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fixed link for
added to KK-theory brief remark and reference to relation to stable -categories / triangulated categories
created microlinear space
One thing I might be mixed up above:
in the literature I have seen it seems to say that
$ X^D x_X X^D \simeq X^{D(2)}$
with
$ D(2) = { (x_1,x_2) \in R \times R | x_i x_j = 0} $.
But shouldn't it be
$ D(2)' = { (x_1,x_2) \in R \times R | x_i^2 = 0} $.
?
See Day convolution
I started writing up the actual theorem from Day’s paper “On closed categories of functors”, regarding an extension of the “usual” Day convolution. He identifies an equivalence of categories between biclosed monoidal structures on the presheaf category and what are called pro-monoidal structures on A (with appropriate notions of morphisms between them) (“pro-monoidal” structures were originally called “pre-monoidal”, but in the second paper in the series, he changed the name to “pro-monoidal” (probably because they are equivalent to monoidal structures on the category of “pro-objects”, that is to say, presheaves)).
This is quite a bit stronger than the version that was up on the lab, and it is very powerful. For instance, it allows us to seamlessly extend the Crans-Gray tensor product from strict ω-categories to cellular sets (such that the reflector and Θ-nerve functors are strong monoidal). This is the key ingredient to defining lax constructions for ω-quasicategories, and in particular, it’s an important step towards the higher Grothendieck construction, which makes use of lax cones constructed using the Crans-Gray tensor product.
wrote an Idea-section at quantum field theory
added pointer to:
removed the following ancient query box discussion:
+–{.query} Left I could understand, but right? —Toby
The way I rewrote it explains it. It is unfortunate that the Eilenberg-Watts theorem treated in Bass was using only right adjoint functors so later they dropped word right. – Zoran
Thanks. —Toby =–
Todd,
you added to Yoneda lemma the sentence
In brief, the principle is that the identity morphism is the universal generalized element of . This simple principle is surprisingly pervasive throughout category theory.
Maybe it would be good to expand on that. One might think that the universal property of a genralized element is that every other one factors through it uniquely. That this is true for the generalized element is a tautological statement that does not need or imply the Yoneda lemma, it seems.
added pointer to yesterday’s
I am giving fiber integration in K-theory a dedicated entry.
One section In operator K-theory used to be a subsection of fiber integration in generalized cohomology, and I copied it over.
Another section In terms of bundles of Fredholm operators I have now started to write.
I have added a little bit to supermanifold, mainly the definition as manifolds over superpoints, the statement of the equivalence to the locally-ringed-space definition and references.
added pointer to:
here and elsewhere
I wanted to be able to use the link without it appearing in grey, so I created a stub for general relativity.
Started an entry in “category:motivation” on fiber bundles in physics.
(prompted by this Physics.SE question)
recorded some recent surveys of the status of MOND at MOND
a bare list of references, to be !include
-ed into relevant entries (such as Witten genus, M5-brane elliptic genus but also inside elliptic cohomology – references) – for ease of harmonizing lists of references
stub for confinement, but nothing much there yet. Just wanted to record the last references there somewhere.
the standard bar complex of a bimodule in homological algebra is a special case of the bar construction of an algebra over a monad. I have added that as an example to bar construction.
I also added the crucial remark (taken from Ginzburg’s lecture notes) that this is where the term “bar” originates from in the first place: the original authors used to write the elements in the bar complex using a notaiton with lots of vertical bars (!).
(That’s a bad undescriptive choice of terminoiogy. But still not as bad as calling something a “triple”. So we have no reason to complain. ;-)
starting a stand-alone Section-entry (to be !include
ed as a section into D=11 supergravity and into D’Auria-Fré formulation of supergravity)
So far it contains lead-in and statement of the result, in mild but suggestive paraphrase of CDF91, §III.8.5.
I am going to spell out at least parts of the proof, with some attention to the prefactors.
changed entry title to full name,
added “category:people” tag,
updated webpage url,
added section “Related nLab entries”, so far with a pointer to Grothendieck construction
created field with one element with two useful references
started self-dual higher gauge theory. Just minimal idea and list of references so far.
am adding references, such as this one:
Added:
The inclusion of the 2-category of monoidal categories into the 2-category of rigid monoidal categories admits a left 2-adjoint functor .
Furthermore, the unit of the adjunction is a strong monoidal fully faithful functor, i.e., any monoidal category admits a fully faithful strong monoidal functor , where is a rigid monoidal category.
See Theorems 1 and 2 in Delpeuch \cite{Delpeuch}.
I’ve added to Eilenberg-Moore category an explicit definition of EM objects in a 2-category and some other universal properties of EM categories, including Linton’s construction of the EM category as a subcategory of the presheaves on the Kleisli category.
Question: can anyone tell me what Street–Walters mean when they say that this construction (and their generalised one, in a 2-category with a Yoneda structure) exhibits the EM category as the ‘category of sheaves for a certain generalised topology on’ the Kleisli category?
Made some some small improvements (ordering of sections, note on how the definition defaults to the usual definition of adjoints, fixing broken link in the references, etc) in relative adjoint functor.
starting a category:reference
-entry.
Just a single item so far, but this entry should incrementally grow as more preprints appear (similar to what we have been doing at Handbook of Quantum Gravity and similar entries).
I know that a soft deadline for submissions of at least one of the sections is this December, so I am guessing this is planned to appear in 2024.
I left a counter-query underneath Zoran’s query at compactly generated space. It may be time for a clean-up of this article; the query boxes have been left dangling and unanswered for quite some time. Either proofs or references to detailed proofs would be welcome.
am in the process of adding some notes on how the D=5 super Yang-Mills theory on the worldvolume of the D4-brane is the double dimensional reduction of the 6d (2,0)-superconformal QFT in the M5-brane.
started a stubby double dimensional reduction in this context and added some first further pointers and references to M5-brane, to D=5 super Yang-Mills theory and maybe elsewhere.
But this still needs more details to be satisfactory, clearly.
I have started on a revision of algebraic K-theory. The old version launched straight into a particular nPOV, which really just summarised the Blumberg et al paper, and did not mention any of the other ideas in the area. At present I have just put in some historical stuff, but given the importance of the subject e.g. in modern C*-algebra the page needs a lot more work.
created tangent category
in order to have a place where to keep just details on the purely 1-categorical "shadow" of tangent (infinity,1)-category.