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I edited the formatting of internal category a bit and added a link to internal infinity-groupoid
it looks like the first query box discussion there has been resolved. Maybe we can remove that box now?
The pages apartness relation and antisubalgebra disagree about the definition of an antiideal: do we assume ¬(0∈A) or ∀p∈A,p#0? Presumably there is a similar question for antisubgroups, etc. In particular, the general universal-algebraic definition at antisubalgebra would give ¬(0∈A) as the definition (since 0 is a constant and ⊥ is a nullary disjunction), contradicting the explicit definition of antiideal later on the same page.
Does this have something to do with whether #-openness is assumed explicitly or not? The page apartness relation claims that, at least for antiideals, openness is automatic as long as the ring operations are strongly extensional. But antisubalgebra assumes openness explicitly, in addition to strong extensionality of the algebraic operations.
Finally, do we ever really need the apartness to be tight?
I have expanded and edited moment map.
A method in integrable systems.
FRT approach to quantum groups
New pages quantum linear group, quantized function algebra (redirects also quantized coordinate ring) and quantized enveloping algebra which refer to certain special cases of a general family of notions of quantum group.
one more from Meissner & Nicolai, last week:
I have expanded the Idea-section at deformation quantization a little, and moved parts of the previous material there to the Properties-section.
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 λRing of λ-rings is monadic and comonadic over the category of CRing of commutative rings.
b) The category λRing¬tor of λ-rings is monadic and comonadic over the category of CRing¬tor 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.
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 R, then by the unit element one usually means the neutral element 1∈R 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
Todd had created subdivision.
I interlinked that with the entry Kan fibrant replacement, where the subdivision nerve∘Face 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 G-gerbes;
classification theorem by AUT(G)-cohomology;
the notion of banded G-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.
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 ℤ.
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 [n]Λ which are freely generated by the graph 0→1→2→…→n→0, and whose morphisms Λ([m],[n])⊂Cat([m],[n]) are precisely the functors of degree 1 (seen either at the level of nerves or via the embedding Ob[n]Λ→R/Z≅S1 given by k↦k/(n+1)modZ on the level of objects, the rest being obvious).
The simplex category Δ can be identified with a subcategory of Λ, having the same objects but with fewer morphisms. This identification does not respect the inclusions into Cat, however, since [n] and [n]Λ are different categories.
started cubical type theory using a comment by Jonathan Sterling