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- Discussion Type
- discussion topicLawvere-Tierney topology
- Category Latest Changes
- Started by Urs
- Comments 18
- Last comment by nLab edit announcer
- Last Active Aug 1st 2024

I have expanded Lawvere-Tierney topology, also reorganized it in the process

- Discussion Type
- discussion topicfree coproduct completion
- Category Latest Changes
- Started by Urs
- Comments 20
- Last comment by varkor
- Last Active Aug 1st 2024

some bare minimum on the free coproduct cocompletion.

The term used to redirect to the entry

*free cartesian category*, where however the simple idea of free coproduct completion wasn’t really brought out.

- Discussion Type
- discussion topiclocally presentable category
- Category Latest Changes
- Started by Tobias Fritz
- Comments 20
- Last comment by qschroed
- Last Active Aug 1st 2024

- In the definition, the article states "every object in C is a small object (which follows from 2 and 3)". The bracketed remark doesn't seem quite right to me, since neither 2 nor 3 talk about smallness of objects. Presumably this should better be phrased as in A.1.1 of HTT, "assuming 3, this is equivalent to the assertion that every object in S is small".

Am I right? I don't (yet) feel confident enough with my category theory to change this single-handedly.

- Discussion Type
- discussion topicMartin Lüscher
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 31st 2024

- Discussion Type
- discussion topicRobert Lee Wilson
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by Urs
- Last Active Jul 31st 2024

Person entry.

Warning: we have a webpage for another algebraist, the group theorist Robert A. Wilson, and elsewhere in the $n$Lab Robert Wilson shortcut is carelessly used for the latter. People tend to shorten links in $n$Lab and the confusion and illegal links might occur in this case. Maybe we should not use version without middle initial for these two guys in links.

- Discussion Type
- discussion topicPhilippe Di Francesco
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by zskoda
- Last Active Jul 31st 2024

- Discussion Type
- discussion topicDodgson condensation
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by zskoda
- Last Active Jul 31st 2024

In statu nascendi.

A method for calculating determinants. It is related to cluster algebras and a special case of Sylvester identity.

- Discussion Type
- discussion topicmatroid
- Category Latest Changes
- Started by Todd_Trimble
- Comments 22
- Last comment by varkor
- Last Active Jul 31st 2024

I have been adding some material to matroid. I haven’t gotten around to defining oriented matroid yet (and of course there’s much besides to add).

- Discussion Type
- discussion topicBoolean algebra
- Category Latest Changes
- Started by Todd_Trimble
- Comments 4
- Last comment by J-B Vienney
- Last Active Jul 31st 2024

Added more material to Boolean algebra, particularly the principle of duality and the connection to Boolean rings, and a wee bit of material on Stone duality.

Stone duality deserves greater expansion, bringing out the dualities via ambimorphic (ahem, schizophrenic) structures on the 2-element set, and mentioning the connection to Chu spaces. Another day, another dollar.

- Discussion Type
- discussion topicglobal equivariant homotopy theory
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by nLab edit announcer
- Last Active Jul 30th 2024

I should say – for those watching the logs and wondering – that I started editing the entry

*global equivariant homotopy theory*such as to reflect Charles Rezk’s account in a coherent way.But I am not done yet. The entry has now some of the key basics, but is still missing the general statement in its relation to orbispaces. Also some harmonizing of the whole entry may be necessary now, as I moved around some stuff.

So better don’t look at it yet. I hope to bring it into shape tomorrow or so.

(In the process I have split off

*global orbit category*now.)

- Discussion Type
- discussion topiccoefficient system
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Jul 30th 2024

- Discussion Type
- discussion topicG-∞-category
- Category Latest Changes
- Started by nLab edit announcer
- Comments 5
- Last comment by nLab edit announcer
- Last Active Jul 30th 2024

- Discussion Type
- discussion topicindexed tensor product
- Category Latest Changes
- Started by nLab edit announcer
- Comments 0
- Last comment by Urs
- Last Active Jul 30th 2024

- Discussion Type
- discussion topicAaron Naber
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 30th 2024

- Discussion Type
- discussion topicMichael T. Anderson
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 30th 2024

- Discussion Type
- discussion topicRicci curvature
- Category Latest Changes
- Started by Daniel Luckhardt
- Comments 7
- Last comment by Urs
- Last Active Jul 30th 2024

- Discussion Type
- discussion topicgraded monad
- Category Latest Changes
- Started by David_Corfield
- Comments 24
- Last comment by varkor
- Last Active Jul 30th 2024

- Discussion Type
- discussion topicStephen Hawking
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 30th 2024

- Discussion Type
- discussion topicPenrose singularity theorem
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Urs
- Last Active Jul 30th 2024

created a minimum at

*Penrose-Hawking singularity theorem*

- Discussion Type
- discussion topicHawking radiation
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 30th 2024

- Discussion Type
- discussion topicGabriel Wong
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 30th 2024

- Discussion Type
- discussion topicPranjal Nayak
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 30th 2024

- Discussion Type
- discussion topicD=3 quantum gravity
- Category Latest Changes
- Started by Urs
- Comments 32
- Last comment by Urs
- Last Active Jul 30th 2024

I have expanded the Idea-section at 3d quantum gravity and reorganized the remaining material slightly.

I feel unsure about the pointer to “group field theory” in the References. Can anyone list results that have come out of group field theory that are relevant here?

I find the following noteworthy, and I am not sure if this is widely appreciated:

the original discussion of the quantization of 3d gravity by Witten in 1988 happens work out to be precisely along the lines that “loop quantum gravity” once set out to get to work in higher dimensions: one realizes

that the configuration space is equivalently a space of connections;

that these can be characterized by their parallel transport along paths in base space;

that therefore observables of the theory are given by evaluating on choices of paths (an idea that goes by the unfortunate name “spin network”).

All this is in Witten’s 1988 article. Of course the point there is that in the case of 3d this can actually be made to work. The reason is that in this case it is sufficient to restrict to

*flat*connections and for these everything drastically simplifies: their parallel transport depends not on the actual paths but just on their homotopy class, rel boundary. Accordingly the “spin networks” reduce to evaluations on generators of the fundamental group, etc.Notice that in 4d the analog of this step that Witten easily performs in 3d was never carried out: instead, because it seemed to hard, the LQG literature

*always*passes to a different system, where smooth connections are replaced by parallel transport that is required to be neigher smooth nor in fact continuous. These are called “generalized connections” in the LQG literature. Of course these have nothing much to do with Einstein-gravity: because there the configuration space does not contain such “generalized” fields.For these reasons I feel a bit uneasy when the entry refers to LQG or spin foams as “other approaches” to discuss 3d quantum gravity. First of all, the existing good discussion by Witten did realize the LQG idea already in that dimension, and it did it correctly. So in which sense are there “other approaches”?

Which insights on 3d quantum gravity do “spin foam”s or does “group field theory”add? If anyone could list some results with concrete pointers to the literature, I’d be most grateful.

- Discussion Type
- discussion topicRozansky-Witten theory
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active Jul 30th 2024

- Discussion Type
- discussion topicmath blogs
- Category Latest Changes
- Started by nLab edit announcer
- Comments 6
- Last comment by nLab edit announcer
- Last Active Jul 29th 2024

- Discussion Type
- discussion topicorthogonal factorization system
- Category Latest Changes
- Started by Urs
- Comments 31
- Last comment by varkor
- Last Active Jul 29th 2024

I have added to

*orthogonal factorization system*in the Definition-section three equivalent explicit formulations of the definition;

in the Properties-section the statement of the cancellability property.

Wanted to add more (and to add the proofs). But have to quit now. Maybe later.

- Discussion Type
- discussion topicpseudomonoid
- Category Latest Changes
- Started by Mike Shulman
- Comments 5
- Last comment by varkor
- Last Active Jul 29th 2024

- Discussion Type
- discussion topicgroup scheme
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Jul 29th 2024

I have edited group scheme and algebraic group slightly. To the latter I added Example-pointers to multiplicative group and additive group

- Discussion Type
- discussion topicHecke algebra
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 29th 2024

added pointer to:

- Garth Warner:
*Elementary Aspects of the Theory of Hecke Operators*, University of Washington (1988) [pdf]

- Garth Warner:

- Discussion Type
- discussion topicgauge theory
- Category Latest Changes
- Started by nLab edit announcer
- Comments 6
- Last comment by Urs
- Last Active Jul 29th 2024

- Discussion Type
- discussion topicHaag-Kastler axioms
- Category Latest Changes
- Started by TobyBartels
- Comments 14
- Last comment by Urs
- Last Active Jul 29th 2024

Tim van Beek has graced us with these: Haag-Kastler axioms.

- Discussion Type
- discussion topicWightman axioms
- Category Latest Changes
- Started by Ian_Durham
- Comments 7
- Last comment by Urs
- Last Active Jul 29th 2024

- I filled in a bit on the Wightman axioms. I also have a query there about adding an "axiom" environment to the LaTeX/CSS style sheets of nLab. I don't know how to do it on nLab but an axiom environment seems like it might be useful.

- Discussion Type
- discussion topicdescriptive set theory
- Category Latest Changes
- Started by Todd_Trimble
- Comments 12
- Last comment by Urs
- Last Active Jul 29th 2024

Some nLab pages had a gray link to descriptive set theory, which now has the following stub:

Descriptive set theory is the study of the structures and hierarchies of subsets of real numbers (or more generally of subsets of Polish spaces) that are definable by formulas with real parameters in second-order arithmetic.

Such subsets include Borel sets and more generally projective sets that are defined by alternating between taking images under projection maps of previously defined sets and taking complements of previously defined sets. Once the domain of topologists of the Polish schools and Russian analysts of the early 20th century, it is now considered a central area of logic in which set theory and computability theory (recursion theory) meet and interact.

- Discussion Type
- discussion topictranscendental number
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Urs
- Last Active Jul 29th 2024

- Discussion Type
- discussion topicGrothendieck fibration
- Category Latest Changes
- Started by Urs
- Comments 47
- Last comment by Urs
- Last Active Jul 29th 2024

I made the former entry "fibered category" instead a redirect to Grothendieck fibration. It didn't contain any addition information and was just mixing up links. I also made category fibered in groupoids redirect to Grothendieck fibration

I also edited the "Idea"-section at Grothendieck fibration slightly.

That big query box there ought to be eventually removed, and the important information established in the discussion filled into a proper subsection in its own right.

- Discussion Type
- discussion topicDoplicher-Roberts reconstruction theorem
- Category Latest Changes
- Started by John Baez
- Comments 4
- Last comment by Urs
- Last Active Jul 29th 2024

- Discussion Type
- discussion topicchoice operator
- Category Latest Changes
- Started by nLab edit announcer
- Comments 5
- Last comment by nLab edit announcer
- Last Active Jul 28th 2024

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 topiclimit
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by nLab edit announcer
- Last Active Jul 28th 2024

I toiuched the formatting and the hyperlinking of the paragraphs on compatibility of limits with other universal constructions.

Merged the previous tiny subsections on this to a single one, now

*Compatibility with universal constructions*.added the hyperlink to the stand-alone entry

*adjoints preserve (co-)limits*.Will create an analogous stand-alone entry for

*limits commute with limits*.

- Discussion Type
- discussion topicETCS plus epsilon
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Jul 28th 2024

starting page on ETCS with a choice operator $\epsilon$. See SEAR plus epsilon for the allegorical set theory analogue.

Anonymouse

- Discussion Type
- discussion topicsupport object
- Category Latest Changes
- Started by nLab edit announcer
- Comments 6
- Last comment by RodMcGuire
- Last Active Jul 27th 2024

- Discussion Type
- discussion topicquantum channel
- Category Latest Changes
- Started by Eric
- Comments 102
- Last comment by Urs
- Last Active Jul 27th 2024

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 topicclassical mechanics
- Category Latest Changes
- Started by Urs
- Comments 59
- Last comment by Urs
- Last Active Jul 27th 2024

I’ll try to start add some actual content to the entries classical mechanics, quantum mechanics, etc. For the time being I added a simple but good definition to classical mechanics. Of course this must eventually go with more discussion to show any value. I hope to be able to use some nice lecture notes from Igir Khavkine for this eventually.

For the time being, notice there was this old discussion box, which I am herby mving to the forum here:

–

+–{.query} Edit: I changed the above text, incorporating a part of the discussion (Zoran).

Zoran: I disagree. Classical mechanics is classical mechanics of anything: point particles, rigid bodies (the latter I already included), infinite systems (mechanics of strings, membranes, springs, elastic media, classical fields). It includes statics, not only dynamics. The standard textbooks like Goldstein take it exactly in that generality.

One could even count the simplified beginning part of the specialized branches like aerodynamics and hydrodynamics (ideal liquids for example), which are usually studied in separate courses and which in full formulation are not just mechanical systems, as the thermodynamics also affects the dynamics. There are also mechanical models of dissipative systems, where the dissipative part is taken only phenomenologically, e.g. as friction terms. Hydrodynamics can also be considered as a part of rheology.

*Toby*: I take your point that ’dynamics’ was not the right word. But do you draw any distinction between ’classical mechanics’ and ’classical physics’? Conversely, what word*would*you use to restrict attention to particles instead of fields, if not ’mechanics’? (Incidentally, I would take point particles as possibly spinning, although I agree that I should not assume that the particle are points anyway.)*Zoran*: you see, in classical mechanics you express all you have by attaching mass, position, velocity etc. to the parfts of mechanical systems. Not all classical physics belongs to this kind of description. The thermodynamical quantities may influence the motion of the systemm, but their description is out of the frame of classical mechanics. If you study liquids you have to take into account both the classical mechanics of the liquid continuum but also variations of its temperature, entropy and so on, which are not expressable within the variables of mechanics. Formally speaking of course, the thermodynamics has very similar formal structure as mechanics, for example Gibbs and Helmholtz free energies and enthalpy are like Lagrangean, the quantities which are extremized when certain theremodynamical quantities are kept constant. To answer the terminological question, there is a classical mechanics of point particles and it is called classical mechanics of point particles, there is also cm of fields and cm of rigid bodies.*Toby*: So ’mechanics’ for you means ‹not taking into account thermal physics›? That's not the way that I learned it! But I admit that I do not have a slick phrase for that (any more than you have a slick phrase for ‹mechanics of point particles›), so I will try to ascertain how the term is usually used and defer to that. =–

- Discussion Type
- discussion topicgeneral relativity
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Urs
- Last Active Jul 27th 2024

I wanted to be able to use the link without it appearing in grey, so I created a stub for general relativity.

- Discussion Type
- discussion topicJoel David Hamkins
- Category Latest Changes
- Started by Mike Shulman
- Comments 3
- Last comment by Urs
- Last Active Jul 27th 2024

- Discussion Type
- discussion topiccontinuum hypothesis
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by nLab edit announcer
- Last Active Jul 27th 2024

- Discussion Type
- discussion topicprinciple of omniscience
- Category Latest Changes
- Started by Mike Shulman
- Comments 21
- Last comment by nLab edit announcer
- Last Active Jul 27th 2024

I added a couple of comments about topos models to principle of omniscience.

- Discussion Type
- discussion topiclimited principle of omniscience
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Jul 27th 2024

moving material about the limited principle of omniscience from principle of omniscience to its own page at limited principle of omniscience

Anonymouse

- Discussion Type
- discussion topicweak limited principle of omniscience
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Jul 27th 2024

moving material about the weak limited principle of omniscience from principle of omniscience to its own page at weak limited principle of omniscience

Anonymouse

- Discussion Type
- discussion topiclesser limited principle of omniscience
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Jul 27th 2024

moving material about the lesser limited principle of omniscience from principle of omniscience to its own page at lesser limited principle of omniscience

Anonymouse

- Discussion Type
- discussion topichigher curvature correction
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Urs
- Last Active Jul 27th 2024

- Discussion Type
- discussion topiccrystalline differential operator
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active Jul 27th 2024

- Discussion Type
- discussion topicD-module
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Jul 27th 2024

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 topicstate on a star-algebra
- Category Latest Changes
- Started by Urs
- Comments 16
- Last comment by Urs
- Last Active Jul 27th 2024

I have expanded the Idea section at

*state on a star-algebra*and added a bunch of references.The entry used to be called “state on an operator algebra”, but I renamed it (keeping the redirect) because part of the whole point of the definition is that it makes sense without necessarily having represented the “abstract” star-algebra as a C*-algebra of linear operators.

- Discussion Type
- discussion topichomotopy theory and algebraic topology -- references
- Category Latest Changes
- Started by Urs
- Comments 33
- Last comment by Urs
- Last Active Jul 27th 2024

following discussion here I am starting an entry with a bare list of references (sub-sectioned), to be

`!include`

-ed into the References sections of relevant entries (mainly at*homotopy theory*and at*algebraic topology*) for ease of updating and syncing these lists.The organization of the subsections and their items here needs work, this is just a start. Let’s work on it.

I’ll just check now that I have all items copied, and then I will

`!include`

this entry here into*homotopy theory*and*algebraic topology*. It may best be*viewed*withing these entries, because there – but not here – will there be a table of contents showing the subsections here.

- Discussion Type
- discussion topicGarth Warner
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jul 27th 2024

- Discussion Type
- discussion topicsheaf and topos theory
- Category Latest Changes
- Started by DavidRoberts
- Comments 14
- Last comment by Urs
- Last Active Jul 27th 2024

Moved talk by McLarty to the History section of the refernces from functorial geometry

- Discussion Type
- discussion topiclist of theses in category theory
- Category Latest Changes
- Started by varkor
- Comments 48
- Last comment by varkor
- Last Active Jul 27th 2024

- Discussion Type
- discussion topicbar
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Jul 26th 2024

- Discussion Type
- discussion topicbar induction
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by nLab edit announcer
- Last Active Jul 26th 2024

- Discussion Type
- discussion topictopological quantum computation with anyons -- references
- Category Latest Changes
- Started by Urs
- Comments 38
- Last comment by Urs
- Last Active Jul 26th 2024

I am giving this bare list of references its own entry, so that it may be

`!include`

-ed into related entries (such as*topological quantum computation*,*anyon*and*Chern-Simons theory*but maybe also elsewhere) for ease of updating and synchronizing