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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.
(…)
removing old comment in query box
+– {: .query} Madeleine Birchfield: Wouldn’t an ordinal number be an object of the decategorification of the category of well-ordered sets, just as a natural number is an object of the decategorification of the category FinSet? =–
Anonymous
Todd,
when you see this here and have a minute, would you mind having a look at monoidal category to see if you can remove the query-box discussion there and maybe replace it by some crisp statement?
Thanks!
I thought to add
started entry on category of monoids. Spelled out the free monoid construction. Stated the construction of pushouts of monoids along free maps with reference to Schwede-Shipley. Will fill in the proof in a moment.
I have added to primitive element the definition of primitive elements in comodules, and their equivalent characterization in terms of cotensor products. Added also a corresponding remark to cotensor product.
I started to greatly expand the entry module
The new toc now looks like this:
Idea
Basic idea
More general perspectives
Enriched presheaves
Stabilized overcategories
Details
Ordinary concept
In enriched category theory
Examples
Related concepts
added pointer to
Added a reference to Langlands functoriality:
Are we any closer to
From this perspective Langlands’s “functoriality” is Lurie’s “global equivariance”…
stub for Heine-Borel theorem
Am starting a write-up (here) of how (programming languages for) quantum circuits “with classical control and/by measurement” have a rather natural and elegant formulation within the linear homotopy type theory of Riley 2022.
Aspects of this have a resemblance to some constructions considered in/with “Quipper”, but maybe it helps clarify some issues there, such as that of “dynamic lifting”.
The entry is currently written without TOC and without Idea-section etc, but rather as a single top-level section that could be !include-ed into relevant entries (such as at quantum circuit and at dependent linear type theory). But for the moment I haven’t included it anywhere yet, and maybe I’ll eventually change my mind about it.
added pointer to:
added to modular form a brief paragraph with a minimum of information on modular forms As automorphic forms. Needs to be expanded.
Added to bimonoid the fact that the category of modules over a bimonoid is monoidal.
The page on inductive-inductive types refers to dialgebras without specifying them. Having a short page as some sort of reference is helpful.
Started an article on monoidal monad. An earlier redirect had sent it over to Hopf monad which is something that Zoran was working on, but I think it deserves an article to itself, with discussion of the relation to commutative monads, etc. (which I have started).
Am trying to get some historical citations straight at linear type theory, maybe somebody can help me:
what are the original sources of the idea that linear logic/type theory should generally be that of symmetric monoidal categories (“multiplicative intuitionistic LL”)?
In order of appearance, I am aware of
de Paiva 89 gives one particular example of a non-star-autonomous SMC that deserves to be said to interpret “linear logic” and clearly identifies the general perspective.
Bierman 95 discusses semantics in general SMCs more generally
Barber 97 reviews this and explores a bit more.
What (other) articles would be central to cite for this idea/perspective?
I am aware of more recent reviews such as
but I am looking for the correct “original sources”.
brief category:people-entry for satisfying links now requested at p-adic Teichmüller theory
added to partial function a new section Definition – General abstract with a brief paragraph on how partial functions form the Kleisli category of the maybe-monad.
New reference entry FAC aka Faisceaux algébriques cohérents. And few improvements to coherent sheaf including historical note.
starting a dedicated entry for the category of vector bundles with homomorphisms allowed to cover non-trivial base maps (while previously we only had VectBund(B) for fixed base ).
For the moment the main point is to record the interesting cartesian- and tensor-monoidal structure (now here)
added pointer also to this recent review in a popular magazine, which is well-informed about the contested status of the “island conjecture”:
Nirmalya Kajuri, Of Islands, Holograms and Saving Quantum Physics From a Black Hole Paradox, Science – The Wire (Nov 2022)
[…] a few sceptics have argued that while the calculations are correct, they don’t help resolve the black hole information-loss paradox. The troubles stem from the reservoir attached to the anti-de Sitter universe. The physicists who authored the island papers assumed that gravity stopped at the boundary of the anti-de Sitter space and didn’t enter the reservoir. This is not an innocuous assumption. […] The key takeaway is that the island way to recover information and save unitarity works perfectly well – if you slightly modify Einstein’s theory of gravity. These criticisms have been around for some two years now, and physicists are yet to resolve them in print. […] physicists continue to publish papers by the hundreds about the entanglement islands but few attempt to answer whether the islands are compatible with the Einsteinian gravity of our universe. […]
I think i will create an include-file now for these references, so that it will be easier to sync with other entrties, such as Bekenstein-Hawking entropy:
Created a little entry Vect(X) (to go along with Vect) and used the occasion to give distributive monoidal category the Examples-section that it was missing and similarly touched the Examples-section at rig category.
Added an article
a bare sub-section with a list of references – to be !included into relevant entries – mainly at confinement and at mass gap problem (where this list already used to live)
in order to satisfy links, but maybe really in procrastination of other duties, I wrote something at quantum gravity
I made étale homotopy and étale homotopy theory be redirects to the entry geometric homotopy groups in an (infinity,1)-topos