> (fg : c --> b), and this assignment clearly satisfies composition. It also preserves identity. So what am I missing here?
• copied over, from overcategory, statement and proof of computing limits in undercategories

• some minimum

• 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 $C$ (which is reasonable), but it then proceeds to give the external formulation of AC for such a $C$, which I think is usually not the best meaning of “AC relative to $C$”. 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.