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I have expanded and edited moment map.
added a sentence to the Idea-section at Kan complex
added to identity type a mentioning of the alternative definition in terms of inductive types (paths).
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…
started function field analogy – table, but didn”t get very far before being interrupted now
was aiming for the table in section 2.6 of
Created left cancellative category. This is a useful technical term.
One natural (and non-posetal) example is the category of fields with ring homomorphisms as the morphisms—provided that the zero ring is removed from it.
Incidentally: is there a usual technical term for, when considering any category , the full subcategory obtained by removing all terminal objects of ?
This is maybe mainly for entertainment. But don’t forget that for newcomers there is a real issue here which may well be worth explaining:
In mathematics it happens at times that one and the same concept is given two different names to indicate a specific perspective, a certain attitude as to what to do whith such objects.
Here are examples:
A quiver is just a directed graph (pseudograph, to be explicit). But one says quiver instead of directed graph when one is interested in studying quiver representations: functors from the free category on that graph to the category of finite-dimensional vector spaces.
A presheaf is just a contravariant functor. But one says presheaf instead of contravariant functor when one is interested in studying its sheafification, or even if one is just intersted in regarding the category of functors with its structure of a topos: the presheaf topos.
(…)
added section about Bézout domains in constructive mathematics, most of the text copied from principal ideal domain
Anonymous
added pointer to today’s
I noticed that “isomorphism class” was just re-directing to decategorification. Just for completeness, I am giving it its own little entry hereby.
Cleaned up partition of unity and fine sheaf a bit, so I could link to them from this MO answer to the question ’Why are there so many smooth functions?’.
I gave the entry logical relation an Idea-section, blindly stolen from a pdf by Ghani that I found on the web. Please improve, I still don’t know what a “logical relation” in this sense actually is.
Also, I cross-linked with polymorphism. I hope its right that “parametricity” may redirect there?
felt the desire to have an entry on the general idea (if any) of synthetic mathematics, cross-linking with the relevant examples-entries.
This has much room for being further expanded, of course.
I just discovered that, all along, the term “quiver representation” was just redirecting to representation. Have started this dedicated page now, with the bare minimum
Linking to the new page torsion points of an elliptic curve.
I have been further working on the entry higher category theory and physics. There is still a huge gap between the current state of the entry and the situation that I am hoping to eventually reach, but at least now I have a version that I no longer feel ashamed of.
Here is what i did:
Partitioned the entry in two pieces: 1. “Survey”, and 2. “More details”.
The survey bit is supposed to give a quick idea of what the set of the scene of fundamental physics is. It starts with a kind of creation story of physics from -topos theory, which – I think – serves to provide a solid route from just the general abstract concept of space and process to the existence and nature of all -model quantum field theories of “-Chern-Simons theory”-type (which includes quite a few) and moreover – by invoking the “holographic principle of higher category theory” – all their boundary theories, which includes all classical phase space physics.
The Survey-bit continues with indicating the formalization of the result of quantizing all these to full extended quantum field theories. It ends with a section meant to indicate what is and what is not yet known about the quantization step itself. This is currently the largest gap in the mathematical (and necessarily higher categorical) formalization of physics: we have a fairly good idea of the mathematics that describes geometric background structure for physics and a fairly good idea of the axioms satisfied by the quantum theories obtained from these, but the step which takes the former to the latter is not yet well understood.
The “More details”-bit is stubby. I mainly added one fairly long subsection on the topic of “Gauge theory”, where I roughly follow the historical route that eventually led to the understanding that gauge fields are modeled by cocycles in higher (nonabelian) differential cohomology.
Apart from this I added more references and some cross-links.
I know that the entry is still very imperfect. If you feel like pointing out all the stuff that is still missing, consider adding at least some keywords directly into the entry.
I worked a little on the entry separated presheaf. Apart from some general editing I
added construction and proof of the separafication functor;
began a section on the full notion of bi-separated presheaf.
More deserves to be done here, but I have to stop for the moment.
Created:
Elements of Mathematics (French: Éléments de mathématique) is a series of books by Nicolas Bourbaki.
Website: https://www.bourbaki.fr/Ouvrages.html
Most of these books were quite influential, some of them especially so.
In particular, Bourbaki’s presentation of multilinear algebra in Chapter III of the second book (Algèbre) was the first detailed expository presentation of the Grassmann algebra.
Another influential book was Groupes et algèbres de Lie, its presentation of root systems and related topics in Chapter 4–6 was the first of its kind.
Some criticisms raised against this series:
Almost complete absence of categories (a deliberate choice);
Théorie des ensembles receives a lot of criticism for its somewhat nonstandard treatment of logic, as well as the rather awkward notion of a structure, which occupies a similar niche to categories.
The use of Daniell’s approach in Intégration, mostly ignoring abstract measure spaces.
Chapter 10 of Algèbre (homological algebra) only covers the very classical results, (intentionally) ignoring derived categories.
Added the statement of the Isbell-Freyd characterization of concrete categories, in the special case of finitely complete categories for which it looks more familiar, along with the proof of necessity.
added textbook pointer:
I gave continuous map a little bit of substance by giving it an actual Idea-paragraph and by writing out the epsilontic definition for the case of metric spaces, together with its equivalence to the “abstract” definition in terms of opens.