User ben sprott - MathOverflow

in what context is is favourable for a category to NOT be locally finitely presentable?

2012-11-16T23:40:17Z

Hey, Sometimes we DON'T want our category to be locally finitely presentable. For instance, maybe we want TOP. In general, why might we want a category to NOT be lically finitely presentable?

Familiar categories are compact in a dcpo of categories

2012-11-16T23:15:27Z

Hi, Suppose we define a dcpo of categories, where the ordering relation is a suitably restricted functor. The functor is restricted specificallly so that our collection and ordering reltion form a dcpo. In this dcpo of categories, the familiar categories, those we find in most any textbook, are the compact elements of this dcpo? I feel I also need to add that the dcpo we want is the largest dcpo of categories so defined. I've heard that terminology before.

Where does Chu fit in with "Nice Categories of Spaces"?

2012-11-03T15:06:33Z

Hi,

Here is a page about categories of spaces with "Nice" properties. When we consider Chu spaces, I believe it was chosen to have certain properties that were "nice" at least to Pratt. How does Chu fit in with the notion of "nice" that is outlined in the NLab site above?

Monoidal category from a monad

2012-08-15T14:42:33Z

Hi,

I realize this question is not research level, but I think this might be just above the abilities of the folks at Mathematics stack. I want a general prescription, or as many as possible, for constructing a monoidal category like FD-Hilb or Hilb given only a monad. I was thinking of using the endo-functor composition as the monoidal product but didn't really get anywhere. We know that the monad axioms are much like the monoidal axioms, though a category like FD-Hilb has many more axioms than can be identified one to one with the monad axioms. Has anyone seen attempts at constructing a basic symmetric monoidal categories from some monad? In particular, I want the monad to be an adjunction of forgetful and free functors, I don't care what the base categories are. Moreover, I want a general prescription: given any free-forgetful adjoint here's how to construct a monoidal category. Help will be most appreciated.

Monads and Comonads that interact

2012-05-20T18:15:54Z

Hi,

I am reading Awodey's text "Category Theory". In section 10.4 he talks about comonads and monads that occur together and interact and he says that "possibility" and "necessity" in propositional modal logic is an example. Can anyone give a few more interesting examples of interacting monads and comonads?

In particular, I am interested to know if interacting (co)monads form a frobenius algebra. Can ee define a frobenius algebra with a suitable monad comonad pair? Also, if anyone has seen a cstar algebra presented as a monad comonad pair, please let me know.

Solving for an operator by minimization

2012-05-04T16:55:13Z

Please note that I am looking for numerical algorithms that will tell me what the operator is that minimizes a problem.

I have a 2x2 complex hermitian operator that is a function of two variables, so $\tilde{O} = \tilde{O}(x,y)$. I will not have this operator in a closed form, but rather as the output of a numberical simulation who's inputs are just $x$ and $y$. I want to find out which values of $x$ and $y$ will allow me to map a given vector $\psi_{in}$ to another given vector $\phi_{1}$. I feel that some kind of gradient descent algorithm is in order. For instance, I can define

$d = | \tilde{O}(x,y)\psi_{in} - \phi_{1} |$

Now I can minimize this value over the two input values $x$ and $y$. So I would have an equation like this:

$ \frac{\partial^2 d}{\partial_x \partial_y} = \frac{\partial^2 | \tilde{O}(x,y)\psi_{in} - \phi_{1} |}{\partial_x \partial_y} $

We might do some minimization from here but I am not sure how to procede and I feel there is a simpler numerical recipe that someone might know of.

Any help will be most appreciated.

Coalgebra for a comonad for a really tiny category

2012-05-04T20:13:31Z

I have a category with only two objects. These objects are just sets, each set has two elements in it. I take the morphisms as all endo-maps of the sets. There are no morphisms between the sets. I take an endo-functor,$F$ that maps one set to another and it is such that $FF = I$ where $I$ is the identity functor. I also map every function on set $S_1$ to its cousin on $S_2$. By that I mean, constant maps go to constant maps and the map $f$ such that $ff=I$ goes to his same kind of function on the other set. I hope this is not too hard to picture. Can anyone say if it will be easy to form a monad/comanad ie I will surely find the right natural transformations? Does anyone want to hazard a guess about the algebra/coalgebra for this monad?

Structural Induction over categories....Just in time for Christmas!

2011-12-25T02:10:46Z

Hi,

I am looking for any kind of paper that uses structural induction where the ordered set is actually a collection of categories and the ordering relation is any kind of functor. I am thinking of the category with only one morphism as a base case. In particular, I am interested in the collection of internal categories in a monoidal category, but any usage of this method (over cats) would be of interest. In the case of internal cats, given any structural inductive proof in the collection of internal categories, what does the proof say about the underlying monoidal category. Can we present an axiom of the underlying monoidal category M as an inductive proof. That is, proven inductively over the collection of internal categories of M, any proven axiom also is an axiom of the underlying category?

a dcpo of categories has limits and compact categories

2011-09-22T22:31:33Z

Hi,

Suppose you are given a dcpo of categories, where the functor is taken as the ordering relation. This collection is a category too and it has limits, or perhaps cartesian products. This comes from the shape of what the supremum looks like. This dcpo might also be a topos. Does anyone have any thoughts on this? Also, what are the compact elements? That is to say, what universal properties do the compact elements have?

Categories presented with Arrows only, no objects: partial monoids

2011-09-21T21:15:34Z

Hi,

I received an answer to a question a while back. The question was about how we can present a category as a collection of arrows and a large list of algebraic relations between them. One of the answers I got was about Freyd's "Categories, Alegories", and here it is:

http://mathoverflow.net/questions/68775/products-in-a-category-without-reference-to-objects-or-sources-and-targets/68847#68847

Can anyone (Wouter maybe?), give much more detail about this presentation. Can anyone give the precise definition of these "kinds" of partial monoids a la Freyd?

As a side note, could someone suggest a good way to define a dcpo of such partial monoids?

functors of string diagrams in a monoidal category

2011-07-26T03:04:12Z

Hi,

If I have a string diagram, can I take its functor easily by drawing a new string diagram and just say the wires go to wires and boxes go to boxes?

products in a category without reference to objects or sources and targets

2011-06-25T02:58:07Z

Hi,

I was thinking about presenting categories with nothing but equations over morphisms. I wondered about products. The definition of a product has its genesis in the following diagram shape

A->B

A->C

You would say that whenever you have this shape, then for any D such that

D->A->B

and

D->A->C

blah blah...the axioms for the product. I thought about how to do this with just words in the arrows without reference to either source and target or objects. You can start with words like

$dab$ and $dac$

where a:D->A and and b: A->B and c:A->C and d:E->D.

You might say: if, the existence of the equations

$dab=e$ and

$dac=f$

implies that $a$ is unique in that if there exists words

$dxb$ $dyc$

then $x=y=a$

then this constitutes a product.

Could something like this work? I mean, can you present a category as just words over morphisms along with equations and a bit of language with quantifiers? Second, could this kind of definition work for products?

Scott topology, but for graphs

2011-06-24T19:09:38Z

Hi,

Would it be possible to define an order theoretic topology on graphs? I am thinking about the scott topology. There would be an associated continuous map on graphs.

linear logic, diagrammatic calculus and foundations

2011-06-23T17:20:53Z

Hi,

I have been interested in foundations for a while, especially categories as foundations. I am of the opinion that, as long as we present the theory of categories in SET, we will not be able to give a reasonable justification for categories as a foundation. (that could be a question: does the persistent presentation of the theory of categories in SET preclude it from taking its place as a foundation?)

How else can we present the theory of categories?

What, in your mind, is a foundation anyway? To me, it is a way to express ideas (language with syntax or diagrammatic calculus or other mysterious thing) and have reasoning that we all agree on because it has precise rules.

My next question is about how to present the theory of categories in a linear logic. I am working on a few intuitions about how to have such a thing. First is the intimate link between linear logic, and the geometry of tensor calculus. Second, we normally think of the geometry of tensor calculus as being only about one particular kind of category. However, the basic reasoning in GTC is about planar graphs. Categories are exactly planar graphs with extra data over the edges. Thus, GTC is also a place to reason about categories in general, not just a particular brand of category.

Is it possible to present the theory of categories in a linear logic and would this presentation not be more of a diagrammatic presentation? GTC comes with a calculational tool. It is a diagrammatic calculus.

Swapping morphisms for arrows in categorical diagrams was a huge turning point for physics: See Markopolou and also see Panangaden, Blut, Ivanov and see Coecke,Abramsky. This swap opened up the diagrammatic calculus. Maybe we should feed this simple change back to category theory as a means of making it more of a foundation.

continuous maps between categories that are not functors

2011-06-10T06:26:56Z

Hey,

Is it possible to define a map between two categories which preserves all products and binary equalizers and yet is not a functor, ie it does not satisfy one or more axioms of a functor? Further, if you replace the functor with maps between categories that preserve all products and binary equalizers, what do the axioms of an adjoint become? That is, can we have continuous maps between categories and adjoints but no functors? I guess I am asking this: if we consider "a category theory" where the notion(or the job) of the functor is replaced by "maps which preserve products and binary equalizers", how much category theory do we loose? Do we simply loose stuff that was just excess baggage that came from the fact that we are presenting the theory of categories in SET?

A knot theory form of the carpenter's ruler question

2011-04-26T00:05:46Z

Hey,

The carpenter's ruler problem is about polygons you can make with planar joints. You can formulate a similar question for knots when you introduce ball joints:

Given m ball joints connected by rods of arbitrary length, which knots can you make?

I found out that you can make the first non trivial knot with your fingers and thumbs and your thumbs are ball joints. This is probably the same question as for stick knots and since the stick number is not known for all stick knots, the answer to the question I must say is, no, the question is an open question.

using some, given category D to talk about all other categories (Topoi?)

2011-04-12T19:29:45Z

Hey,

Suppose I want to establish a theory of the category $C$ (vector spaces or whatever), but what I really have is $D$, some precisely known category. This is to say, I know all the axioms of $D$, but I only have an intuition for $C$ and I want to develop a theory of $C$. I would normally have diagrams in $C$ by mapping cpos or Domains into $C$. But instead, I want to do it with $D$. What I do is define the largest category, $J$, of Domains in $D$. I do this by defining a dcpo with objects as elements in $D$ and relations as arrows in $D$. Then any functor will map the domains in $J$ to diagrams in $C$.

It seems like I am just inserting a category $D$ in the normal diagram functor $J \rightarrow C$ resulting in $J \rightarrow D \rightarrow C$ which seems to miss the point of the exercise. The point of the exercise, I think, is to try to do a lot of category theory when you have to live in some category $D$.

We start by saying that we "have access to" all diagrams in $D$. Further, we say that we have access to none of the morphisms in any other category. So if we want to talk about a category $C$, it will have to be in terms of diagrams in $D$. Next, we intuit the existence of a category $C$ (I am using this restricted language to reflect the notion that we do don't have access to $C$). Next, we consider endofuntors of $D$, but we really see them as diagrams in $D$ indexed by the domains we constructed in $D$ by $J$. These endofunctors are meant to mimic functors from $D$ to $C$. We are pretending to have access to $C$, by attempting a construction of $C$ in $D$.

Sorry that this is so unclear, especially the idea of having an "intuition of C" and "attempting a construction of". I think that this is an expression of a Topos, and so I have some questions. Firstly, what kind of minimum structure do we need in $D$ to really start doing some work? Second, if we really want to say that we only have access to $D$, then we cannot present $D$ as a set of morphisms and a set of objects because that would imply we are actually in SET, not $D$. Is there any way to start working only in $D$? This goes back to the first question (although thinking about this too much is a bit of a morass).

compact elements and continuous functors

2011-04-08T00:41:33Z

Hi,

I am interested in abstracting the Scott topology from Domains to Categories. One can find a definition of a continuous functor which is just such an abstraction:

A functor F:C→D is continuous if it preserves all small (weighted) limits that exist in C, i.e. if for every small category diagram A:E→C in C there is an isomorphism F(limA)≃lim(F∘A).F(\lim A) \simeq \lim (F\circ A).

I found this on the n-Lab. What I am interested in is a notion of compact element which is defined for Domains as this:

if k is less than the sup of any directed subset D, then there is an element x in D such that k is less than x.

Any reference would be good. My intuitions are telling me that if a category is compact, in this sense, it should have a kind of finite presentation. This would be an abstraction from a dcpo of groups where the compact elements are finitely generated. I realize this might not make sense so any help would be great.

what is bottom in a dcpo of groups

2011-02-24T23:27:19Z

Hi,

Given a dcpo of finitely presented groups, I wanted to say that the free group is bottom, but I don't think that is right. Can anyone say what is a typical bottom in a dcpo of groups? Furthermore, are the finitely presentable groups the finite or compact elements?

Best,

Ben

transformation properties of divergence (of a vector field)

2010-12-07T23:18:40Z

Hi,

If I have a divergence free vector field defined on a smooth manifold, and I apply some diffeomorphism, what can I say about what happens to the vector field? The example I am using is of an open or closed (pick one) disk embedded in 3 dimensions and we contract the disk to a line. Perhaps the question could be answered by giving first only the most general transformation that could be applied to the vector field (probably general linear transformation) followed by others that may depend increasingly on the metric (which I may want to ignore since I am more interested in diffeomorphisms rather than smooth maps) and yet further depending on the properties of the vector field.

thanks

Answer by Ben Sprott for Why do categorical foundationalists want to escape set theory?

2010-11-16T23:00:11Z

My opinion may not be of interest to readers here since I am a physicist by training and as such am most interested in how mathematics and reality come together. That said, it is possible that my journey through foundations of mathematics, entwined with an interest in quantum gravity and especially background independence, perhaps sheds light in both directions. After all, physics has been known to introduce interesting ideas to mathematics.

I think the short answer to the question of categories, sets and foundations comes from, as Todd pointed out, intrinsic meaning versus relational meaning. For instance, I understand a functor from A to B to be the giving of meaning to some collection of objects of type A in terms of objects and morphisms of type B. Contrast this with some of the axioms of set theory:

Two sets are equal (are the same set) if they have the same elements

We rely on some basic "knowledge" of the meanings of terms. This meaning is provided by the reader. Thus, we agree on the meaning of terms.

This relates to physics in the following way. If I were God, I could see the universe from the outside. Just as if I were reading that axiom 1, I would give meaning to all aspects of the universe. Contrast this to the far more realistic vantage point of human beings who are decidedly embedded within the universe. In this case, meaning comes not from an external viewer (the God looking at the set of all sets), but rather in how a change in one system induces a change in another system. Think, for example, of a distant star, collapsing millions of years ago and turning slightly reddish. This event is interpreted on earth on two photographic plates, or by a bit string being flipped from "white" to "red".

Rather than using Sets, which come with a sort of pre-arranged meaning, physics seeks to use a causal relation. Further, this causal relation is seen as morphism in a category. Up we go in terms of structure from nothing but "morphism structure" to Category-A structure. The "meaning" is provided at each step by how the previous phase is interpreted in the latest phase.

More pointedly, in physics we cannot afford to build into the foundations of physics the ontology of set theory simply because it seems like a foundation for doing the math we love.

Answer by Ben Sprott for How and how much do the notations and diagrams influence our understanding of mathematical concepts?

2010-11-12T19:23:57Z

I think a good answer to this comes from category theory, linear logic, diagrams and the geometry of tensor calculus (Joyal-street). We often talk about category theory through the use of diagrams which are planar graphs (objects as nodes morphisms as arrows). Written down, these diagrams can be seen as fishing nets, kind of embedded in a plane, so we don't care about how one line crosses over another. These diagrams can be rewritten in different ways that respect the topology of the network. These deformations are exactly what Joyal and Street were talking about in the geometry of tensor calculus. We know from further work that, (and excuse my poor explanation ) the geometry of tensor calculus is a model of linear logic. This would mean that the axioms of a symmetric monoidal category can support the axioms of linear logic...(please excuse the poor understanding, I have come to these thoughts with little help). The long and short is that, if we talk about category theory in terms of diagrams, then we are most likely thinking in terms of linear logic. This would be in contrast to a model of the theory of categories in Set. In that case, we have a set of objects and set of morphisms, the axioms of (some kind of ) Set theory and further the axioms of the category we want to talk about. This would be thinking in a different kind of logic.

approximating categories with continuous functors

2010-10-12T21:59:20Z

I am wondering about approximation and idealization. Specifically, I am wondering if anyone has seen some work on the following. In the semantics of programming languages we find Domains as a place to talk about iteration and approximation. We can define a Scott Topology on the Domain and now our Domain-maps are continuous maps. Next,we can see our Domains as categories and turn the continuous Domain-maps into continuous functors. If we push the idea further, we have continuous functors and a notion of approximation which is now over categories. Lambek ponders the existence of the category of Sets. What about approximations to the category of sets. For that matter, what might it look like to approximate any well-known category like that of manifolds or FDVec? I realize the question is vague. I wish I had a more concrete question, but my grasp of the material is too weak. Naturally, any thoughts on this would be most appreciated.

Comment by Ben Sprott

2012-11-14T02:32:50Z

My deepest thanks for the responses here.

Comment by Ben Sprott

2012-08-15T16:25:31Z

Thanks for the suggestions :)

Comment by Ben Sprott

2012-07-26T15:29:59Z

ok. I think I want to change CAT. I might want to change it into a kind of background for a kind of "structural induction for categories" that I have been thinking of. For example, consider CAT as all categories but with just one arrow between any two cats, so a partial order: a dcpo of categories where a notion of "compact category" is taken straight out of domain theory. Just imagine presenting every true statement about "categories" via a (structural) inductive proof over some "suitable collection of categories" (think CAT) but with good properties in the relations between categories.

Comment by Ben Sprott

2012-07-26T15:23:42Z

Thanks for the comment Toby. Maybe I want to think more about atoms than about initial objects or least objects. If I chose one atom, then everything that it points to obeys that axiom or has that structure. If I choose three atoms, likewise, everything they point to has that structure or those axioms hold. Are these atoms diagram shapes? Small diagram shapes? Can we even abstract away from our reliance on the notion of the "small axiomatic diagram shape" like 0,1,2?

Comment by Ben Sprott

2012-07-26T13:39:01Z

Hi, I think I might be under the impression that an initial object in CAT is an artifact of the presentation of the theory of Categories in SET. If we don't consider a category as consisting of a set of objects and a set of morphisms, perhaps CAT has no initial object. So the question is, is the existence of a initial object in CAT dependant on the presentation of the theory of categories?

Comment by Ben Sprott

2012-06-03T14:56:13Z

Adamek talks about the initial algebra for the identity functor so perhaps there is more to say about it?

Comment by Ben Sprott

2012-05-29T18:06:18Z

Hi Michal, Could you also see how to create a Cstar algebra from the frobenius algebra?

Comment by Ben Sprott

2012-05-27T17:39:40Z

Thanks Michal! The set theory example sounds very interesting. Perhaps you are right about the necessity for a single carrier (which I assume is the endofunctor). Are you saying that one could not ever have a comanad and monad from the same endofunctor?

Comment by Ben Sprott

2012-05-17T14:54:37Z

What happens when I am not evaluating a "function at a point" but an operator at a point and in particular, an operator that can be written as a noncommuting product of operators each defined on different variables or components of the "point". The "point" is (x,y) and the operator is $\tilde{O_1}(x) \tilde{O_1}(y)$. If you have any thoughts, they would be helpful.

Comment by Ben Sprott

2012-05-04T20:05:42Z

Will gradient descent be fine in the case I have mentioned?

Comment by Ben Sprott

2012-05-04T20:04:32Z

I should mention that I do not need to minimize for more than one pair at a time. I will want to take a single pair and find x,y that form a solution, and then a new pair will come along and new x,y values will do. Also, $\psi$ is fixed.

Comment by Ben Sprott

2012-05-04T19:10:24Z

Thanks very much for the suggestion!

Comment by Ben Sprott

2012-05-04T19:08:34Z

Hi, Yes $\phi$ and $\psi$ are 2 element complex vectors. Elements of a 2-d Hilbert space.

Comment by Ben Sprott

2011-09-23T22:25:27Z

Here is a more pointed form of this question. A dcpo can be seen as a category where there is only one map between any two objects. Does this category have limits?

Comment by Ben Sprott

2011-09-23T22:18:20Z

I am sorry for being so brief in the question. This is a serious post. I am not sure what the confusion is. If I have an ordered structure constisting of a set of categories with an ordering relation then I have to describe that ordering relation. So here is the ordering reltion: a functor. How this has caused any confusion, I a not sure. If this is still unclear then let me describe it differently. Suppose I have a set of categories and also some set of funtors between the categries. I may have ten categories and only one functor, say between cats A and B. Ok. So now suppose that th