User peter arndt - MathOverflow

Is the injective model structure on symmetric spectra Bousfield localizable?

2010-10-19T13:23:38Z

I am interested in injective model structures on both symmetric spectra as exposed in Hovey/Shipley/Smith and motivic symmetric spectra as in Jardine's article. Both authors take a model structure on the underlying spaces - simplicial sets in Hovey/Shipley/Smith and motivic spaces in Jardine - with monomorphisms as cofibrations. Then both establish two level model structures on symmetric (resp. motivic symmetric) spectra with weak equivalences given levelwise, an injective one with levelwise cofibrations and a projective one with levelwise fibrations.

However, both then proceed using only the projective model structure, and Bousfield localizing it with respect to an appropriate class of maps. Presumably they do so because it is much more straightforward to verify that the projective model structure satisfies the prerequisites for a Bousfield localization, in particular one has easy candidates for generating cofibrations which then turn out to do the job.

My question is: Is the injective level model structure on symmetric spectra (resp. motivic symmetric spectra) known or expected to be cellular or combinatorial? How would you try to prove this?

Tangent space of the moduli stack of Drinfeld modules

2013-01-19T16:18:05Z

I am going through the proof of Thm 1.5.1 of Laumon, Cohomology of Drinfeld modular varieties, which says that a certain map of stacks is smooth. To prove this, Laumon considers the tangent space of a moduli stack:

Very roughly, a Drinfeld module over $k$ is a certain ring homomorphism $A \to k[\tau]$, the moduli stack of Drinfeld modules (of a fixed rank, but never mind) associates to a ring $k$ the groupoid of such ring homomorphisms.

The tangent space of this stack at a $k$-rational point corresponding to a Drinfeld module $A \to k[\tau]$ parametrizes deformations of this, i.e. Drinfeld modules $A \to (k[\epsilon])[\tau]$ ($k[\epsilon]$ the ring of dual numbers), who become the given module after modding out $\epsilon$.

So deformations of a Drinfeld module are lifts of this homomorphism $A \to k[\tau]$ to $A \to (k[\epsilon])[\tau]$ and these are controlled by Hochschild cohomology: Let $m[\tau]:=\epsilon \cdot k[\tau]$ denote the ideal of $(k[\epsilon])[\tau]$.

The existence of lifts has an obstruction in $HH^2(A,m[\tau])$ and this cohomology group vanishes, so lifts always exist. The existing lifts are then parametrized by $HH^1(A,m[\tau])$ and so that is the tangent space of the moduli stack of Drinfeld modules of rank $d$ at the $k$-rational point given by the Drinfeld module $A \to k[\tau]$.

Hochschild cohomology is given by the following Ext-groups $HH^n(A,m[\tau]) \cong Ext^n_{A \otimes_{\mathbb{F}_p} A}(A,m[\tau])$ and the Ext-groups are the cohomology groups of the complex $RHom_{A \otimes_{\mathbb{F}_p} A}(A, m[\tau])$. One knows that these $Ext^n$-groups vanish for $n \geq 2$.

This is all fine but then Laumon asserts the following quasi-isomorphism:

$$ RHom_{A \otimes_{\mathbb{F}_p} A}(A, m[\tau]) \cong (T_{A/ \mathbb{F}_p} \otimes^L_{A \otimes_{\mathbb{F}_p} A} m[\tau])[-1] $$

where $T_{A/\mathbb{F}_p} := Hom_A(\Omega^1 _{A/\mathbb{F}_p}, A)$ and where this is said to be considered as an ${A \otimes_{\mathbb{F}_p} A}$ -module "via the augmentation map" ${A \otimes_{\mathbb{F}_p} A} \to A$.

This is where I am stuck. I would be grateful for an explanation of the above quasi-isomorphism or a reference which provides one. Thanks!

Answer by Peter Arndt for Several Topos theory questions

2009-10-28T22:28:45Z

About 1: Yes!

About 2: (Internal logic of Zariski topos) I don't think it has been done systematically. A glimpse of it is in Anders Kock, Universal projective geometry via topos theory, if I remember well, and certainly in some other places. But one point is that it is not at all easy to find formulas in the internal language which express what you have in mind. See my answer at http://mathoverflow.net/questions/606/synthetic-reasoning-applied-to-algebraic-geometry

About 3:You can indeed glue all sorts of things:

Things fitting into the axiomatic framework of "geometric contexts": Look at the "master course on Algebraic stacks" here: http://www.math.univ-montp2.fr/~toen/m2.html This one is great reading to understand the functorial point of view on schemes and manifolds!
Commutative Monoid objects in good monoidal (model) categories: http://arxiv.org/abs/math/0509684
Commutative monads (here you can glue monoids, semirings and other algebraic structures mixing them all): http://arxiv.org/abs/0704.2030
In Shai Haran's "Non-Additive Geometry" you can even glue the monoids and semirings etc. with relations (although I wouldn't know why)
You can also glue things "up to homotopy instead" of strictly - this is roughly what Lurie's infinity-topoi are about, and also the model catgeory part of the 2nd point, or any oter approaches to derived algebraic geometry

One of several good points of view on what a Grothendieck topology does, is to say it determines which colimits existing in your site should be preserved under the Yoneda embedding, i.e. what glueing takes already place among the affine objects. So, if you insist on glueing groups it could be a good idea to look e.g. for a topology which takes amalgamated products (for me this means glueing groups, you may want only selected such products, e.g. along injective maps) to pushouts of sheaves... Then feel free to develop a theory on this and send me a copy!

About 4: (Why don't people work with sheaves instead of schemes) They do. One situation where they do is when taking the quotient of a scheme by a group action. The coequalizer in the category of schemes is often too degenerate. One answer is taking the coequalizer in the category of sheaves, the "sheaf quotient" (but sometimes better answers are GIT quotients and stack quotients).

Is there a general setting for self-reference?

2010-09-22T15:30:52Z

This is a question about self-reference: Has anyone established an abstract framework, maybe a certain kind of formal language with some extra structure, which makes it possible to define what is a self-referential statement?

Answer by Peter Arndt for Is there a general setting for self-reference?

2012-10-07T20:02:51Z

I found another one:

John Bell, “Incompleteness in a General Setting”. Bulletin of Symbolic Logic 13, 2007. It's paper number 66 here

When are localizations of Lawvere theories flat?

2012-04-18T08:05:34Z

Suppose we have a Lawvere theory $L$, i.e. a category with finite products and objects $[n]$ given by the natural numbers such that $[n] \cong [ 1 ] ^n $, and localize it to a Lawvere theory $S^{-1}L$ by inverting an arrow in it (this works: one can e.g. get a small monoidal category with the arrow inverted as in this question, then make it into a category with finite products as in this question).

Now I consider models in $Set$ of the theories so obtained:

The morphism $L \to S^{-1}L$ induces a forgetful functor $S^{-1}L-Alg \to L-Alg$ which has a left adjoint (given by left Kan extension). Analogously to the case of localization of rings this left adjoint preserves finite limits, if our arrow had domain and codomain $[ 1 ]$, i.e. we just inverted an operation with one input and one output. My question is this:

Does anybody know conditions under which the inversion of more general arrows also induces a finite limit preserving left adjoint? Simple examples are welcome. What about the free Lawvere theory on one operation $[n] \to [n]$ which we then invert? Most importantly: What about the free commutative such Lawvere theory?

Answer by Peter Arndt for Elementary mathematical books

2012-04-24T22:27:08Z

Fearless Symmetry by Ash and Gross

Answer by Peter Arndt for Localization of a symmetric monoidal category at a single morphism

2012-04-03T00:42:45Z

I'll try to bend David White's answer towards the actual situation of your question. The outcome is somewhat clumsy and it totally looks like model structures can be eliminated from it, but anyway:

Assume your category C is closed monoidal and locally presentable. Then it is a monoidal model category with cofibrations and fibrations all morphisms and weak equivalences the isomorphisms. This model category is cofibrantly generated: One can take the identity of the initial object as generating trivial cofibration and the set of all morphisms between the objects $G$ from some generating set as generating cofibrations.

This model category then satisfies the hypotheses of Barwick's Thm. 4.46 in this article for a Bousfield localization at the one element set containing $f$. The homotopy category for the localized model structure has the universal property you want and lives in the same universe. You have an adjunction between the homotopy category of the original model structure, which is the category itself, and the localization.

This adjunction is a reflection to an orthogonal subcategory as in Adamek/Rosicky, 1.35-1.38, namely to the full subcategory of all objects from whose point of view $f$ "was already an isomorphism" (i.e. $f$-orthogonal objects; precise definition via a unique-lifting-condition). This is much like in your example (but with the condition on the twist removed and without the domain of f having to be special). If you chase through the proofs, you also get an expression of the reflection functor as a colimit via the small object argument, resp. via Adamek/Rosicky's proof...

Barwick's Prop. 4.47 gives then a criterion for the homotopy category to be closed monoidal again: It suffices that any object $X$ which satisfies the unique right lifting condition with respect to $f$ also satisfies it with respect to $f \otimes G$ for every generating object $G$ (remember the category was locally presentable now) i.e. if $f$ induces an iso $Hom(f,X)$ then $Hom(f \otimes G,X)$ is an iso, too, for every generating object $G$.

edit: Sorry, I am no longer sure that the homotopy category of the Bousfield localization is in fact the localization along $f$ in the sense you asked for: When you localize with respect to an arrow you automatically invert together with it a bunch of other arrows. When you do plain category theory it is somewhat uncontrollable what those other arrows are, it seems to me. When you do Bousfield localization these other arrows are those having the left lifting property with respect to the $f$-local objects. Now I don't see a reason why the class of additionally inverted arrows should be the same in both cases. What Bousfield localization as sketched here probably yields, is the universal colimit preserving functor which inverts $f$.

Left adjoint to the forgetful functor from finite product categories to symmetric monoidal categories

2012-04-02T14:22:24Z

I recall reading that the forgetful functor $FinProdCat \to SymMonCat$ from categories with finite products and product preserving functors to symmetric monoidal categories and tensor preserving functors has a left adjoint.

To make this precise one has to insert lax, weak or strict in several places -- I am interested in any combination of these (but most in a 2-adjunction between the categories with weakly product, resp. tensor, preserving functors).

Is something like this true at all? If so, can anyone give a true and precise statement and/or a reference? I wouldn't mind getting a concrete description of the left adjoint, but a confirmation of its existence would already be a treat.

Thanks!

(This is not a case of google laziness: I spent half a day looking for reference. I would imagine that the statement emerges after inserting the right things into long known results about enriched base change or 2-monads, but I wasn't able to find the right one)

EDIT: What would be nice would be an argument along these lines: Both Symmetric monoidal categories and finite product categories are algebras for certain pseudomonads. Algebras for pseudomonads are are finite copower preserving functors from Cat-enriched Lawvere theories, see Power's Enriched Lawvere Theories, Thm 3.4. There should be a map (sort of an inclusion, since we demand less structure for a symmetric monoidal category) from the Lawvere theory for symmetric monoidal categories to that for finite product categories and the forgetful functor should be precomposition with it. Now the left adjoint could be obtained by taking Cat-enriched left Kan extensions along this map.

One problem is that I only know that left Kan extensions of product preserving functors along product preserving functors are product preserving again, but I don't know the corresponding statement for copowers. This could either be true or for our special Lawvere theories it could be enough to ask for product preserving functors, then the above might have a chance to work.

Answer by Peter Arndt for Massey Products vs. $A_\infty$-Structures

2012-03-26T23:10:40Z

This is in Loday/Vallette's new book on operads, in particular sections 9.4.10 to 9.4.12.

Answer by Peter Arndt for quadratic forms over fields of characteristic 2

2012-03-21T00:07:55Z

This book by Manfred Knebusch starts with the limerick

$$\begin{array}{l}\text{A Mathematician Said Who}\cr\text{Can Quote Me a Theorem that’s True?}\cr\text{For the ones that I Know}\cr\text{Are Simply not So,}\cr\text{When the Characteristic is Two!}\end{array}$$

It gives a uniform treatment of quadratic forms in all characteristics including two.

Answer by Peter Arndt for Pointer to literature on double enrichment and functors among enriching categories?

2012-03-09T09:35:49Z

The thesis of Dominic Verity treats base change in a very abstract 2-categorical setting - looks like a lot of specialization is necessary until you arrive at the sort of situation sketched in the question.

Answer by Peter Arndt for Connections between various generalized algebraic geometries (Toen-Vaquié, Durov, Diers, Lurie)?

2012-02-29T00:33:46Z

First, here are some things about the four generalizations you mention:

Monoids don't fall into Diers' framework: By his Proposition 1.4.1 the terminal object in his framework is strict, i.e. any morphism $1 \to A$ is an isomorphism, which is definitely not the case for monoids. I also wouldn't expect Diers' examples to be instances of Toen/Vaquie's framework in general, Diers' example 1.3.16, the category of pairs (ring, module over it), might be a counterexample. I don't know about Durov's setting.

Durov's geometry is in no obvious way an instance of Toen/Vaquie's framework. If you want to force it into that framework, this might be an idea to go after: Monads are monoids in the monoidal category of endofunctors. Commutative monads, however, are not commutative monoids in that category; indeed it doesn't even make sense to say that since the category is not symmetric monoidal. So first you have to find a symmetric monoidal ambient category in which Durov's generalized rings live. Seeing monads as Lawvere algebraic theories or as (things presented by) sketches might do the job - a commutative theory is probably exactly a sketch with an isomorphism from its tensor square. Another idea could be to consider a category of monads where morphisms are natural monad transformations together with distributive laws...

Derived algebraic geometry on the other hand is an instance of the homotopical version of Toen/Vaquie's framework, also contained in that article - see also below.

Second let me point out that there are many more generalizations of algebraic geometry than those four:

° Rings with extra structure can count as generalization, if one can endow any usual ring with such an extra structure, e.g.

Not Borger's geometry with lambda-rings: Not any ring can be endowed with the trivial lambda-ring structure - see his comment
Berkovich's analytic geometry: Any ring can be endowed with the trivial metric

° One can replace rings by first order structures in several ways:

Several Russian authors do this in somewhat similar ways, a recent reference is this one by Daniyarova, Myasnikov, Remeslennikov which has many references to other work in this direction; see also this one by B. Plotkin.
First order structures can be described by sketches and there is an outline of algebraic geometry along this line in this text by R. Guitart.

° There are hyperrings (used for algebraic geometry by Connes/Consani) and fuzzy rings (by Walter Wenzel and Andreas Dress, e.g. this), which are certain second order structures.

° There are two generalizations of rings used by Shai Haran to compactify the integers, F-rings and the one given in his "Non-additive prolegomena".

° There is another generalization made by Shai Haran in his article on "dyslectic geometry" in which rings are endowed with gradings over general monoids (see here). Something quite similar seems to be going on in James Dolan's generalization of algebraic geometry (unpublished, but see here, there also was a series of videos of talks somewhere)

° derived algebraic geometry doesn't necessarily have to be based on simplicial rings; dg-algebras and $E_\infty$-ring spectra are equally important inputs, and there are many others, captured in

Toen-Vezzosi's HAG-contexts (these are homotopically additive)
Lurie's structured spaces from DAG 5 which capture about everything based on the idea of glueing together homotopical algebraic structure.

° note that Toen/Vaquie in their relative geometry do not stop at commutative monoids in some monoidal category but also give a homotopical version - this is something like a non-additive version of HAG-contexts and covers e.g. geometry over the "spectrum with one element" whose input are simplicial monoids.

° replacing rings by groupoid objects in rings together with an appropriate notion of equivalence gives you stack theory. Of course you can go on to higher stacks.

° of course there are the several approaches to non-commutative algebraic geometry - see Mahanta's survey for some of them, many related to the next point

° Chirvasitu/Johnson-Freyd's 2-schemes

° Takagi's generalized schemes

° Deitmar's congruence schemes

° Lorscheid's blue schemes

° I am sure I forgot several things...

Third, since I am at it, let me note that there are also generalizations of algebraic geometry which do not exactly build upon a generalized notion of ring, e.g.

° Hrushovski/Zilber's Zariski Geometries (see here) capturing the essential structure which is used in the applications of model theory to arithmetic geometry

° Rosenberg's noncommutative geometry. It doesn't have to be non-commutative, and also not additive, as Z. Skoda pointed out here

° in particular: schemes as dg-categories (Kontsevich, Rosenberg, Tabuada)

° Balmer's triangulated geometry

...

To summarize: Carefully mapping AG-land will keep you busy for a while. I have gathered quite some material for a rudimentary map (or maybe a low resolution satellite photo) accompanied by a few selected closer snapshots, but I won't start writing it before the second half of the year...

Answer by Peter Arndt for Higher categories in logic

2012-02-23T22:05:25Z

Another group of related applications are those in rewriting theory, concurrency theory, directed homotopy theory etc. For this check out the pages of Yves Lafont and his collaborators.

Answer by Peter Arndt for Is there a nice application of category theory to functional/complex/harmonic analysis?

2011-12-13T20:55:52Z

Here are four articles on category theory and analysis (see section IV of that collection). I liked this one: R. Boerger, Fubini's Theorem from a Categorical Viewpoint, manages to conclude Fubini's theorem from the fact that left adjoints preserve coproducts...

Answer by Peter Arndt for On Sketches and Institutions

2011-12-05T23:16:02Z

Well, as you said yourself the category of signatures is left unspecified in the general setup of institution theory - it just tells you what, given some notion of signature, can be said about the relation of syntax and semantics. It's quite amazing how many meaningful things can be said at such a level of abstraction, see Razvan Diaconescu's book "Institution Independent Model Theory".

Now, as you said yourself again, sketches are a specific type of signature, and they come with a specific type of semantics, so they form an instance of the general concept of institution. An outline of a proof of this is given in section 10.3 of Barr/Wells' book "Categories for Computing Science".

About your question why you find more references to institution theory I can only speculate. It probably is because you are looking through computer science literature and would be the the other way round if you were scanning math literature. The reason might be that mathematics often cares for qualitative statements while in computer science things have to be effectively spelled out: Given, for example some specification of a structure, a mathematician is able to say "Okay, those structures form an accessible category, so by Makkai/Paré they are the models of some sketch", while for a computer scientist the actual syntax of the specification counts, and this might not easily be translatable into sketch language - thus he resorts to institution theory which can immediately acommodate any sort of specification. But I'm just guessing...

Answer by Peter Arndt for Is there a mathematical axiomatization of time (other than, perhaps, entropy)?

2011-11-28T23:07:40Z

Again it's not only about time, but rather about space time, but quite different from the other mentioned approaches:

The Hungarian logicians Andreka, Madarasz, Nemeti, Toke and Sain have actually explored the possibilities of treating space-time and relativity in terms of model theory, so they really give axiomatizations in the technical sense and not just mathematical models. See the preprints on this page here, which include also a survey of previous axiomatizations of space-time.

Answer by Peter Arndt for What is the intuition of connections for cubical sets?

2011-11-27T20:59:06Z

A very concrete instance where you can see the meaning and usefulness of connections is this article by Brown and Mosa: They show that double categories (which do have an underlying (truncated) cubical set) with connections are the same as (globular) 2-categories.

The reason is that the connection allows to fold the four different edges of a 2-cell in the cubical double category structure into just two edges, leaving degenerate edges at the other sides, and this can as well be captured in the data of a globular 2-category where 2-cells have just one source and one target 1-cell - see the definition of he folding map right before Proposition 5.1 in the above article.

Answer by Peter Arndt for Do simplicial objects in a Topos form a model category?

2011-11-09T01:00:55Z

As has been established in the comments the answer for a general topos is no, while for a Grothendieck topos it is yes, by the work of Joyal.

The general question of when one can transfer a model structure on a category based on Sets to an arbitrary Grothendieck topos is beautifully adressed in Tibor Beke's articles on "Sheafifiable Homotopy Model Categories", available here. The examples include simplicial objects, cyclic objects and groupoid and category objects.

The proofs really use the assumption that you are in a Grothendieck topos (e.g. he chooses a site defining the given topos and uses the existence of morphisms to the topos of sets, plus the existence of the necessary colimits to interpret infinitary geometric logic, possibly even accessibility somewhere) and I don't think his arguments can be saved for more general toposes. Anyway the papers are worth a look and contain several interesting remarks about the role of accessibility in establishing model structures, giving a good perspective on your original question.

Answer by Peter Arndt for Learning Arakelov geometry

2011-10-19T00:26:42Z

I second the suggestion of the book "Lectures on Arakelov Geometry" by Soulé, Abramovich, Burnol and Kramer. There is this nice text by Demailly which motivates the treatment of intersection theory on the infinite fibers and probably suits you with your background. With this in mind the analytic part of the above book should be ok to read.

Answer by Peter Arndt for Do finite groups acting on a ball have a fixed point?

2011-10-18T23:37:40Z

The answer is "yes" (it has a fixed point) if the action is affine, i.e. if it satisfies for all $g \in G, x,y \in B^n$ and all $0 \leq t \leq 1$: $$g(tx+(1-t)y)=tgx+(1-t)gy$$.

In that case one can construct a fixed point by taking an $x \in B^n$ and averaging over its $G$-orbit: $$p:=\frac{1}{|G|}\Sigma_{g \in G}\ gx$$ By convexity of $B^n$ the point $p$ is again in $B^n$ and by the affineness of the action $p$ is indeed afixed point. Linear actions are of course affine, now with your piecewise linear action you have to see whether you can find an orbit which falls into a linear piece, for example.

The groups which allow the above kind of argument are called "amenable groups", as I just learned on monday...

Answer by Peter Arndt for Means of Promoting Mathematics in Young Countries!

2011-10-15T23:16:02Z

As it was said in an other answer it seems like a good idea to send students abroad to study at strong places and have them come back, or if they don't, keep contact, have them give courses in their home country etc.

A scientifically excellent place helping with precisely such a policy is the ICTP in Trieste, see the "Education and Training" and the "Research "sections of their website (they are called institute of theoretical physics, but they have a strong mathematics section). Concretely they provide training programmes for undergrads to prepare them for a PhD - see here - and they also offer short and longer term stays for working scientists from developing countries, as well as institution partnerships, see here.

Answer by Peter Arndt for Free, high quality mathematical writing online?

2011-10-05T18:15:18Z

The Comptes Rendus de l'Académie des Sciences are available online.

Semi-simplicial versus simplicial sets (and simplicial categories)

2011-09-10T12:58:45Z

Hi,

Let's denote by "semi-simplicial set" a simplicial set without degeneracies, i.e. a contravariant functor from the category $\Delta_{inj}$ of finite linearly ordered sets and order preserving injections to sets (this is also known as $\Delta$-set).

We have an inclusion functor $j: \Delta_{inj} \rightarrow \Delta$ giving us an adjunction between semi-simplicial sets and simplicial sets: The right adjoint is $j^*: Set^{\Delta^{op}} \rightarrow Set^{\Delta_{inj}^{op}}$ given by precomposition with $j$ (i.e. forgetting degeneracies), the left adjoint is $j_!:Set^{\Delta_{inj}^{op}} \rightarrow Set^{\Delta^{op}} $ given by left Kan extension (i.e. taking realization in simplicial sets). It is known from the article referred to in this MO-answer that boosting a semi-simplicial set up to a simplicial one and then forgetting degeneracies again results in an equivalent semi-simplicial set (in the sense that their realizations are equivalent).

I would like to know about the other direction: Does anyone know conditions on a simplicial set $K$ which ensure that the map $j_!j^*K \rightarrow K$ is a weak equivalence?

I am especially interested in the case where $K$ is the nerve of a category.

A related question is the following: Given a Reedy cofibrant semi-cosimplicial object in the category of simplicial categories, which is equivalent to the underlying semi-cosimplicial object of the usual cosimplicial object, one can by the usual yoga set up an adjunction between semi-simplicial sets and simplicial categories.

Starting with a simplicial set $K$ one can view it as a semi-simplicial set, then produce a simplicial category. I would like to know under which conditions on $K$ this simplicial category is equivalent to the simplicial category which one gets by applying the usual functor from simplicial sets to simplicial categories (the left adjoint to the coherent nerve) directly to $K$. Again the case of biggest interest for me is when $K$ is the nerve of a category.

Answer by Peter Arndt for What is the analog of a topos in quantum logic?

2011-09-07T20:12:32Z

While the other two answers referred to very interesting connections of topos theory and quantum physics I think the following is going more into the direction the OP was imagining: In non-commutative topology one considers quantales, which are, roughly, an axiomatization of what you get when you replace open sets with projection operators - in particular intersection of open sets becomes composition of operators and does no longer have to be commutative. Here a few approaches to this idea are listed. One short definition is: A quantale is a monoid object in the monoidal category of lattices.

One can define sheaves over quantales (e.g. as done by Miraglia and Solitro in this article, or by Mulvey and Nawaz in this nicely written paper) and the categories of such sheaves would be the analogon to the notion of Grothendieck topos (but: to my knowledge no Giraud type characterization of such categories is known, let alone an elementary one). One can interpret logic in such categories, as e.g. done by Coniglio here for the Miraglia/Solitro setting.

In some ways such categories of sheaves over quantales connect back to actual topos theory as for example seen in the last corollary of the above Mulvey/Nawaz paper and more impressively in this article by Pedro Resende.

Edit: After having looked into the article pointed out by Urs Schreiber I would like to add a comment on what the authors see as drawbacks of quantum logic. They write that, on the logical side, quantum logic is not distributive and thus "difficult to interpret as a logical structure", that no satisfactory implication operator has been found "so that there is no deductive system in quantum logic" and that no satisfactory first order quantum logic has been found.

I cannot judge what would be satisfactory, but Coniglio's interpretation linked to above has an implication operator and first order quantifiers. In any case, deduction systems can be built without implication operators being present inside the language, one just has to devise rules which say when one can infer a formula from a set of hypotheses (this is done in many nonclassical logics). The non-distributivity presents undeniable and annoying technical difficulties, the difficulty to interpret a non-distributive system "as a logical structure" on the other hand may be a matter of reading it in an appropriate way, see the next point.

The authors also see a physical drawback: They see the law of excluded middle $x \vee x^\bot = 1$, valid in quantum logic, as not reflecting the probabilistic spirit of quantum physics, because there it is not the case that either a proposition $x$ or its complement $x^\bot$ are true - they both may have intermediate "degrees of truth". I think an answer to this is that the interpretation of $\vee$ should not be that one of its arguments is true, but rather that its arguments together span the space of all possibilities. Non-distributivity then makes some sense also.

In a similar vein many things in intuitionistic logic make more sense if one interprets it as talking not about the truth of assertions but about whether they are known or provable (and an observer-centered perspective on quantum physics seems very appropriate). So the choice of an underlying logic is the choice of a point of view. Maybe one can see the two topos approaches and the quantale approach as talking about quantum physics from different perspectives - one could even consider creating a bigger formal language containing the connectives from both interpretations and allowing to relate the several points of view (giving a formal semantics for this language might be quite a challenge, though).

Disclaimers: 1. My background in the things discussed above lies in categorical and non-classical logic - my knowledge of quantum physics is quite superficial. 2. Although I jumped to the defense of quantales here, I really like both topos approaches...

Answer by Peter Arndt for What does the "category" of $(\infty,1)$ category look like.

2011-08-26T10:28:27Z

You can see the collection of $(\infty,1)$-categories as forming themselves an $(\infty,1)$-category, which is sufficient to see where weak associativity shows up: There are many models for the intuiti9ve concept of $(\infty,1)$-category, the simplest is that of a usual 1-category endowed with a class of weak equivalences (see Barwick/Kan's "Relative Categories: Another model for the homotopy theory of homotopy theories" here).

With the weak Kan complexes - together with the notion of equivalence between them - you happen to have found a strictly associative model for the $(\infty,1)$-category of $(\infty,1)$-categories. You can transform it into different other models, e.g. into simplicially enriched categories or quasicategories or Segal categories, as exposed e.g. in Bergner's "Three models for the homotopy theory of homotopy theories" (available here).

The Segal category and the quasicategory of $(\infty,1)$-cats do no longer have strict associativity and the fact that they are equivalent descriptions of the $(\infty,1)$-cat of $(\infty,1)$-cats reflects that the strict associativity in your model was an accident and not an essential feature...

Edit (in response to the comment) About the significance of having a strict model: Well, different models have different advantages. Your strict one is certainly good for computing compositions of functors between $(\infty,1)$-cats. The quasicategory of $(\infty,1)$-cats on the other hand is e.g. a better model to relate the $(\infty,1)$-cat of $(\infty,1)$-cats to other $(\infty,1)$-cats - examples are the relations between the quasicategories of (small) $(\infty,1)$-cats, presentable $(\infty,1)$-cats and stable $(\infty,1)$-cats given in Lurie's "Higher Topoi" and in DAG 1 (now "Higher Algebra"): There are $(\infty,1)$-adjunctions between these - e.g. between $(\infty,1)$-cats and stable $(\infty,1)$-cats given by taking spectra in an $(\infty,1)$-cat, forgetting the stability of a stable $(\infty,1)$-cat, respectively - and these facts would be hard to express using your model.

Answer by Peter Arndt for Discrete version of Nullstellensatz?

2011-08-25T08:55:14Z

Joyal and Tierney have done Algebraic Geometry with lattices (in topoi) somewhat along the lines you describe in their landmark monograph An extension of the Galois Theory of Grothendieck

What do higher Chow groups mean?

2009-11-10T02:09:06Z

Let $z^i(X, m)$ be the free abelian group generated by all codimension $i$ subvarieties on $X \times \Delta^m$ which intersect all faces $X \times \Delta^j$ properly for all j < m. Then, for each i, these groups assemble to give, with the restriction maps to these faces, a simplicial group whose homotopy groups are the higher Chow groups CH^i(X,m) (m=0 gives the classical ones).

Does anyone have an intuition to share about these higher Chow groups? What do they measure/mean? If I pass from the simplicial group to a chain complex, what does it mean to be in the kernel/image of the differential?

Could one say that the higher Chow groups keep track of in how many ways two cycles can be rationally equivalent (and which of these different ways are then equivalent etc.)?

Finally: I don't see any reason why the definition shouldn't make sense over the integers or worse base schemes. Is this true? Does it maybe still make sense but lose its intended meaning?

Answer by Peter Arndt for What are the different theories that the motivic fundamental group attempts to unify?

2011-07-29T19:29:45Z

A short answer: The different Weil cohomology theories do not only provide cohomology groups or rings but also come with extra structure; e.g. l-adic cohomology comes with an action from a Galois group and de Rham cohomology comes with a Hodge structure (and Hodge structures can also be expressed as being an action from some group). This extra structure varies with the cohomology theory.

The motivic fundamental group should unify these extra structures -- they all should be shadows of an action of the motivic fundamental group.

For a start see e.g. the "motivic Galois group" section on the wikipedia page here and these notes by Sujatha Ramdorai. The book by Yves Andre referenced there is maybe a good next step.

Answer by Peter Arndt for Extremely messy proofs

2011-03-16T16:46:08Z

The traditional way of proving Grothendieck duality is to first show it for proper maps and for open immersions, which already is quite a labour. Then one uses that any morphism of Noetherian schemes factors into such and pastes the partial results together. This requires an awful lot of non-trivial checking that certain diagrams are commutative. The extension of the result to Non-Noetherian schemes then requires yet more work.

In contrast, Neeman's proof of Grothendieck duality via Brown representability is slick, short (30 pages) and conceptual and a pure pleasure to read.

(but: the first approach gives you more insight into what the functors from Grothendieck duality actually do, so it is by no means worthless)

Comment by Peter Arndt

2013-05-30T17:42:50Z

Bemvindo ao MO!

Comment by Peter Arndt

2013-05-15T22:20:54Z

Thanks! If I ever come back to the project which drove me to this question, it will be nice to have this recorded.

Comment by Peter Arndt

2013-05-05T21:24:54Z

You might enjoy section 1.1 of Clark Barwick's Göttingen talks, which serves exactly the purpose of motivating homotopy theory for number theorists. There's not so much specifically on K(Z) though... dl.dropbox.com/u/1741495/papers/barwick.pdf

Comment by Peter Arndt

2013-05-03T23:30:16Z

(Click on my name to find my email address)

Comment by Peter Arndt

2013-05-03T23:29:54Z

The link doesn't work. Could you send those notes to me?

Comment by Peter Arndt

2013-01-21T10:25:57Z

Oh, of course, thanks!

Comment by Peter Arndt

2013-01-20T00:46:04Z

Ah, thanks! This might actually be enough to continue the proof. Do you have a reference for this fact?

Comment by Peter Arndt

2013-01-19T16:35:22Z

Sure, thanks. I was struggling with the Latex-displaying...

Comment by Peter Arndt

2012-08-11T21:55:47Z

@DavidWhite: 1. Yes, you need more than one object, G is just one generating object. 2. My answer was only about the first part, not about the Hopf algebra part. I don't think H-mod is locally presentable; it seems that you need more than just finite limits to describe that structure.

Comment by Peter Arndt

2012-05-09T17:51:10Z

No, I haven't thought about it for two years. Do you have an idea?

Comment by Peter Arndt

2012-05-03T21:42:36Z

I don't know the original proof, but I heard that the trick of Rabinovich provided a drastic improvement of the proof of Hilbert's Nullstellensatz.

Comment by Peter Arndt

2012-05-02T21:17:15Z

If your question is whether the above left Kan extension of a product preserving functor produces a product preserving functor again, the answer is yes: math.mq.edu.au/~street/MitchB.pdf

Comment by Peter Arndt

2012-05-02T09:51:08Z

Come on, last chance to grab that bounty :-)

Comment by Peter Arndt

2012-04-04T21:12:46Z

Apparently the original proof is in P. J. Freyd and G. M. Kelly, Categories of continuous functors. I, J. Pure Appl. Algebra 2 (1972), 169-191, but I have no access to it right now.

Comment by Peter Arndt

2012-04-03T15:29:17Z

Here we have the Adamek-Rosicky theorem and the small object argument united: math.yorku.ca/~tholen/ahrt2.ps