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Added reference
note that the website linked on this page doesn’t work anymore
Anonymouse
I created a stub page for Douglas Bridges. I linked to his home page but also to a page on FAQs in constructive mathematics. He seems to have other stuff there and there may be other useful links worth creating.
I made “constructive logic” redirect to here (“constructive mathematics”) instead of to “intuitionistic mathematics”, as it used to
added to partial function a new section Definition – General abstract with a brief paragraph on how partial functions form the Kleisli category of the maybe-monad.
Added reference
Anonymouse
I added a new section to algebraically closed field, on the classification in terms of characteristic and transcendence degree.
I looked at real number and thought I could maybe try to improve the way the Idea section flows. Now it reads as follows:
A real number is something that may be approximated by rational numbers. Equipped with the operations of addition and multiplication induced from the rational numbers, real numbers form a number field, denoted . The underlying set is the completion of the ordered field of rational numbers: the result of adjoining to suprema for every bounded subset with respect to the natural ordering of rational numbers.
The set of real numbers also carries naturally the structure of a topological space and as such is called the real line also known as the continuum. Equipped with both the topology and the field structure, is a topological field and as such is the uniform completion of equipped with the absolute value metric.
Together with its cartesian products – the Cartesian spaces for natural numbers – the real line is a standard formalization of the idea of continuous space. The more general concept of (smooth) manifold is modeled on these Cartesian spaces. These, in turnm are standard models for the notion of space in particular in physics (see spacetime), or at least in classical physics. See at geometry of physics for more on this.
I edited Trimble n-category:
added table of contents
added hyperlinks
moved the query boxes that seemed to contain closed discussion to the bottom. I kept the query box where I ask for a section about category theory for Trimble n-categories, but maybe we want to remove that, too. Todd has more on this on his personal web.
I wrote out a proof which uses very little machinery at fundamental theorem of algebra. It is just about at the point where it is not only short and rigorous, but could be understood by an eighteenth-century mathematician. (Nothing important, just fun!)
added a bit more text to the Idea-section at Wick rotation and in particular added cross-links with Osterwalder-Schrader theorem.
Created:
[…]
Fredrik Dahlqvist, Vincent Danos, Ilias Garnier, and Alexandra Silva, Borel kernels and their approximation, categorically. In MFPS 34: Proceedings of the Thirty-Fourth Conference on the Mathematical Foundations of Programming Semantics, volume 341, 91–119, 2018.
Tobias Fritz, A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics. Adv. Math., 370:107239, 2020. arXiv:1908.07021.
Noé Ensarguet, Paolo Perrone, Categorical probability spaces, ergodic decompositions, and transitions to equilibrium. arXiv.
definition as described in
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. =–
definition as per
added an Idea-section to moduli stabilization
Stub. For now just recording links which go beyond the scope of arithmetic cryptography.
added pointer to
for discussion of monopole correlation functions.
Really I am looking for discussion of caloron correlation functions, though…
stub for N=4 D=3 super Yang-Mills theory…
… for the moment just so as to be able to reference the literature on the flavor of S-duality needed at symplectic duality.
moved information on extension types from type theory with shapes to its own page
Added more material to Boolean algebra, particularly the principle of duality and the connection to Boolean rings, and a wee bit of material on Stone duality.
Stone duality deserves greater expansion, bringing out the dualities via ambimorphic (ahem, schizophrenic) structures on the 2-element set, and mentioning the connection to Chu spaces. Another day, another dollar.
felt like adding a handful of basic properties to epimorphism
added pointer to today’s
created computational trinitarianism, combining a pointer to an exposition by Bob Harper (thanks to David Corfield) with my table logic/category-theory/type-theory.
I have started adding references to string field theory , in particular those by Jim Stasheff et al. on the role of L-infinity algebra and A-infinity algebra. Maybe I find time later to add more details.
Added a cross-reference to Dedekind finite object.
Here is an old discssion box from finite object which hereby I am moving from there to here.
+–{.query}
Toby: I think that I'll move the internal stuff to finite object, to keep each page relatively short.
By the way, do I understand you correctly that ’finite object’ in topos theory by default means ’decidable K-finite object’?
Mike: Okay (to the move). To the question, I’m realizing more and more that I don’t really have the background to be able to say what “topos theorists” say. My only source for this material is the Elephant (and what I’ve been able to deduce on my own, which of course tells us nothing about terminology). The Elephant never says “finite object” unqualified; only “finite cardinal” or “K-finite object” or “decidable K-finite object” or “-finite object.” If “projective” means “externally projective,” and likewise for “choice” and (maybe) “inhabited,” then “finite object” should mean “finite cardinal,” but I wouldn’t use it that way myself out of fear of ambiguity and since “finite cardinal” means the same thing. I don’t see any objection to “internally finite object” meaning “decidable K-finite object,” though.
=–
Made a start on this page, which seems like a useful reference to have. The format might work better as a table than a list. It’s missing a lot of cases at the moment. Some might be easily derivable from existing literature, but I think others have not been considered (e.g. the fully bicategorical case).
Added a table of contents to topos, a section on "special classes" and one on "higher toposes".
following Zoran’s suggestion I added to the beginning of the Idea-section at monad a few sentences on the general idea, leading then over to the Idea with respect to algebraic theories that used to be the only idea given there.
Also added a brief stub-subsection on monads in arbitrary 2-categories. This entry deserves a bit more atention.
the entries dependent type and indexed set did not know of each other.
I have now cross-linked them minimally in their “Related entries”-sections. But this would deserve to be expanded on for exposition…
added to E7 the statement of the decomposition of the smallest fundamental rep under and (here) and used this then to expand the existing paragraph on As U-duality group of 4d SuGra