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$\begingroup$ +1. Henry, that is quite a remarkable result. Your notion of equivalence is stronger than the one I had proposed, since you say two expressions are equivalent if it is true that they define the same function, whereas I had said they are equivalent if it is provable in some fixed theory that they define the same function.
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First of all, the "extensive list of axioms" is largely a list of definitions (of rather natural terminology). ). Anyway, he seems to be just getting at the issue of the use of fiber squares in arguments involving properties of morphisms, such as come up in many of the basic constructions in algebraic geometry (Hilbert schemes, quotients by group actions, descent theory, et
A definition due to Devaney (see his text, An Introduction to Chaotic Dynamical Systems, Benjamin/Cummings 1986, MR0811850 (87e:58142)) seems to be the one most commonly accepted today. By the Devaney definition, period three does not imply chaos; for example, a function with a stable 3-cycle is not Devaney-chaotic.
(This won't simplify the definition of commutative algebra, but will be a much shorter account which doesn't muck about with general operads.) $\endgroup$ – Jacob Lurie Commented Mar 29, 2018 at 11:25
I would like a precise definition/construction of such a measure on a general symmetric space. I have seen the example of the upper half-plane where the unit tangent bundle can be identified with a group and one uses the Haar measure. Setup: Assume $(G,\mathfrak{g})$ is a non-compact, real, semi-simple, Lie group with finite center.
I feel that for a fully elegant definition of integers one needs to readdress the definition of an Abelian group. For the sake of communication I will even introduce a synonym minusop for abelian groups to stress the independence of their definition from the general groups. The general groups have an elegant definition (perhaps more than one).
It was verified by computer. It is big but it fits in a page. I am not sure there is an obvious simplifying move, it would be nice to know. By the way it would be interesting to have a definition of what a 'generalized Reidemeister move' is. A 'good' definition could perhaps give interesting diagrams of the unknot that are not easy to simplify.
Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.