Proof for one component: As 0-cycles are synonimous with 0-chains, they have <latex>\alpha_{0}</latex> many dimensions-of-freedom for sure. How many the 0-boundaries of our component have? Since the incidence matrix is a <latex>\alpha_{0}</latex> by <latex>\alpha_{1}</latex> matrix (see my previous posting with ‘homology’ tag), they have at least <latex>\alpha_{1} - \alpha_{0}</latex> many for sure. And <latex>1</latex> more for the rank deficiency! The proof follows from this arithmetic. ]]>

I want to record a certain lemma on the nlab that is rarely made explicit (slight variation here: https://arxiv.org/abs/1112.0094).

The theorem is that if you have a profunctor $R$ from $C$ to $D$ and it is representable as a functor $F : C \to D$ then the action of $F$ on morphisms is uniquely determined by $R$ and the representation isomorphism.

This explains the initially strange feature of type theoretic connectives, which have semantics as functors but we never actually define their action on morphisms because we can derive the action from the introduction and elimination rules.

My question is should this get a page on the nlab, called I guess “Representability determines Functoriality” or something like that? Or should I try to find some existing page?

And a meta-question is should I err on the side of just making a page with whatever name I feel like or come to discuss the name here on the nforum? Is it easy to change names of pages after the fact?

]]>I am running into the following simple question and am wondering if there is anything useful to be said.

Let

$\mathcal{A} \in dgcAlg_\mathbb{Q}$be a differential graded-commutative algebra in characteristic zero, whose underlying graded algebra is free graded-commutative on some graded vector space $V$:

$\mathcal{A} = (Sym(V), d) \,.$Consider an odd-graded element

$c \in \mathcal{A}_{odd} \,,$and write $(c)$ for the ideal it generates.

In this situation I’d like to determine whether it is true that

there is an inclusion $\mathcal{A}/(c) \hookrightarrow \mathcal{A}$;

for every element $\omega \in \mathcal{A}$ there is a decomposition

$\omega = \omega_0 + c \omega_1$for

*unique*$\omega_0, \omega_1 \in \mathcal{A}/(c) \hookrightarrow \mathcal{A}$.

For example if $c \neq 0 \in V_{odd} \hookrightarrow \mathcal{A}_{odd} \hookrightarrow \mathcal{A}$ is a generator, then these conditions are trivially true.

On the other extreme, if $c$ is the product of an odd number $\gt 1$ of odd generators, then it is not true. For example if $c = c_1 c_2 c_3$, with $c_1, c_2, c_3 \in V_{odd} \hookrightarrow \mathcal{A}_{odd}$, then for instance $c (1 + c_1) = c (1 + c_2) = c$ and so the coefficient $\omega_1$ is not unique.

Is there anything useful that one can say in general?

]]>added a few lines and original references to *supersymmetry breaking*.

Took a stab at a general formulation of Poisson summation formula, although the class of functions to which it is supposed to apply wasn’t nailed down (yet).

(Some of the ingredients of Tate’s thesis are currently on my mind.)

]]>started something at *splitting principle*

(wanted to do more, but need to interrupt now)

]]>As many of you know, the nLab is currently hosted on a server at CMU, financed by a grant for HoTT there. I am not sure how long there is to go on that grant, but I am not sure it is longer than a couple of years. I would suggest that it might be a good idea try to begin applying for some further grants fairly soon. How shall we go about this?

I am also a little bit worried about whether there are any backups of the nLab or nForum currently if the server at CMU is hit by lightning. To safeguard years of work, we need some good safety net in place. One option, if we have the money, is to use cloud storage, e.g. Amazon’s. Their set up is such that it is basically impossible that the data could be lost. It is pay per use, and a priori quite cheap, but if we host the actual database there, since it is up 24/7, costs might not be completely trivial. We could perhaps use the cloud only for backup, say once a day, and maybe use an S3 bucket rather than a database, which would be a lot cheaper, but still not completely negligible.

Another option is to try to initiate some kind of ’goodwill system’ amongst a collection of universities across the world to call our server once a day, say, to get backups of the data. We could have a minimal web application hosted at CMU which monitors these, so we know they are taking place.

]]>I have spent some minutes starting to put some actual expository content into the Idea-section on *higher gauge theory*. Needs to be much expanded, still, but that’s it for the moment.

A *lot* of new visitors have trouble entering math on the forum. It ought to be easy to add some more descriptive help text to the formatting options below the post. Could it say something like

To use (La)TeX mathematics in your post, make sure “Markdown+Itex” is selected and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted; see here for a list.

?

]]>After a long time I looked at the Authors page. It lists 7 pages of authors, but the page 7 is empty. Page 6 ends with Y and the names under Z are not listed (including my own name). I guess the page 7 as the last has been somehow skipped in generation as the last in the queue due a bug in generation script.

]]>added a brief historical comment to *Higgs field* and added the historical references

Let $X$ be a set with an apartness relation. We can then define a subset $U$ of $X$ to be open if and only if

$\forall x, y. (x \in U \Rightarrow y \in U \vee x # y).$In that way, $X$ becomes a topological space. As detailed at *apartness relation*, the induced locale $X'$ can be described as a certain quotient of the discrete locale over $X$ in the category of locales.

I’m trying to work out what the points of this locale are. The canonical continuous map

$X \longrightarrow Pt(X'), x \mapsto \{ U \subseteq X | U open, x \in U \}$factors over $X/\sim$, where $({\sim})$ is the equivalence relation induced by the apartness, and I can show that the resulting map ${X/{\sim}} \to Pt(X')$ is injective. Is it surjective?

(Assuming classical logic, it is. In this case furthermore $X'$ is a locally compact locale. But I need an intuitionistic proof.)

My motivation is as follows. It’s possible to pass from $X$ to the quotient set $X/{\sim}$; however, as we know, this loses information and is badly behaved in some sense. If the points of $X'$ turned out to be the elements of $X/{\sim}$, we could subscribe to the following philosophy: “Sure you can consider the quotient. Just make sure to consider it in the category of locales instead of sets. The localic quotient will have exactly the points you expect, but will come with additional ’glue’ to create a well-behaved object.”

]]>at surjective geometric morphism I have spelled out in detail most of the proof of the various equivalent characterizations, and all of the proof of the statement that geometric surjections are comonadic.

]]>If one solves

${b}_{1}\u27e8{a}_{1}\u27e9+\dots +{b}_{m}\u27e8{a}_{m}\u27e9=\delta [\sum \sum {x}_{i,j}\u27e8{a}_{i},{a}_{j}\u27e9]$for x’s when b’s are given, one ends up with

$\mathit{\eta}\overrightarrow{x}=\overrightarrow{b}$where \eta is the INCIDENCE MATRIX! Then all proofs about 0-homology groups are just about the ranks of square-submatrices of this matrix. Am I right?

]]>I have added the Fierz identities that give the $S^2$-valued supercocycle in 5d here.

Added this briefly also at *Fierz identity*: here

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.

]]>This was long overdue: I have created a page *lattice (disambiguation)* and added corresponding warnings on terminology to the relevant entries.

In the course of this I have created stubs for *lattice in a vector space*, *integral lattice* and *modular integral lattice*.

Not of any direct relevance to the nLab, but I thought one or two of you might be interested in this note. It will appear on the arXiv on Friday (European time).

]]>created ball

]]>Has anyone here encountered one of these "in nature"?

I can see why the first situation would be much more common with cartesian closed categories, as "preserving finite products" happens under simple conditions, but I don't see why one should be more common than the other with general monoidal closed categories. ]]>

I hope that Urs sees this, but anyone is welcome to answer, of course! I am wondering if we can recover Tate twists in, say, singular cohomology or Deligne cohomology as a special case of twisted cohomology? If so, could someone unwrap how?

]]>I have been adding and editing a bit at *axion* in the section *In string theory*.

The axion fields in string theory form a curious confluence point relating

- abstract concepts related to higher gauge theory

with

- fundamental questions in particle physics/cosmology phenomenology

as indicated schematically in this table (now also in the entry):

$\,$

$\array{ \mathbf{\text{higher gauge theory}} && && \mathbf{\text{particle physics/cosmology phenomenology}} \\ \\ \left. \array{ \text{higher gauge fields} \\ \text{higher characteristic classes} \\ \updownarrow \\ \text{non-perturbative QFT/string effects} \\ \text{in HET: Green-Schwarz anomaly cancellation} \\ \text{in IIA/B: higher WZW term for Green-Scharz D-branes} } \right\} &\longrightarrow& \array{ \text{axion fields} \\ \text{in the string spectrum} } &\longrightarrow& \left\{ \array{ \text{solve strong CP-problem as with P-Q robustly} \\ \text{solve dark matter problem by FDM} } \right. }$$\,$

I will be trying to expand on this a little more.

]]>I have touched formal group a bit, but don’t have time to do anything substantial.

I need to adjust some of the terminology that I had been setting up at cohesive (infinity,1)-topos related to *infinitesimal cohesion* : the abstract notion currently called “$\infty$-Lie algebroid” there should be called “formal cohesive $\infty$-groupoid”. The actual L-infinity algebroids are (just) the first order formal smooth $\infty$-groupoids.

While on the train I started expanding some other entries on this point, but I need to quit now and continue after a little interruption.

]]>created *worldline formalism* to go with this Physics.SE answer