I was looking over Urs’s page for his talk On the nLab and noted in relation to the idea of renaming it –

Maybe MathPhysPhilLab or MathPhysLab (we never had substantial contribution on Phil).

There are positions in philosophy which make sense of this. Take Peter Hacker’s version of Wittgenstein’s understanding of the subject (p. 45):

(i) Philosophy has no subject matter of its own—in the manner in which the natural, social, and human sciences have a subject matter of their own.

(ii) There are no philosophical propositions—in the sense that there are propositions of physics or chemistry, economics or history.

(iii) There are no theses in philosophy.

(iv) There are no theories in philosophy—in the sense in which there are theories in the sciences of nature and of man.

(v) There is no philosophical knowledge—comparable to the knowledge achieved in the sciences. Philosophy is not part of the quest for knowledge of the world. The philosopher is not a citizen of any republic of ideas.

(vi) Philosophy is an activity of conceptual clarification the purpose of which is to resolve philosophical problems.

There’s something to this, although it does excludes the proto-scientific style of philosophy (say, Hegel foreseeing cohesive HoTT). But then this sort of thing doesn’t lend itself well to the established results of an encyclopaedia.

]]>I came across bimorphism recently. Since from the discussion on the page it seems very few, if any, people think this is reasonable terminology (it’s unclear to me that the concept itself is interesting at all), might it be reasonable to delete the page? There doesn’t seem any advantage to propagating bad terminology.

(I’m not sure how to link this thread to the relevant nLab page. I thought “Discuss this page” would be sufficient, but apparently not.)

]]>thanks in advance. ]]>

https://ncatlab.org/nlab/show/Récoltes+et+semailles ]]>

My name is Flavio, and I represent SearchOnMath (searchonmath.com), a search engine designed especially for mathematical content. This means that SearchOnMath is able to search for mathematical formulas and/or text.

The purpose of my message is to inform you that our latest version includes the nLab content pages. To submit a query to be searched only on nLab pages, just click on "Limit Search Domain" button, placed at our start page, and choose 'nLab'.

Finally, formulas must be written in TeX (LaTeX) and placed between the ${ }$ delimiters, as in:

Einstein's formula, ${E = mc^2}$, revolutionized physics.

Thank you very much for your time.

Regards,

Flavio ]]>

In this article it says that the groupoid cardinality of the 2D hypercomplex number systems is $\frac 3 2$ because there are three 2D hypercomplex number systems (up to isomorphism) each equipped with one non-trivial algebra automorphism. But surely, the dual numbers have infinitely many automorphisms satisfying $1 \mapsto 1, \epsilon \mapsto k\epsilon$ (where $k$ is an arbitrary non-zero real number), so the value of the groupoid cardinality is actually $\frac 1 2 + \frac 1 2 + \frac 1 \infty = 1$.

Who is right?

]]>Is Chaitins constant which is frequently given as an example of an undefinable number really even a number?

Kleene and Skolem seem to think the Skolem paradox and model theory in general show that there is no absolute notion of counting. How is this categorically understood?

Thank you for all pointers ahead of time ]]>

I would like to suggest that the page of page-categories of the nLab https://ncatlab.org/nlab/page_categories should have a more strict structure.

I wanted to propose something along the lines of, each page_category should be either:

1. a mathematical object (all the theorems, definitions and constructions go there)

2. a person or people (some 3163 pages--this is mostly ok already),

3. books and papers, (have 3 overlapping pages, it seems: https://ncatlab.org/nlab/all_pages/Paper%20References, https://ncatlab.org/nlab/all_pages/reference and https://ncatlab.org/nlab/all_pages/references) - some 200 pages, perhaps?

4. fields of mathematics (Wikipedia has 72 of these) AND

5. MISC, for whatever else people wanted to have, like jokes, or things difficult to classify.

This would help when relating the nLab to WikiData using the property https://www.wikidata.org/wiki/Property:P4215

Does this sound sensible to you? There are only 60 page_categories, so these we could do by hand very easily, if you think this is a good idea.

Maybe this has been discussed in the nForum before, maybe it was discarded because of traditional rules that I don't know about. If so, I would like to know the rationale. Thanks! ]]>

Consider either the category of sheaves over the big zariski site, or some setting for Lawvere’s synthetic differential geometry.

What does the category of objects $X$ with a fixed reduction $X_0$ look like. The morphisms are morphisms $X \rightarrow Y$ such that $X_{red} \rightarrow Y_{red}$ is an isomorphism.

These are like the fibers of $\text{Type} \rightarrow \text{Type}_{red}$ where $\text{Type}_{red}$ consists of the reduced objects of $\text{Type}$, under modal homotopy type theory.

In the geometric setting, we should get some kind of linear category, like the category of quasicoherent sheaves. But I can’t find a place where this is discussed.

It’s a bit like how $\text{QuasicoherentSheaves} \rightarrow \text{Scheme}$ is like a $2$-fibration, and the fiber of a scheme is its category of quasicoherent sheaves.

Or the “2-fibration” of a tangent category over a given category, possibly adjoint to a certain $2$-context extension.

One thing to note is that all of these fibers of categories are settings for some linear type theory (possibly without objects being dualizable).

]]>Regarding this article, I noticed a few possible discrepancies that I was hoping someone could help me understand.

The page says that a closed immersion of schemes determines a homeomorphism of the underlying spaces. But Lurie instead says that it a homeomorphism onto a closed subspace. Could this be a typo?

This page states that the comorphism, oriented as $\mathcal{O}_Y\to f_*\mathcal{O}_X$, is an epimorphism; this agrees with the definition from classical scheme theory as far as I know. But Lurie instead requires $f^*\mathcal{O}_Y\to\mathcal{O}_X$ to be an epimorphism. The classic condition obviously implies Lurie’s condition (using the fact that the inverse image functor preserves epis and the counit of the adjunction is an iso), but I did not see how to show the converse.

I wonder what’s going on here? Thanks in advance!

]]>Could categories be considered generalized uniform structures, extending this table by the following row?

- proarrow -> directed graph
- monad -> category
- pro-monad -> ?
- symmetric proarrow -> undirected graph
- symmetric monad -> groupoid
- symmetric pro-monad -> ?

Proposition 2.26. (closure of a finite union is the union of the closures)

In the proof , it says:

“because if every neighbourhood of a point intersects all the $U_i$, then every neighbourhood intersects their union.”

maybe it should be “because if every neighbourhood of a point intersects some $U_i$, then every neighbourhood intersects their union.”

]]>Is there any results in this direction? ]]>

the coverage over the point \Delta[0] is empty, and for any sheaf its fiber over \Delta[0] is a singleton.

Should any coverage instead be the union of all the maps from the interval \Delta[1] and all the maps from the point \Delta[0]? ]]>

Reading through this I found a ‘spring’ dating from October 2012.

]]>It seems like there’s some confusion in the definition section of the 2-limit page – it asserts the existence of a pseudonatural equivalence $K(X,L)\simeq Cat^\mathcal{C}(J,K(X,F-)),$ but these are hom-categories and not 2-categories themselves so pseudonatural transformations aren’t defined on them. The two obvious routes I see to rectify this are to require the equivalence on the full $2$-categories instead of the hom-categories, or to promote the hom-categories to discrete $2$-categories, but I’m not sure which (or if either) is implicitly intended. Any clarification is greatly appreciated.

]]>In the discussion at the bottom of monoidal category, we read:

In fact a strict monoidal category is just a monoid internal to the category Cat. Unfortunately this definition is circular, since to define a monoid internal to Cat, we need to use the fact that Cat is a monoidal category!

And then later

For example, you can define a monoidal category to be a pseudomonoid internal to the 2-category Cat — but nobody knew how to define these concepts until they knew what a monoidal category is!

Doesn’t the same circularity afflict the definition of monoidal category that’s on the page? For example, the associator is given as

$a \;\colon\; ((-)\otimes (-)) \otimes (-) \overset{\simeq}{\longrightarrow} (-) \otimes ((-)\otimes(-))$But this doesn’t make sense if taken literally. You cannot have a natural transformation between functors on different domains, and the domain of these functors are not the same. The domain of the left functor is $(\mathcal{C}\times \mathcal{C})\times\mathcal{C}$ whereas the domain of the right functor is $\mathcal{C}\times (\mathcal{C}\times\mathcal{C})$. Of course those two categories are isomorphic, and using that isomorphism, we can make sense of the definition. But that’s using the monoidal structure of $\text{Cat}$! We’re being circular in exactly the same way as we would if we defined a monoidal category as a (pseudo)monoid in the monoidal (2-)category $\text{Cat}$!

I guess it’s not circular in any formal sense, since we can just observe that any cartesian category has canonical isomorphisms $(\mathcal{C}\times \mathcal{C})\times\mathcal{C}\cong \mathcal{C}\times (\mathcal{C}\times\mathcal{C})$ and we can just insert that into the definition as needed, without commenting that it is part of a monoidal structure on the ambient category. The same applies to any monoid in any cartesian category. In particular, I think it’s not formally circular to define a (weak/strict) monoidal category as a (pseudo) monoid in $\text{Cat}$.

Shouldn’t that equivalent definition be mentioned higher in the article, since it’s valid and not really circular?

And shouldn’t the article be more explicit about this, about using the cartesian associator and unitors of $\text{Cat}$, given that it’s basically an article about the need to be careful and rigorous about the axioms of associators and monoidal structures?

Also, is there some more coherent, higher categorical way out of this circularity, other than than just capping it with a cartesian structure at some level of the higher categorical ladder?

]]># Haar measure

* table of contents

{: toc}

## Idea

If $G$ is a [[topological group]], a _Haar measure_ is a translation-invariant measure on the [[Borel set]]s of $G$. The archetypal example of Haar measure is the [[Lebesgue measure]] on the (additive group underlying) [[cartesian space]] $\mathbb{R}^n$.

## Definition

The proper generality in which to discuss Haar measure is where the topological group $G$ is assumed to be [[locally compact space|locally compact]] [[Hausdorff space|Hausdorff]], and from here on we assume this. (For [[topological group]]s, the Hausdorff assumption is rather mild; it is equivalent to the $T_0$ separation condition. See the discussion at [[uniform space]].)

Let $C_c(G)$ denote the vector space of continuous real-valued functionals with compact support on $G$. This is a [[locally convex topological vector space]] where the locally convex structure is specified by the family of seminorms

<latex>\rho_K(f) = \sup_{x \in K} |f(x)|,</latex>

$K$ ranging over compact subsets of $G$. Recall that a [[Radon measure]] on $G$ may be described as a continuous linear functional

<latex>\mu: C_c(G) \to \mathbb{R}</latex>

which is _positive_ in the sense that $\mu(f) \geq 0$ whenever $f \geq 0$. This defines a measure $\hat{\mu}$ on the $\sigma$-algebra of Borel sets in the usual sense of [[measure theory]], where

<latex>\hat{\mu}(B) = sup \{\mu(f): supp(f) = K \subseteq B, \rho_K(f) = 1\}</latex>

By abuse of notation, we generally conflate $\mu$ and $\hat{\mu}$.

A **left Haar measure** on $G$ is a nonzero Radon measure $\mu$ such that

<latex>\mu(g B) = \mu(B)</latex>

for all $g \in G$ and all Borel sets $B$.

### The Haar Integral

Let $G$ be a topological group, and let $\mathbb{C}[G]$, the group ring over $G$. Let $G \text{-Ban}$ be the category of Banach representations of $G$. Objects in $G \text{-Ban}$ are banach spaces $X$ over $\mathbb{C}$ with a continuous action $G \times X \rightarrow X$. Maps in $C$ are bounded, $G$-equivariant maps. (Alternatively, $G \text{-Ban}$ can be viewed as a category of certain $\mathbb{C}[G]$-modules.)

Let $\text{Top}$ be the category of topological spaces, and consider $[G, \mathbb{C}]_{ \text{Top}}$, a Banach representation of $G$ with action $G \times [G, \mathbb{C}]_{ \text{Top}} \rightarrow [G, \mathbb{C}]_{ \text{Top}}$.

We may view $\mathbb{C}$ as a Banach representation of $G$ where $gz = z$ for each $z \in \mathbb{C}$ and each $g \in G$. $\mathbb{C}$ embeds into $[G, \mathbb{C}]_{\text{Top}}$ as constant functions. We may then consider the exact sequence

<latex>0 \rightarrow \mathbb{C} \rightarrow [G, \mathbb{C}]_{\text{Top}} \rightarrow [G, \mathbb{C}]_{\text{Top}}/ \mathbb{C} \rightarrow 0</latex>

A Haar integral on the $G$-representation $[G, \mathbb{C}]_{\text{Top}}$ is a retract $\int_G : [G, \mathbb{C}]_{\text{Top}} \rightarrow \mathbb{C}$ for the injection $\mathbb{C} \rightarrow [G, \mathbb{C}]_{\text{Top}}$. In other words, it is a function $\int_G : [G, \mathbb{C}]_{\text{Top}} \rightarrow \mathbb{C}$ such that

<latex> \int_G (f_1 + f_2) = \int_G f_1 + \int_G f_2 \ \ \ \forall f_1, f_2 \in [G, \mathbb{C}]_{\text{Top}}</latex>

<latex> \int_G a f = a \int_G f \ \ \ \forall f \in [G, \mathbb{C}]_{\text{Top}}, a \in \mathbb{C}</latex>

<latex> \int_G f^g = \int_G f \ \ \ \forall f \in [G, \mathbb{C}]_{\text{Top}}, g \in G</latex>

<latex> \exists C \in \mathbb{R}_{\geq 0 } : \left| \left| \int_G f \right| \right| \leq C \int_G ||f|| \ \ \ \forall [G, \mathbb{C}]_{\text{Top}}</latex>

The last of these requirements, given the others, is equivalent to continuity of $\int_G$.

It is a fundamental theorem, which we will now show, that there is precisely one Haar Measure.

**Remark:** In some sense, we might wish to show that $\text{Ext}^1_{\mathbb{C}[G]}([G, \mathbb{C}]_{\text{Top}}, \mathbb{C})$ vanishes in an appripriate category; this would show that the sequence

<latex>0 \rightarrow \mathbb{C} \rightarrow [G, \mathbb{C}]_{\text{Top}} \rightarrow [G, \mathbb{C}]_{\text{Top}}/ \mathbb{C} \rightarrow 0</latex>

splits by the usual characterization of extensions via $\text{Ext}^1$. On further contemplation, however, it is sufficient only to show that the trivial $G$-representation $\mathbb{C}$ is an injective object in $G \text{-Ban}$. This could be seen as an equivariant Hahn-Banach theorem.

**Proof:** From the remark, it is sufficient to show that $\mathbb{C}$ is an injective object in $G \text{-Ban}$. Take an injection of Banach representations of $G$, $X \rightarrow Y$. Let $f : X \rightarrow \mathbb{C}$ be a map of Banach representations of $G$. By the (usual) Hahn-Banach theorem, there exists a functional $g : Y \rightarrow \mathbb{C}$ extending $f$, though it may lack $G$-invariance.

Consider the subset of all extensions of $f$ to $Y$. Let $S$ be the collection of $G$-invariant compact convex subsets of this set. $S$ contains the convex hull of $G g$, where $g$ is some chosen extension of $f$ to $Y$, so $S$ is nonempty. Using compactness and Zorn's lemma, we may find a minimal element of $S$ in this collection, where $S$ is ordered where $A \leq B$ when $A \subset B$. Call this element $H$. $H$ must be a singleton. If $H$ contains a point which is not extremal then it contains the convex hull of the orbit of that point, which would be a proper $G$-invariant compact convex subset of $H$ (see Krein Milman theorem).

Therefore $H$ is a singleton, and its unique element is a $G$-invariant functional extending $f$.

In particular, since $\mathbb{C}$ has been shown to be injective, the map $\text{Id}_{\mathbb{C}} : \mathbb{C} \rightarrow \mathbb{C}$ lifts along the inclusion

<latex>0 \rightarrow \mathbb{C} \rightarrow [G, \mathbb{C}]_{\text{Top}}</latex>

**Remark:** this alone does not show uniqueness. However, uniqueness is not hard.

**Remark:** by the Riesz-Markov-Kakutani representation theorem, it follows that there is a unique Haar measure on $G$. This result was first proven by Weil. A proof along different lines can be found in these online [notes](http://simonrs.com/HaarMeasure.pdf) by Rubinstein-Salzedo.

### Left and Right Haar Measures that Differ

The left and the right Haar measure may or may not coincide, groups for which they coincide are called **unimodular**.

Consider the matrix subgroup

<latex>

G := \left\{ \left.\, \begin{pmatrix} y & x \\ 0 & 1 \end{pmatrix}\,\right|\, x, y \in \mathbb{R}, y \gt 0 \right\}

</latex>

The left and right invariant measures are, respectively,

<latex>

\mu_L = y^{-2} \,\mathrm{d}x \,\mathrm{d}y,\quad \mu_R = y^{-1} \,\mathrm{d}x \,\mathrm{d}y

</latex>

and so G is not unimodular.

[[Abelian groups]] are obviously unimodular; so are [[compactum|compact]] groups and [[discrete topology|discrete]] groups.

[1]: https://arxiv.org/abs/math/0606794

[[!redirects Haar measure]]

[[!redirects Haar measures]]

[[!redirects haar measure]]

[[!redirects haar measures]] ]]>

I feel that, while the concept of algebra over a monad is of course an instance of the more general module over a monad, it could benefit from having its own dedicated page, with motivation from other fields and dedicated examples.

If I get a green light from enough people here, I can create the page myself (and replace the redirection by links both ways).

]]>I would like to add content to the nLab about the costrength of a monad. I could either (significantly) modify the strong monad page, or create a separate dedicated page. Any thoughts?

]]>I believe commutative monad should redirect to monoidal monad and not to commutative algebraic theory (but of course have a link to the latter). Is anyone disagreeing?

]]>We don’t have an article on Peter Freyd’s “algebraic real analysis”. This is related to Tom Leinster’s characterization of $L^1$ as well. It’s all very interesting work, and it leads to clean proofs of e.g. existence and uniqueness of Haar measure. I was hoping to make an article about these things on Wednesday if that is ok.

]]>There is a question and answer in MO here Are they right that the n-Lab entry is wrong?

]]>In the page dependent sum type, the presented “term introduction rule” seems not really dependent, since the typing derivation of the second element is not aware of the first. I would expect $\frac{\vdash t:A\quad \vdash u:B[t/x]}{\vdash (t,u):\Sigma(x:A).B}$ which corresponds to what is presented in the text on positive and negative versions.

]]>