Not signed in (Sign In)

A discussion forum about contributions to the nLab wiki and related areas of mathematics, physics, and philosophy.

Want to take part in these discussions? Sign in if you have an account, or apply for one below

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

- Discussion Type
- discussion topicset theory - contents
- Category Latest Changes
- Started by nLab edit announcer
- Comments 5
- Last comment by nLab edit announcer
- Last Active Nov 27th 2022

- Discussion Type
- discussion topicconstructive set theory
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by nLab edit announcer
- Last Active Nov 27th 2022

- Discussion Type
- discussion topictaboo
- Category Latest Changes
- Started by Mike Shulman
- Comments 24
- Last comment by nLab edit announcer
- Last Active Nov 27th 2022

Inspired by a discussion with Martin Escardo, I created taboo.

- Discussion Type
- discussion topicsyntomic cohomology
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Anton Hilado
- Last Active Nov 27th 2022

stub for syntomic cohomology

- Discussion Type
- discussion topicNP
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 27th 2022

a stub entry — for the moment just to make links work at

*nondeterministic computation*and at*complexity class*

- Discussion Type
- discussion topicrecursive function
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by nLab edit announcer
- Last Active Nov 27th 2022

- Discussion Type
- discussion topiccomplexity theory
- Category Latest Changes
- Started by Corbin
- Comments 5
- Last comment by Urs
- Last Active Nov 27th 2022

Unstub the page with a basic but original blurb. Compare and contrast my article list of complexity classes on esowiki, which has a very different audience.

- Discussion Type
- discussion topiccomputation
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Urs
- Last Active Nov 27th 2022

In looking for texts that would address the question “What is computation?” and arrive at an answer vaguely akin to path lifting/transport, I found (and have now added pointer to) this text:

- Jan van Leeuwen, Jiří Wiedermann:
*Knowledge, Representation and the Dynamics of Computation*, pp. 69-89 in:*Representation and Reality in Humans, Other Living Organisms and Intelligent Machines*, Studies in Applied Philosophy, Epistemology and Rational Ethics**28**, Springer (2017) [doi:10.1007/978-3-319-43784-2_5, pdf]

which gets pretty close, in particular in and around their Figure 1.

- Jan van Leeuwen, Jiří Wiedermann:

- Discussion Type
- discussion topicnondeterministic computation
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Urs
- Last Active Nov 27th 2022

- Discussion Type
- discussion topicdifferential cobordism cohomology
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 27th 2022

added pointer to this recent preprint:

- Knut Bjarte Haus, Gereon Quick,
*Geometric Hodge filtered complex cobordism*[arXiv:2210.13259]

- Knut Bjarte Haus, Gereon Quick,

- Discussion Type
- discussion topicproof theory
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Nov 27th 2022

I have made some trivial edits to the wording, hoping to make it flow more nicely.

By the way, this entry is linking (at least now that I adjusted the plural redirect) to

*recursive function*. This is only natural, but – unfortunately – our entry*recursive function*is empty (and always has been)!Much of the material needed there is at

*partial recursive function*. We should either put redirects or (better) add a little bit of content to*recursive function*.

- Discussion Type
- discussion topicendomorphism monoid object
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Nov 27th 2022

- Discussion Type
- discussion topicChristos Papadimitriou
- Category Latest Changes
- Started by J-B Vienney
- Comments 1
- Last comment by J-B Vienney
- Last Active Nov 27th 2022

- Discussion Type
- discussion topiccomputability
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by J-B Vienney
- Last Active Nov 27th 2022

- Discussion Type
- discussion topicMichael Sipser
- Category Latest Changes
- Started by J-B Vienney
- Comments 2
- Last comment by J-B Vienney
- Last Active Nov 27th 2022

- Discussion Type
- discussion topiccomma category
- Category Latest Changes
- Started by Urs
- Comments 24
- Last comment by Urs
- Last Active Nov 27th 2022

while bringing some more structure into the section-outline at

*comma category*I noticed the following old discussion there, which hereby I am moving from there to here:

[begin forwarded discussion]

+–{.query} It's a very natural notation, as it generalises the notation $(x,y)$ (or $[x,y]$ as is now more common) for a hom-set. But personally, I like $(f \rightarrow g)$ (or $(f \searrow g)$ if you want to differentiate from a cocomma category, but that seems an unlikely confusion), as it is a category of arrows from $f$ to $g$. —Toby Bartels

Mike: Perhaps. I never write $(x,y)$ for a hom-set, only $A(x,y)$ or $hom_A(x,y)$ where $A$ is the category involved, and this is also the common practice in nearly all mathematics I have read. I have seen $[x,y]$ for an internal-hom object in a closed monoidal category, and for a hom-set in a homotopy category, but not for a hom-set in an arbitrary category.

I would be okay with calling the comma category (or more generally the comma object) $E(f,g)$ or $hom_E(f,g)$

*if*you are considering it as a discrete fibration from $A$ to $B$. But if you are considering it as a*category*in its own right, I think that such notation is confusing. I don’t mind the arrow notations, but I prefer $(f/g)$ as less visually distracting, and evidently a generalization of the common notation $C/x$ for a slice category.*Toby*: Well, I never stick ‘$E$’ in there unless necessary to avoid ambiguity. I agree that the slice-generalising notation is also good. I'll use it too, but I edited the text to not denigrate the hom-set generalising notation so much.*Mike*: The main reason I don’t like unadorned $(f,g)$ for either comma objects or hom-sets is that it’s already such an overloaded notation. My first thought when I see $(f,g)$ in a category is that we have $f:X\to A$ and $g:X\to B$ and we’re talking about the pair $(f,g):X\to A\times B$ — surely also a natural generalization of the*very*well-established notation for ordered pairs.*Toby*: The notation $(f/g/h)$ for a double comma object makes me like $(f \to g \to h)$ even more!*Mike*: I’d rather avoid using $\to$ in the name of an object; talking about projections $p:(f\to g)\to A$ looks a good deal more confusing to me than $p:(f/g)\to A$.*Toby*: I can handle that, but after thinking about it more, I've realised that the arrow doesn't really work. If $f, g: A \to B$, then $f \to g$ ought to be the set of transformations between them. (Or $f \Rightarrow g$, but you can't keep that decoration up.)Mike: Let me summarize this discussion so far, and try to get some other people into it. So far the only argument I have heard in favor of the notation $(f,g)$ is that it generalizes a notation for hom-sets. In my experience that notation for hom-sets is rare-to-nonexistent, nor do I like it as a notation for hom-sets: for one thing it doesn’t indicate the category in question, and for another it looks like an ordered pair. The notation $(f,g)$ for a comma category also looks like an ordered pair, which it isn’t. I also don’t think that a comma category is very much like a hom-set; it happens to be a hom-set when the domains of $f$ and $g$ are the point, but in general it seems to me that a more natural notion of hom-set between functors is a set of natural transformations. It’s really the

*fibers*of the comma category, considered as a fibration from $C$ to $D$, that are hom-sets. Finally, I don’t think the notation $(f,g)$ scales well to double comma objects; we could write $(f,g,h)$ but it is now even less like a hom-set.Urs: to be frank, I used it without thinking much about it. Which of the other two is your favorite? By the way, Kashiwara-Schapira use $M[C\stackrel{f}{\to} E \stackrel{g}{\leftarrow} D]$. Maybe $comma[C\stackrel{f}{\to} E \stackrel{g}{\leftarrow} D]$? Lengthy, but at least unambiguous. Or maybe ${}_f {E^I}_g$?

Zoran Skoda: $(f/g)$ or $(f\downarrow g)$ are the only two standard notations nowdays, I think the original $(f,g)$ which was done for typographical reasons in archaic period is abandonded by the LaTeX era. $(f/g)$ is more popular among practical mathematicians, and special cases, like when $g = id_D$) and $(f\downarrow g)$ among category experts…other possibilities for notation should be avoided I think.

Urs: sounds good. I’ll try to stick to $(f/g)$ then.

Mike: There are many category theorists who write $(f/g)$, including (in my experience) most Australians. I prefer $(f/g)$ myself, although I occasionally write $(f\downarrow g)$ if I’m talking to someone who I worry might be confused by $(f/g)$.

Urs: recently in a talk when an over-category appeared as $C/a$ somebody in the audience asked: “What’s that quotient?”. But $(C/a)$ already looks different. And of course the proper $(Id_C/const_a)$ even more so.

Anyway, that just to say: i like $(f/g)$, find it less cumbersome than $(f\downarrow g)$ and apologize for having written $(f,g)$ so often.

*Toby*: I find $(f \downarrow g)$ more self explanatory, but $(f/g)$ is cool. $(f,g)$ was reasonable, but we now have better options.=–

- Discussion Type
- discussion topic2-pullback
- Category Latest Changes
- Started by Tim_Porter
- Comments 4
- Last comment by Urs
- Last Active Nov 27th 2022

Query from Stephen on 2-pullback.

- Discussion Type
- discussion topiccomma object
- Category Latest Changes
- Started by Tim_Porter
- Comments 6
- Last comment by Urs
- Last Active Nov 27th 2022

At comma object, Eduardo Pareja-Tobes put a query box which does not seem to have been ‘answered’. This was some time ago it seems.

- Discussion Type
- discussion topicLorenzo Tortora de Falco
- Category Latest Changes
- Started by J-B Vienney
- Comments 2
- Last comment by Urs
- Last Active Nov 27th 2022

- Discussion Type
- discussion topicToby Gee
- Category Latest Changes
- Started by Anton Hilado
- Comments 2
- Last comment by Urs
- Last Active Nov 27th 2022

- Discussion Type
- discussion topiccompletely distributive category
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by Urs
- Last Active Nov 27th 2022

- Discussion Type
- discussion topicmoduli stack of L-parameters
- Category Latest Changes
- Started by Anton Hilado
- Comments 5
- Last comment by Anton Hilado
- Last Active Nov 27th 2022

- Discussion Type
- discussion topicchoice operator
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active Nov 27th 2022

removing query box from page

+– {: .query} Mike Shulman: Is there a formal statement in some formal system along the lines of “a non-extensional choice operator does not imply AC”?

*Toby*: I don't know about a formal statement, but I can give you an example.Recall: In Per Martin-Löf's Intuitionistic Type Theory (and many other systems along similar lines), the basic notion axiomatised is not really that of a set (even though it might be called ’set’) but instead a preset (or ’type’). Often one hears that the axiom of choice

*does*hold in these systems, which doesn't imply classical logic due to a lack of quotient (pre)sets. However, if we define a set to be a preset equipped with an equivalence predicate, then the axiom of choice fails (although we have COSHEP if presets come with an identity predicate).A lot of these systems (including Martin-Löf's) use ’propositions as types’, in which $\exists_{x:A} P(x)$ is represented as $\sum_{x:A} P(x)$, which comes equipped with an operation $\pi: \sum_{x:A} P(x) \to A$. That is not going to get us our choice operator, but since a choice operator is constructively questionable anyway, then let's throw in excluded middle. This is known to not imply choice, but we do have, for every preset $A$, an element $\varepsilon_A$ of $A \vee \neg{A}$, that is of $A \uplus \empty^A$. It's not literally true that $\varepsilon_A$ is of type $A$, of course, but that would be unreasonable in a structural theory; what we do have is a fixed $\varepsilon_A$ such that, if $A$ is inhabited, then $\varepsilon_A = \iota_A(e)$ for some (necessarily unique) $e$ of type $A$ (where $\iota_A$ is the natural inclusion $A \to A \uplus \empty^A$), which I think should be considered good enough. This is for presets (types), but every set has a type of elements, so that gets us our operator.

How is this nonextensional? We do have $\varepsilon_A = \varepsilon_B$ if $A = B$ (which is a meaningful statement to Martin-Löf, albeit not a proposition exactly), but if $A$ and $B$ are given as subsets of some $U$, then we may well have $A = B$ as subsets of $U$ without $A = B$ in the sense of identity of their underlying (pre)sets. In particular, if $f: U \to V$ is a surjection and $A$ and $B$ are the preimages of elements $x$ and $y$ of $V$, then $x =_V y$ will not imply that $\varepsilon_A = \varepsilon_B$, and the proof of the axiom of choice does not go through. It

*will*go through if $x$ and $y$ are identical, that is if $x = y$ in the underlying preset of $V$, so again we do get choice for presets (again), but not for sets.I'm not

*certain*that a nonextensional global choice operator won't imply excluded middle in some other way, but I don't see how it would. You'd want to do something with the idea that $\varepsilon_A$ always exists but belongs to $A$ if and only if $A$ is inhabited, but I don't see how to parse it (just by assuming that it exists) to decide the question.Mike Shulman: That’s a very nice explanation/example, and it did help me to understand better what’s going on; thanks! (Did you mean to say “excluded middle” and not “AC” in your final paragraph?) What I would really like, though, is a statement like “the addition of a nonextensional global choice operator to ____ set theory is conservative” (i.e. doesn’t enable the proving of any new theorems that doen’t refer explicitly to the choice operator). Of course I am coming from this comment, wondering whether what you suggested really is a way to get a choice operator without implying the axiom of choice.

*Toby*: Yeah, I really did mean to say ’excluded middle’; remembering that comment, I assume that the real question is whether the thing is OK for a constructivist. I just argued $\mathbf{ITT} + EM \vDash CO$, and I know the result $\mathbf{ITT} + EM \not\vDash AC$, so I conclude $\mathbf{ITT} + CO \not\vDash AC$; but I don't know $\mathbf{ITT} + CO \not\vDash EM$ for certain. I certainly don't have $\mathbf{ITT} + CO$ conservative over $\mathbf{ITT}$, nor with any other theory (other than those that already model $CO$, obviously).Mike Shulman: Where should I look for a proof that $\mathbf{ITT} + EM$ doesn’t imply AC?

*Toby*: I'm not sure, it's part of my folk knowledge now. Probably Michael J. Beeson's*Foundations of Constructive Mathematics*is the best bet. I'll try to get a look in there myself next week; I can see that it's not exactly obvious, and perhaps my memory is wrong now that I think about it.Mike Shulman: I’m trying to prove the sort of statement I want over at SEAR+?.

*Toby*: No, I can't get anything at all out of Beeson (or other references) about full AC (for types equipped with equivalence relations) in $\mathbf{ITT}$.*Harry Gindi*: I have references for this discussion that should settle the issue at hand:Bell, J. L., 1993a. ’Hilbert’s epsilon-operator and classical logic’, Journal of Philosophical Logic, 22:1-18

Bell, J. L., 1993b. ’Hilbert’s epsilon operator in intuitionistic type theories’, Mathematical Logic Quarterly 39:323-337

Meyer Viol, W., 1995a. ’A proof-theoretic treatment of assignments’, Bulletin of the IGPL, 3:223-243

*Toby*: Thanks, Harry! Now I just have to find these journals at the library. =–Anonymous

- Discussion Type
- discussion topicMichele Pagani
- Category Latest Changes
- Started by J-B Vienney
- Comments 1
- Last comment by J-B Vienney
- Last Active Nov 26th 2022

- Discussion Type
- discussion topictermination
- Category Latest Changes
- Started by J-B Vienney
- Comments 2
- Last comment by J-B Vienney
- Last Active Nov 26th 2022

- Discussion Type
- discussion topicset theory versus dependent type theory
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active Nov 26th 2022

- Discussion Type
- discussion topiccut rule
- Category Latest Changes
- Started by Urs
- Comments 24
- Last comment by J-B Vienney
- Last Active Nov 26th 2022

some bare minimum at

*cut rule*

- Discussion Type
- discussion topiccondensed mathematics
- Category Latest Changes
- Started by David_Corfield
- Comments 32
- Last comment by Anton Hilado
- Last Active Nov 26th 2022

- Discussion Type
- discussion topicuniversal quantifier
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Nov 26th 2022

- Discussion Type
- discussion topicfan theorem
- Category Latest Changes
- Started by MarkSaving
- Comments 5
- Last comment by nLab edit announcer
- Last Active Nov 26th 2022

Change 1: Original page describes the fan theorem as requiring the bar to be decidable, claims that the “classical” fan theorem contradicts Brouwer’s continuity principle. The latter claim is not true; I corrected the error. I have stated the result as two separate theorems: the decidable fan theorem, about decidable bars, and the fan theorem, about bars in general.

Change 2: Slightly more information is provided about the relationship between the Fan Theorem and Bar Induction. Eventually, we should make a page about the latter.

Change 3: the section on equivalents to the fan theorem has been fixed somewhat. The section originally asserted that all of the statements provided were equivalent to the decidable fan theorem; in fact, some are equivalent to the decidable fan theorem and some to the full fan theorem.

- Discussion Type
- discussion topicfunction monad
- Category Latest Changes
- Started by Urs
- Comments 18
- Last comment by Urs
- Last Active Nov 26th 2022

created a minimum at

*function monad*(aka “reader monad”, “environment monad”)

- Discussion Type
- discussion topicequivariant cohomology
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by nilesjohnson
- Last Active Nov 26th 2022

I have removed the bulk of the Idea section that I had written, starting way back when the entry was created in 2013.

I kept the other material that Mike (Shulman) had written.

I see now that mine was really besides the point.

Now that I finally understand this topic more deeply, maybe I find time to write a better explanation of what’s really going on.

- Discussion Type
- discussion topicrepeat-until-success computing
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 26th 2022

- Discussion Type
- discussion topicdynamic lifting
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Nov 26th 2022

- Discussion Type
- discussion topicValentin Perrelle
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 26th 2022

- Discussion Type
- discussion topicDongho Lee
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 26th 2022

- Discussion Type
- discussion topicrecursion scheme
- Category Latest Changes
- Started by maxsnew
- Comments 6
- Last comment by Urs
- Last Active Nov 26th 2022

- Discussion Type
- discussion topicadic space
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 2
- Last comment by Anton Hilado
- Last Active Nov 26th 2022

- Discussion Type
- discussion topicRiemann-Hilbert correspondence
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Anton Hilado
- Last Active Nov 26th 2022

When I was about to create it for

*flat connection*I notice that we already did have*Riemann-Hilbert correspondence*. So now I have cross-linked it with*flat connection*,*flat infinity-connection*,*local system*,*Riemann-Hilbert problem*and the latter with*Hilbert’s problems*

- Discussion Type
- discussion topicD-module
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Anton Hilado
- Last Active Nov 26th 2022

I tried to implement the connection between D-modules and quasicoherent sheaves a bit more

added to D-module the alternative definition in terms of quasicoherent sheaves of the deRham space

added to deRham space accordingly a pointer to D-modules (also fixed wrong notation in the formulas there)

added to quasicoherent sheaf at the very bottom a pointer to D-modules.

This needs improving. Notably good references should be given.

- Discussion Type
- discussion topicreal polynomial function
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by nLab edit announcer
- Last Active Nov 25th 2022

- Discussion Type
- discussion topiccosine
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Nov 25th 2022

- Discussion Type
- discussion topicsine
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active Nov 25th 2022

- Discussion Type
- discussion topicexponential map
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active Nov 25th 2022

- Discussion Type
- discussion topicsolid module
- Category Latest Changes
- Started by Anton Hilado
- Comments 14
- Last comment by Anton Hilado
- Last Active Nov 25th 2022

- Discussion Type
- discussion topicsix operations
- Category Latest Changes
- Started by Urs
- Comments 25
- Last comment by Urs
- Last Active Nov 25th 2022

I am starting something at six operations.

(Do we already have an nLab page on this? I seemed to remember something, but can’t find it.)

- Discussion Type
- discussion topicconvolution algebra
- Category Latest Changes
- Started by J-B Vienney
- Comments 1
- Last comment by J-B Vienney
- Last Active Nov 25th 2022

Added two propositions about the general case of monoids and comonoids in a closed monoidal categories. It looks that it could work in such a general context and thus be used in models of linear logic or in differential categories, so that’s exciting. I’ll try to prove them.

- Discussion Type
- discussion topicbit flip channel
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Nov 25th 2022

- Discussion Type
- discussion topicAlberto Gandolfi
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 25th 2022

- Discussion Type
- discussion topicFederico Camia
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 25th 2022

- Discussion Type
- discussion topicValentino Foit
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 25th 2022

- Discussion Type
- discussion topicBrownian loop soup
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 25th 2022

- Discussion Type
- discussion topicStéphane Attal
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 25th 2022

- Discussion Type
- discussion topicquantum mechanics
- Category Latest Changes
- Started by Urs
- Comments 20
- Last comment by Urs
- Last Active Nov 25th 2022

in analogy to what I just did at classical mechanics, I have now added some basic but central content to quantum mechanics:

Quantum mechanical systems

States and observables

Spaces of states

Flows and time evolution

Still incomplete and rough. But I have to quit now.

- Discussion Type
- discussion topicquantum operation
- Category Latest Changes
- Started by Eric
- Comments 92
- Last comment by Urs
- Last Active Nov 25th 2022

Wikipedia has a nice article on quantum operations.

The nLab also had a page quantum operations and channels (cache bug?), but I’ve renamed this to simply quantum operation since a quantum channel seems to be nothing but a quantum operation when viewed from the perspective of quantum information theory. Eventually, this page might need some disambiguation since there may be several uses of the term, but for now I think it is “ok”.

I think this page can be cleaned up. I started, but don’t think I will be able to finish.

In particular, there is some background material that might be better on separate pages. I’ll continue trying to clean things up, but family might be calling soon and I’ll need to run quickly whatever state it is in.

I also made the simple statement

In quantum mechanics, a

*quantum operation*is a morphism in the category of density matricesat the beginning of the Idea section motivated by O’Loan’s comment

A quantum channel is a mapping which sends density matrices to density matrices.

This seems innocent enough, but someone might check the statement. For one, I’ve never seen a category of density matrices, but the idea seems obvious enough. Maybe a word on density matrix would be good.

- Discussion Type
- discussion topicquantum probability theory
- Category Latest Changes
- Started by Urs
- Comments 31
- Last comment by Urs
- Last Active Nov 25th 2022

I started a bare minimum at

*quantum probability*(redirecting*noncommutative probability space*etc.)Some entries have long been secretly referencing such an entry, and I have cross-linked accordingly, for instance from

*von Neumann algebra*and*quantum computing*.I had the feeling somewhere we already had a detailed account of probability theory dually in terms of von NNeumann algebras, but if we do I didn’t find it(?)

- Discussion Type
- discussion topicstochastic process
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 25th 2022

- Discussion Type
- discussion topicquantum noise
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Nov 25th 2022

a stub for the moment, to go alongside

*noise*and*quantum error correction*

- Discussion Type
- discussion topicquantum reader monad
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Nov 25th 2022

now creating this entry.

The technical material under “Details” (here) is copied over from what I had written at

*reader monad – Examples – quantum reader monad*. (There may still be room left to adjust the wording in order to reflect that this material moved to a new entry.)To this I have now added an Idea-section (here) which highlights the relation to (equivalence with) Bob Coecke’s “classical structures” (which term I made redirect to here now)

- Discussion Type
- discussion topicpseudocode
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 25th 2022