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I think the second sentence below needs to have the phrase “torsion-free” added to it twice. Right? I’m going to do that.
a) The category of -rings is monadic and comonadic over the category of of commutative rings.
b) The category of -rings is monadic and comonadic over the category of of commutative rings.
added reference to dendroidal version of Dold-Kan correspondence
Stub a page for what has been called “the most important law”, “the only unbreakable law”, and a generalization of both Amdahl’s and Brooks’ laws. While this is important to software engineering, it’s applicable to any engineered system, and Conway 1968 uses all sorts of infrastructure to make their point alongside software-specific examples.
stub for braid group statistics (again, for the moment mainly in order to record a reference)
Have added to cyclic set a pointer to notes from 1996 by Ieke Moerdijk where the theory classified by the topos of cyclic sets is identified (abstract circles).
This is an unpublished note, but on request I have now uploaded it to the nLab
I have also added a corresponding brief section to classifying topos.
By the way, there is an old query box with an exchange between Mike and Zoran at cyclic set. It seems to me that this has been resolved and the query box could be removed (to make the entry read more smoothly). Maybe Mike and/or Zoran could briefly look into this.
have added pointers to Alex Hoffnung’s preprint to tetracategory, tricategory, span and (infinity,n)-category of spans.
Created a stub for this concept, as I think it’s important to distinguish between coherence theorems and strictification theorems, as, while they are related, they are not the same, and their relationship can be quite subtle. I plan to expand this page and move some content over from coherence theorem soon.
added an Idea-section to coherence theorem for monoidal categories just with the evident link-backs and only such as to provide a minimum of an opening of the entry
added to gravity references discussing the covariant phase space of gravity, as part of a reply to this TP.SE-question
I have tried to brush-up existential quantifier a little more. But not really happy with it yet.
The entry used to start out with the line “not to be confused with neutral element”. This was rather suboptimal. I have removed that sentence and instead expanded the Idea-section to read now as follows:
Considering a ring , then by the unit element one usually means the neutral element with respect to multiplication. This is the sense of “unit” in terms such as nonunital ring.
But more generally a unit element in a unital (!) ring is any element that has an inverse element under multiplication.
This concept generalizes beyond rings, and this is what is discussed in the following.
expanded concrete sheaf: added the precise definition and some important properties.
stub for Hilbert’s sixth problem
added pointer to:
added to the entry on David Hilbert a pointer to this remarkable recording:
Added this pointer also, cross-link wise, at Galileo Galilei and at The Unreasonable Effectiveness of Mathematics in the Natural Sciences
Adding reference
Anonymouse
Unfortunately, there are two entries on the same topic, both created by Urs: quantum Hall effect (redirecting also fractional quantum Hall effect what should eventually split off) with some substance, and the microstub quantum hall effect. I would like to create quantum spin Hall effect and I think I should rename/reclaim the stub quantum hall effect for this. Do others agree ? Urs ?
As the action is now delayed I record here the reference which I wanted to put there
Somewhat surprisingly, the authors and roughly this work of them are mentioned (though not in the list of references) in a paper in algebraic geometry
which considers the mirror symmetry and topological states of matters (topological insulators in particular) as main applications.
Todd had created subdivision.
I interlinked that with the entry Kan fibrant replacement, where the subdivision appears.
created a minimum at function monad (aka “reader monad”, “environment monad”)
mathematical physics with a slight distinction from physical mathematics which points to the same entry. The relation to theoretical physics has been discussed, but I am not sure yet if we should have theoretical physics as a separate entry so I do not put is as another redirect.
added to gerbe
definition of -gerbes;
classification theorem by -cohomology;
the notion of banded -gerbes.
I gave the category:people entry Daniel Freed a bit of actual text. Please feel invited to edit further. Currently it reads as follows:
Daniel Freed is a mathematician at University of Texas, Austin.
Freed’s work revolves around the mathematical ingredients and foundations of modern quantum field theory and of string theory, notably in its more subtle aspects related to quantum anomaly cancellation (which he was maybe the first to write a clean mathematical account of). In the article Higher Algebraic Structures and Quantization (1992) he envisioned much of the use of higher category theory and higher algebra in quantum field theory and specifically in the problem of quantization, which has – and still is – becoming more widely recognized only much later. He recognized and emphasized the role of differential cohomology in physics for the description of higher gauge fields and their anomaly cancellation. Much of his work focuses on the nature of the Freed-Witten anomaly in the quantization of the superstring and the development of the relevant tools in supergeometry, and notably in K-theory and differential K-theory. More recently Freed aims to mathematically capture the 6d (2,0)-superconformal QFT.
am starting curvature characteristic form and Chern-Simons form.
But still working…
I have begun cleaning up the entry cycle category, tightening up definitions and proofs. This should render some of the past discussion obsolete, by re-expressing the intended homotopical intuitions (in terms of degree one maps on the circle) more precisely, in terms of “spiraling” adjoints on the poset .
Here is some of the past discussion I’m now exporting to the nForum:
The cycle category may be defined as the subcategory of Cat whose objects are the categories which are freely generated by the graph , and whose morphisms are precisely the functors of degree (seen either at the level of nerves or via the embedding given by on the level of objects, the rest being obvious).
The simplex category can be identified with a subcategory of , having the same objects but with fewer morphisms. This identification does not respect the inclusions into , however, since and are different categories.
started cubical type theory using a comment by Jonathan Sterling
Inspired by a discussion with Martin Escardo, I created taboo.
Created polymorphism.
I added this to the entry for Nima Arkani-Hamed.
Urs (or anyone else) do you know anything about Nima’s recent interest in category theory?
“six months ago, if you said the word category theory to me, I would have laughed in your face and said useless formal nonsense, and yet it’s somehow turned into something very important in my intellectual life in the last six months or so” (@ 44:05 in The End of Space-Time July 2022)
A combinatorial notion in the study of total positivity.
for completeness, to go with the other entries in coset space structure on n-spheres – table
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.