I think JB used to stay on top of this and delete such comments, but the nCafe currently has 2 questionable recently new entries:

1) This crackpot entry.

2) This entry with the only purpose being to include a url to “cures for candida”.

The 2nd should be deleted and probably the 1st unless crackpots are kept for amusement purposes.

Who has the privileges to delete such these days?

]]>I have a few questions related to the super-algebra page, some of which I’ve also asked at MathOverflow. I think there’s some value on asking them here too (in particular because I’m hoping Urs might see this, who I think might have thought about this before, or, if not, might find this interesting nonetheless).

The “abstract idea” section of the super-algebra page reads:

Superalgebra is universal in the following sense. The crucial super-grading rule (the “Koszul sign rule”, Grassmann 1844, §37, §55)

$a \otimes b = (-1)^{deg(a) deg(b)} b \otimes a$in the symmetric monoidal category of $\mathbb{Z}$-graded vector spaces is induced from the subcategory which is the abelian 2-group of metric graded lines. This in turn is the free abelian 2-group (groupal symmetric monoidal category) on a single generator. (This point of view is amplified in the first part of (Kapranov 13), whose second part is about super 2-algebra, more details in Kapranov 15). Generally then super-grading and hence super-algebra arises from the 2-truncation (3-coskeleton) of the free abelian ∞-group on a single generator, which is the sphere spectrum $\mathbb{S}$.

and then:

This suggests (as indicated in (Kapranov 13, Kapranov 15)) that in full generality higher supergeometry is to be thought of as $\mathbb{S}$-graded geometry, hence dually as higher algebra with ∞-group of units augmented over the sphere spectrum.

A similar point of view is put forward in the spectral super-scheme page:

Definition.Spectral/$E_\infty$ super-geometryis simply the E-∞ geometry over even periodic ring spectra.

I’ve been exploring these ideas a bit the last few days, and I think I found a possible notion of $\mathbb{E}_{\infty}$-superalgebras and $\mathbb{E}_{\infty}$-supergeometry that differs from the ones suggested by the above quoted pages.

It starts with the following definition, taken from Bunke–Nikolaus, Section 2: given a monoidal category $\mathcal{C}$ and a ring $R$, we define a **$\mathcal{C}$-graded $R$-algebra** to be a lax monoidal functor $M\colon(\mathcal{C},\otimes_{\mathcal{C}},\mathbf{1}_{\mathcal{C}})\to(\mathsf{Mod}_R,\otimes_R,R)$. If moreover $\mathcal{C}$ is braided, then a $\mathcal{C}$-graded $R$-algebra $M$ is **$\mathcal{C}$-graded-commutative** if $R$ is a *braided* lax monoidal functor. More generally, one could allow $\mathcal{C}$ to be a symmetric monoidal $\infty$-category; this is relevant when working with spectra, but for rings only the $1$-truncation is relevant.

These recover monoid-graded algebras as the $A_{\mathsf{disc}}$-graded ones, but there’s also a number of other interesting notions arising from this definition. In particular, following Kapranov’s idea, one can consider rings graded by $k$-truncations of the sphere spectrum. For $k=0$, this recovers $\mathbb{Z}$-graded algebras, but the situation is considerably more interesting for $k=1$:

A

$\tau_{\leq1}\mathbb{S}$-graded ringconsists of a pair $(R_\bullet,\{\sigma_k\}_{k\in\mathbb{Z}})$ with

- $R_\bullet$ a $\mathbb{Z}$-graded ring;
- $\{\sigma_k\colon R_k\to R_k\}_{k\in\mathbb{Z}}$ a family of order $2$ automorphisms, one for each $R_k$;
which is moreover

$ab=\begin{cases}ba &\text{if deg(a)deg(b) is even,}\\\sigma_{\deg(a)+\deg(b)}(ab) &\text{if deg(a)deg(b) is odd}\end{cases}$$\tau_{\leq1}\mathbb{S}$-graded commutativeif we havefor each $a,b\in R_\bullet$.

By choosing $\sigma_k(a)=-a$ for all $a\in R_k$ and all $k\in\mathbb{Z}$, these recover $\mathbb{Z}$-graded-commutative algebras as a special case, and in this definition the Koszul rule comes from $\pi_1(\mathbb{S})\cong\mathbb{Z}_2$, while the $\mathbb{Z}$-grading (vs. $\mathbb{Z}_2$-grading issue that Kapranov mentions) is explained by $\pi_0(\mathbb{S})\cong\mathbb{Z}$!

“Higher supersymmetry” now corresponds to higher and higher $k$-truncations of $\mathbb{S}$ as suggested by Kapranov; though at each $n$-categorical level one can only really see “$n$-supersymmetry” (e.g. for $n=1$ this corresponds to $\infty$-functors $\mathbb{S}\to\mathrm{N}_{\bullet}(\mathsf{Mod}_R)$ being the same as $1$-functors $\mathsf{Ho}(\mathbb{S})\to\mathsf{Mod}_R$, and in this case we have $\mathsf{Ho}(\mathbb{S})\cong\mathsf{Ho}(\tau_{\leq k}\mathbb{S})$ for all $k\geq1$, so only $\tau_{\leq1}\mathbb{S}$ appears here).

This suggests a general definition of “($k$-super)-$\mathbb{E}_{n}$-$R$-algebras” as the $\mathbb{E}_{n}$-monoids in $\mathsf{Fun}(\tau_{\leq k}\mathbb{S},\mathsf{Mod}_R)$, where for $n=\infty$ we recover lax symmetric monoidal functors $\tau_{\leq k}\mathbb{S}\to\mathsf{Mod}_R$. Finally, one possible notion of $\mathbb{E}_{\infty}$-supergeometry would then be the variant of SAG obtained by replacing $\mathbb{E}_{\infty}$-rings as the basic building blocks with these “($k$-super)-$\mathbb{E}_{\infty}$-rings”.

(P.S. There are also other very interesting “universal gradings” leading to variants a bit similar to these. In particular, recalling the characterisation of the sphere spectrum as the free $\mathbb{E}_{\infty}$-group on a point, we may think of considering instead the free $\mathbb{E}_{n}$-group on a point, $\Omega^n S^n$. While $\tau_{\leq k}\mathbb{S}$-gradings give a $\mathbb{Z}$-grading together with actions induced by the first $k$ stable homotopy groups of spheres, $\tau_{\leq k}\Omega^{n}S^{n}$-gradings give a $\mathbb{Z}$-grading again, though this time the induced actions correspond to the *unstable* homotopy groups of spheres $\pi_{n+1}(S^n)$, $\pi_{n+2}(S^n)$, $\ldots$, $\pi_{n+k}(S^n)$! For ordinary rings, this means that gradings by the free braided $2$-group are given by a $\mathbb{Z}$-grading together with a $\mathbb{Z}$-indexed family of $\mathbb{Z}$-actions, corresponding to $\pi_0(\Omega^2 S^2)\cong\pi_2(S^2)\cong\mathbb{Z}$ for the gradings and $\pi_1(\Omega^2S^2)\cong\pi_3(S^2)\cong\mathbb{Z})$ for the actions!)

Has this construction combining this *specific* notion of grading with truncations of the sphere spectrum been considered before? (Have you perhaps already thought about this, Urs?)

Also, are there other nice classes of $\tau_{\leq1}\mathbb{S}$-graded commutative rings besides ordinary rings and $\mathbb{Z}$-graded-commutative ones?

]]>Hi! Corbin here with yet another ridiculous idea.

Quoting from the infamous database of categories discussion:

I remember someone at nLab could write the database in Ruby, is this offer still on?

I want to propose something quite different.

I’ve prototyped what I’m thinking of as a catabase. This is a plain SQL database which encodes some relations between categories, considering them as first-order objects. For example, there is a table called `categories`

with a column for category names, and two human-readable columns describing objects and arrows. There’s a table called `subcategories`

with a column for subcategory names, and a column for parent category names. Indeed, for all properties and structures listed by Baez as “the basic properties one instantly wants to know about any category one meets,” there is a table or view. For example, in the case of finite limits, there is a table `categorical_structure`

which pairs categories with universal objects (and a table `universal_objects`

) and a view `has_finite_limits`

which queries that table to enumerate categories which have sufficient universal objects to have all finite limits.

Further, I have prototyped the basics of user-friendly display using Datasette. For each category, and some categorical structures, a nice blurb is generated. This is similar to the Abstract Wikipedia proposal. For example, the table `karoubi_envelopes`

has formatting so that Datasette prints the following upon viewing a row:

The category Man is the Karoubi envelope of Op(FinCartSp). Put another way, Op(FinCartSp) is a wide subcategory of Man and every idempotent in Man is split.

While terse (and probably wrong, as I’m not actually good at maths!) this is well-formatted and can be arbitrarily extended with basic SQL and HTML knowledge.

It’s not at all perfect. A short summary of issues:

- Properly-factored tables require many associated queries to look up linked data. This is the N+1 antipattern in Web application design. The template for the
`categories`

table has 25 queries, of which four fit the N+1 antipattern. - I don’t think that the tables are properly factored.
`groupoids`

is effectively a view on`categories`

, but`topoi`

is its own table, because topoi have some associated properties (whether it has an NNO, is Boolean, etc.) while groupoids are merely categories which satisfy a property. This leads to… - Strange monomorphism. Should Cat get a row in
`categories`

, or should there be a table`2-categories`

? Or more pointedly, every core is an essentially wide subcategory, but some cores are only known as special cases; how should the tables`cores`

and`essentially_wide_subcategories`

be factored?- For bonus points, SQLite doesn’t permit taking foreign keys of views, so it’s not possible to preserve our custom SQL types; we also can’t take tables from views, so we can’t compute facts in a way which propagates through the SQL structure.

- A lot of data is redundant. There’s a table
`categorical_structure`

for all universal properties, and a table`enrichments`

for al enriched categories; they could be finer-grained.

How could this be productionized? The main concern is multiple writers to a single source of truth, and so I imagine that SQLite would be replaced with Pg. Otherwise, though, I’m not sure that it’d be different from any other Datasette deployment; there’s a Docker image, a database that has to be backed up, and a list of contributor keys to authenticate writers.

What would be better? Wagn (now Decko) is pretty nifty, although it used to be a headache to administer and I’m not sure whether it’s gotten easier.

Thanks for reading! I am extremely interested in folks’ feelings about this.

~ C.

]]>The Karoubian envelope is also used in the construction of the category of pure motives,

and in K-theory.

Although there is a lot of online notes/courses available where is precisely explaned how taking Karoubian envelope

is involved in the construction of (pure) motives, there seems to be a serious lack of sources where is explained how

the Karoubian envolope is involved in construtions in K-theory.

(appart from the 'basic' construction of algebraic K-theory K_0 (A) for a ring A as K_0 (P_A), where P_A

is the category of finitely generated A-modules, where P_A can also be recognized as Karoubi completion of the

category F_A of finite generated free A-modules.

Nevertheless this construction of algebraic K_0 might be considered as a 'toy' example.

Is there in the quoted sentence above also referred to certain constructions in K-theory in more general setting (eg for K-groups of exact or Waldhausen-categories)

which make use of the Karoubian completion? ]]>

http://en.wikipedia.org/wiki/Wheel_theory

This was pointed out by quid on MO, and details what some people may find interesting from the viewpoint of constructive algebra (as in, it might be interesting to give a constructive approach to wheels). I’d never heard of this before, but that probably shouldn’t be surprising.

At worst, something for a moment’s entertainment. It may go on the nLab, it may not, depending on how interested I feel about it :-)

]]>Current tex file (surely with some mistakes): (output: https://imgur.com/a/BxcuqGE )

% !TeX engine = lualatex

\documentclass[tikz]{standalone}

\usetikzlibrary{cd, arrows, graphs, graphdrawing}

\usegdlibrary{layered}

\newcommand{\cat}[1]{\ensuremath{\mathbf{#1}}}

\begin{document}

\begin{tikzpicture}[>={Stealth}, rounded corners]

\graph [layered layout, nodes={font=\bfseries}]{

Vect -> Mod -> Ab -> {Grp, CMon -> Mon -> Cat};

Grp -> {Loop, ISGrp, Mon};

SGrp -> SGrpd;

Grp -> Grpd -> {SGrpd, pSet, Cat -> SGrpd -> Set};

{UMag, Loop} -> pSet -> Set;

{{ISGrp, Mon} -> SGrp, {Loop, ISGrp} -> QGrp, {Loop, Mon} -> UMag} -> Mag -> Set;

Field -> CRing -> Ring -> Grp;

CRing -> Ab;

Poset -> Set;

Hilb -> IPS -> Norm -> TVect -> Top -> Set;

Hilb -> Ban -> CMet -> Met -> "Top\textsubscript{Haus}" -> Top;

Ban -> Norm -> Met;

TVect -> Vect;

Set -> Rel;

};

\end{tikzpicture}

\end{document} ]]>

A number of nLab pages (e.g. ring, associative algebra, graded algebra, differential graded algebra, differential graded-commutative algebra) define (or mention) the notion in question as a monoid in some monoidal category.

Is this possible in the case of divided power algebras?

]]>Is there a natural categorical way to capture the “adjunction” involving bimodules in a monoidal category? That is: given a biclosed monoidal category $(\mathcal{C},\otimes,1,[-,-]^{\mathrm{L}}_{\mathcal{C}},[-,-]^{\mathrm{R}}_{\mathcal{C}})$ which is also bicomplete and whose tensor product respects equalisers and coequalisers, and given also monoids $A$, $B$, and $C$ in $\mathcal{C}$, we have functors

together with isomorphisms

of $(D,A)$-bimodules and $(C,D)$-bimodules with

- $M$ an $(A,B)$-bimodule;
- $N$ a $(B,C)$-bimodule;
- $P$ a $(D,C)$-bimodule;
- $Q$ an $(A,D)$-bimodule.

Due to this annoying combination of bimodule structures, it seems two-variable adjunctions don’t quite capture this concept. Is there some other notion which does?

]]>Hi,

I have a beginner’s question. In $Vect_{R}$, the linear application $f:R \rightarrow E$ ($E$ a vector space over $R$) defined such as $f(1)=u$ allows to see the vector obtained by multiplication $\lambda u$ as the image of $\lambda$ : $f(\lambda)=\lambda u$. Is there a similar way to define the addition $u+v$ between two vectors $u$ and $v$ ?

Thank you for your help.

Marc

]]>In the real numbers field, when you know a function (from the reals to the reals) on the rationals and that this function is continuous (over the reals), then you know the function. In vector spaces, when you know an application on a basis and that this application is linear, then you fully know the application over the whole vector space.

My question(s) : is there a categorical explanation to this fact ? Can this be generalized ?

Thanks for your reading.

Marc ]]>

The entry for constructive fields contains a discussion about different options to define a field constructively, but the most obvious definitions is not among the options. One could simply say that a constructive field is a commutative ring $R$ such that

$(x\neq 0)\Leftrightarrow\mathsf{isinvertible}(x)$.

for all $x\in R$. Strangely, in several of the definitions on the nlab page it is proposed to replace this condition, but it is never said why this is necessary.

On the other hand, the definition of a residue field is included. Here the condition is

$(x=0)\Leftrightarrow\neg\mathsf{isinvertible}(x)$

for all $x\in R$. This condition implies that the predicate $x=0$ is double-negation stable, which seems to reduce the applicability of the definition more than the more natural condition which seems to be dismissed.

So I was wondering why the condition $(x\neq 0)\Leftrightarrow\mathsf{isinvertible}(x)$ is not considered as an option for a definition of constructive field. Perhaps there is something obviously wrong with it, that I’m missing.

]]>For a bit of fun, here are a few stats from last year!

Number of distinct pages edited on the nLab in 2020: 4055

Number of distinct edits on the nLab in 2020: 10441

Number of distinct edits per author (minimum 5):

```
c author
6978 Urs Schreiber
460 Anonymous
429 David Corfield
291 Dmitri Pavlov
189 Thomas Holder
139 Tim Porter
136 Richard Williamson
135 Paolo Perrone
116 Toby Bartels
102 John Baez
100 Daniel Luckhardt
100 David Roberts
99 Luigi Alfonsi
89 Todd Trimble
64 Matt Oliveri
57 Mike Shulman
43 Oscar Cunningham
39 alexis.toumi
36 Hurkyl
35 varkor
34 Zoran Škoda
22 Jesus Lopez
22 Jon Awbrey
22 Yuxi Liu
20 Fabian
18 Théo de Oliveira S.
16 Valeria de Paiva
15 Felipe Ponce
15 mattecapu
14 Egbert Rijke
14 Daniele Palombi
14 Mark S Davis
14 Yuri Ximenes Martins
12 Sam Staton
11 Ben Moon
11 tphyahoo
10 Elves
9 Chase Bednarz
8 AnodyneHoward
8 Jake Bian
8 mmanu F
8 Max S. New
8 Viet
8 Jin
7 arsmath
7 Lucas Immanuel Janz
7 ziggurism
7 Theresa May
7 Vikraman Choudhury
7 edeany@umich.edu
7 Luidnel Maignan
6 AlexisHazell
6 Bartosz Milewski
6 NikolajK
6 Younesse Kaddar
6 J Koizumi
6 VitalyR
6 Adam
6 Jonas Frey
6 Ulrik Buchholtz
5 Antonin Delpeuch
5 David Jaz Myers
5 DavidWhitten
5 Ivan
5 Tobias Fritz
5 Mark John Hopkins
5 Théo
```

Most edited pages in 2020 (minimum 30 distinct edits):

```
c name
202 Sandbox
70 flavour anomaly
70 AdS-QCD correspondence
52 skyrmion
52 chiral perturbation theory
49 Critique of Pure Reason
36 meson
36 CompLF/HOAS
32 Adams e-invariant
```

]]>
Is there any existing literature on what the internal logical of a groupoid might look like?

Please excuse the syntax bashing of this complete amateur, but i came up with:

abstraction introduction:

$\frac{ \Gamma, x : A \vdash A : e : B \dashv B : x, \Delta }{ \Gamma \vdash A \leftrightarrow B : \lambda x. e : A \leftrightarrow B \dashv \Delta }$abstraction elimination:

$\frac{ \Gamma \vdash A \leftrightarrow B : f : A \leftrightarrow B \dashv \Delta \qquad \Gamma \vdash B: e : A \dashv \Delta }{ \Gamma \vdash A : f e : B \dashv \Delta }$for a closed grouoid, which “seems truthy”.

]]>The category of sheaves of sets on a topological space is a topos that need not satisfy the law of excluded middle or the axiom of choice.

Doing mathematics in the topos of sheaves of sets on a topological space amounts to doing mathematics in continuous families over this topological space. Likewise, doing mathematics in the topos of G-sets amounts to doing mathematics G-equivariantly.

Many geometers do not care about logic, but they do care about doing mathematics in families or G-equivariantly. Thus, such a hypothetical geometer would still care about constructive mathematics.

Furthermore, the failure of the law of excluded middle and the axiom of choice admits really simple geometric explanations: the union of an open set and the interior of its complement need not be equal to the entire space; not every surjective etale map has a continuous section.

Suppose now that this hypothetical geometer encounters predicative mathematics. Is there an easy geometric way to convince him that he should care about predicative mathematics?

For instance, is there a geometric construction that naturally produces a ΠW-pretopos that is not a topos?

Is there an easy geometric way to see how the existence of powerobjects can fail in this construction?

If geometric examples for ΠW-pretoposes are unavailable, I would also be interested in geometric examples for pretoposes together with a simple geometric explanation for the nonexistence of (local) internal homs.

]]>I am periodically asked what has become of Joyal’s Cat Lab, which seems to be dormant or moribund. Has anyone asked him about this recently?

There would need to be a certain amount of clean-up, but we might need his permission…

]]>Given an (adjoint) equivalence in a 2-category, does anyone know if it is possible to replace the objects and/or 1-arrows in some reasonable way (up to equivalence/isomorphism) so that either the unit or the co-unit becomes an identity, not just a natural isomorphism? I don’t have time to think about it just now, and maybe someone knows something off the top of their head.

]]>I’ve had reason to think about locally internal categories/locally small fibrations over a base topos lately, and I was asked to what extent one can view these as categories of families of objects of a “locally small category” in a structural axiomatic set theory. To me it seems like one should take the fibration to be a stack, since given compatible families of objects on some cover, then one should definitely be able to glue them. Maybe I’m looking in the wrong places, but I don’t see any statements to this effect in the various papers on locally internal categories (in all their various guises and names), by Penon, Benabou, Paré–Schumacher the Baby Elephant, and The Elephant. I didn’t read them thoroughly, but I also didn’t see it in Mike’s *Sets for category theory* or *Enriched indexed categories*.

Does anyone else concur, or know of a result in the literature close to this?

]]>added the explicit definition at *localization of a commutative ring*

The nLab page on Gray’s *formal category theory: adjointness for 2-categories*
reads:

The book was supposed to be the first part of a four volume work, but unfortunately later volumes/chapters never appeared.

Do drafts of these later volumes exist?

]]>We have an entry Smith space which deals with some kind of generalized smooth space. But there’s a more important notion Wikipedia: Smith space in functional analysis, due to a different Smith.

At the very least we disambiguate, but I’m wondering whether the later should have the plain title – Smith space, and the former some qualification. Does anyone ever use the former concept?

]]>Suppose A: Cart^op → Ab is a sheaf of abelian groups on the cartesian site.

Denote by A[0]: Cart^op → Ch the 1-sheaf of unbounded chain complexes that sends S∈Cart to A(S)[0].

Under what conditions is A[0] an ∞-sheaf?

This is true if A is a constant sheaf or A is the representable sheaf of an abelian Lie group, or A is a sheaf of real vector spaces. However, I do not know a general criterion.

As a special case, for the vanishing of the first cohomology group the map A(U)⊕A(V)→A(U∩V) must be surjective.

I do not know any examples of A where this map is not surjective.

]]>It would be good to agree on a definition. ]]>

On the one hand, we have BG the space, or the BG the “delooping” one-object groupoid with morphisms G, which we might write G => *. These are closely related; the former is the geometric realization of the nerve of the latter.

On the other hand we have BG the classifying topos or stack, which is (I think) the category of all principal G-bundles.

The notation and similar role played by those objects suggest they are versions of the same thing. On nLab, we find in classifying topos it reads says that the correspondence of toposes GBund(X) = Topos(Sh(X), G) is analogous to the correspondence pi0 GBund(X) = pi Top(X,BG).

Ok but is it just an analogy, or is there some kind of stackification or Yoneda process that turns BG the space/groupoid into BG the topos/stack? Or is there some kind of truncation or geometric realization process that turns BG the stack back into BG the space?

In the article moduli stack it says the moduli stack *//G is the base of the universal principal bundle. Does that mean in the category of stacks? What’s the “total space” over *//G? Is it a stackified version of EG?

It’s hard to believe that these theories are so utterly parallel just by coincidence without a literal connection.

]]>