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• 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.

• The following sentence was at inaccessible cardinal:

A weakly inaccessible cardinal may be strengthened to produce a (generally larger) strongly inaccessible cardinal.

I have removed it after complaints by a set theorist, and in light of the below discussion box, which I have copied here and removed.

Mike: What does that last sentence mean? It seems obviously false to me in the absence of CH.

Toby: It means that if a weakly inaccessible cardinal exists, then a strongly inaccessible cardinal exists, but I couldn't find the formula for it. Something like $\beth_\kappa$ is strongly inaccessible if $\kappa$ is weakly inaccessible (note that $\aleph_\kappa = \kappa$ then), but I couldn't verify that (or check how it holds up in the absence of choice).

Mike: I don’t believe that. Suppose that the smallest weakly inaccessible is not strongly inaccessible, and let $\kappa$ be the smallest strongly inaccessible. Then $V_\kappa$ is a model of set theory in which there are weakly inaccessibles but not strong ones. I’m almost certain there is no reason for the smallest weakly inaccessible to be strongly inaccessible.

JCMcKeown: Surely $\beth_\kappa$ has cofinality at most $\kappa$, so it can’t be regular. Maybe the strengthening involves some forcing or other change of universe? E.g., you can forcibly shift $2^\lambda = \lambda^+$ for $\lambda \lt \kappa$, and then by weak inaccessibility, etc… I think. Don’t trust me. —- (some days later) More than that: since the ordinals are well ordered, if there is any strongly inaccessible cardinal greater than $\kappa$, then there is a least one, say $\theta$. Then $V_\theta$ is a universe with a weakly inaccessible cardinal and no greater strongly inaccessible cardinal. Ih! Mike said that already… So whatever construction will have to work the other way around: if there is a weakly inaccessible cardinal that isn’t strongly inaccessible, and if furthermore a weakly inaccessible cardinal implies a strongly inaccessible cardinal, then the strongly inaccessible cardinal implied must be less than $\kappa$. And that sounds really weird.

I was also supplied with a AC-free proof that weakly inaccessibles are $\aleph_{(-)}$-fixed points:

First, a quick induction shows that $\alpha\leq\omega_\alpha$ is always true. If $\kappa$ is a limit cardinal, then the set of cardinals below $\kappa$ is unbounded; but since it’s also regular there are $\kappa$ of them. So $\omega_\kappa\leq\kappa\leq\omega_\kappa$, and equality ensues.

I can edit this into the page if desired.

• Some important references added, including recent geometric study

• Leonid O. Chekhov, Marta Mazzocco, Vladimir N. Rubtsov, Painlevé monodromy manifolds, decorated character varieties, and cluster algebras, IMRN 24 (2017) 7639–7691 doi

and a section of papers studying the mathematics of Painlevé equations and of Fuchsian $j_\Gamma$-functions from model theoretic perspective (strongly minimal structures), including a new paper in Annals 2020.

• removing query boxes

+– {: .query} Madeleine Birchfield: Wouldn’t a cardinal number be an object of the decategorification of the category Set, just as a natural number is an object of the decategorification of the category FinSet? =–

+– {: .query} Roger Witte First of all sorry if I am posting in the wrong place

While thinking about graphs, I wanted to define them as subobjects of naive cardinal 2 and this got me thinking about the behaviour of the full subcategories of Set defined by isomorphism classes. These categories turned out to be more interesting than I had expected.

If the background set theory is ZFC or similar, these are all large but locally small categories with all hom sets being isomorphic. They all contain the same number of objects (except 0, which contains one object and no non-identity morphisms) and are equinumerous with Set. Each hom Set contains $N^N$ arrows. In the finite case $N!$ of the morphisms in a particular hom set are isomorphisms. In particular, only 0 and 1 are groupoids. I haven’t worked out how this extends to infinite cardinalities, yet.

If the background theory is NF, then they are set and 1 is smaller than Set. I haven’t yet worked out how 2 compares to 1. I need to brush up on my NF to see how NF and category theory interact.

I am acutely aware that NF/NFU is regarded as career suicide by proffesional mathematicians, but, fortunately, I am a proffesional transport planner, not a mathematician.

Toby: Each of these categories is equivalent (but not isomorphic, except for 0) to a category with exactly one object, which may be thought of as a monoid. Given a cardinal $N$, if you pick a set $X$ with $N$ elements, then this is (up to equivalence, again) the monoid of functions from $X$ to itself. The invertible elements of this monoid form the symmetric group, with order $N!$ as you noticed. Even for infinite cardinalities, we can say $N^N$ and $N!$, where we define these numbers to be the cardinalities of the sets of functions (or invertible functions) from a set of cardinality $N$ to itself.

From a structural perspective, there's no essential difference between equivalent categories, so the fact that these categories (except for 0) are equinumerous with all of Set is irrelevant; what matters is not the number of objects but the number of isomorphism classes of objects (and similarly for morhpisms). That doesn't mean that your result that they are equinumerous with Set is meaningless, of course; it just means that it says more about how sets are represented in ZFC than about sets themselves. So it should be no surprise if it comes out differently in NF or NFU, but I'm afraid that I don't know enough about NF to say whether they do or not.

By the way, every time you edit this page, you wreck the links to external web pages (down towards the bottom in the last query box). It seems as if something in your editor is removing URLs. =–

Anonymous

• Clarify the categorical description of the partial trace construction.

Jake Bian

• null edit to start discussion page

Anonymous

• added to identity type a mentioning of the alternative definition in terms of inductive types (paths).

• Create a new page to keep record of PhD theses in category theory (with links to the documents where possible), particularly older ones that are harder to discover independently. At the moment, this is just a stub, but I plan to fill it out more when I have the chance.

• brief category:people-entry for hyperlinking references

• brief category:people-entry for hyperlinking references

• 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.

• Init fixed point operator

• somebody on the nForum thread on the identity type article brought up objective type theory so I thought I would start an article on that subject

Anonymous

• Added a reference to Seely’s 2-categorical analysis of eta expansion.

• Created a page on Manuel Rivera who has some interesting papers on the cobar construction.

• created smooth topos on the axioms on toposes used in synthetic differential geometry.

• Page created, but author did not leave any comments.

Anonymous

• Created, with so far just an overview of all the possibilities.

• I made a little addition to opposite category, pointing out some amusing nuances regarding the opposite of a $V$-enriched category when $V$ is merely braided. This remark could surely be clarified, but I think you’ll get the idea.

(In case you’re wondering why I did this, it’s because I needed a reference for “opposite category” in a blog entry I’m writing.)

• Started this since it was mentioned elsewhere. So it’s just the opposite one object 2-category, isn’t it?

• Add redirects for “co” and “coop”.

• I have given necessity and possibility (which used to redirect to S4-modal logc) an entry of their own.

The entry presently

• first recalls the usual axioms;

• then complains that these are arguably necessary but not sufficient to characterize the idea of necessity/possibility;

• and then points out that if one passes from propositional logic to first-order logic (hyperdoctrines) and/or to dependent type theory, then there is a way to axiomatize modalities that actually have the correct interpretation, namely by forming the reflection (co)monads of $\exists$ and $\forall$, respectively.

You may possibly complain, but not necessarily. Give it a thought. I was upset about the state of affairs of the insufficient axiomatics considered in modal logic for a long time, and this is my attempt to make my peace with it.

• Added result discussed at the Cafe that tensor products of symmetric pseudomonoids are a weak 2-coproduct.

• Created:

## Definition

A graded vector bundle is simply a graded object in the category of vector bundles, or, equivalently, a bundle of graded vector spaces.

## Related concepts

• Page created, but author did not leave any comments.

Anonymous

• A query box has been added:

I suspect there is a variant of the definition involving a transformation $R^Z_{X Y} \colon [X,Y] \to [[Y,Z],[X,Z]]$ rather than $L$. Is this correct? If so, how do these two definitions relate? Can one of them be expressed in terms of the other? Or is there a refined definition which comprises both $L$ and $R$?

• Add a students section (very sparse at the moment).

• I corrected an apparent typo:

A 2-monad $T$ as above is lax-idempotent if and only if for any $T$-algebra $a \colon T A \to A$ there is a 2-cell $\theta_a \colon 1 \Rightarrow \eta \circ a$

to

A 2-monad $T$ as above is lax-idempotent if and only if for any $T$-algebra $a \colon T A \to A$ there is a 2-cell $\theta_a \colon 1 \Rightarrow \eta_A \circ a$

It might be nice to say $\eta_A$ is the unit of the algebra….

• Removed an incorrect historical claim (Dwyer and Kan did throughly investigate relative categories already in 1980s, way before 2000s).

• quickly added at accessible category parts of the MO discussion here. Since Mike participated there, I am hoping he could add more, if necessary.

• Page created, but author did not leave any comments.

• I added a short abstract description of the Cartesian fibration over the interval, and commented that the section describes a construction with additional strictness properties in the quasi-category model.

• changed entry title to full name,

updated webpage url,

added section “Related nLab entries”, so far with a pointer to Grothendieck construction

• Updated affiliation, area of work and link to homepage.

Severin Bunk

• added to path space object an Examples-section with some model category-theoretic discussion, leading up to the statement that in a simplicial model category for fibrant $X$ the powering $X^{\Delta[1]}$ is always a path space object.

• split off from precategory into its own article

Anonymous

• Page created, but author did not leave any comments.

Anonymous

• Created a small page to describe the different usages of the term locally.

• Page created, but author did not leave any comments.

Anonymous

• A late-night edit for entertainment – I’ll try to polish it up tomorrow when I am more awake:

I have added a translated original quote from Leibniz, as given by Gries & Schneider 1998:

Two terms are the same (eadem) if one can be substituted for the otherwithout altering the truth of any statement (salva veritate). If we have $P$ and $Q$, and $P$ enters into some true proposition, and the substitution of $Q$ for $P$ wherever it appears results in a new proposition that is likewise true, and if this can be done for every proposition, then $P$ and $Q$ are said to be the same;

Interestingly – and this is what I was searching for – Leibniz ends this paragraph with stating the converse:

conversely, if $P$ and $Q$ are the same, they can be substituted for one another.

I was chasing for a historical reference on this “principle of substitution of equals” (or what do people call it?) since this is the logical seed of path induction.

I’d like to find a more canonical reference. But not tonight.

• Quite a few references added, some of which are also scattered in $n$Lab, some not.

• I have added to flat functor right after the very first definition (“$C \to Set$ is flat if its category of elements is cofiltered”) a remark which spells out explicitly what this means in components. Just for convenience of the reader.

• just the evident minimum

• Page created, but author did not leave any comments.

• Page created, but author did not leave any comments.

• a bare list of references, to be !include-ed into the references lists of relevant entries

• brief category:people-entry for hyperlinking references

• brief category:people-entry for hyperlinking references

• Page created, but author did not leave any comments.

• brief category:people-entry for hyperlinking references

• example of nominal sets with separated tensor added, see Chapter 3.4 of Pitts monograph Nominal Sets

Alexander Kurz

• I added to decidable equality some remarks on the difference between the propositions-as-types version and the propositions-as-some-types version.