Re: 'Directed Algebraic Topology'

Re: 'Directed Algebraic Topology'

[Urs Schreiber]

Marco Grandis wrote:

> My book
>
>    'Directed Algebraic Topology'
>    Models of non-reversible worlds
>
> has appeared, at Cambridge University Press.

In that context I am wondering about the following:

it would be nice to have a notion of directed topological space that
would extend the relation between (nice) topological spaces and
oo-groupoids to one between (nice) directed topological spaces and
(oo,1)-categories.

More generally, it would be nice to have a notion of "r-directed
topological space" for r in N that would extend the relation between
(nice) topological spaces and oo-groupoids to one of (nice)
"r-directed spaces" and (oo,r)-cateories.

(Probably such a notion of directed spaces can't be supporrted by
plain topological spaces with direction information, but requires
filtered directed spaces or the like. )

Has anything like this been considered?


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Re: 'Directed Algebraic Topology'

[Marco Grandis]

Dear George,

There is indeed such a connection between asymmetric distances
and directed algebraic topology, but is not entirely trivial. The
obvious solution,
by 'left' and 'right topologies' would not work well. The good
solution, in my
opinion, is constructing a 'symmetric topology' and adding
distinguished paths.

All this is in my book, and also in a paper:
M. Grandis,
The fundamental weighted category of a weighted space: From directed
to weighted algebraic topology,
Homology Homotopy Appl. 9 (2007), 221-256.

I am presently working with a colleague. Later, I will be able to
comment more precisely on
these points.

All the best

Marco


On 28 Sep 2009, at 20:43, George Janelidze wrote:

> Dear Colleagues,
>
> In addition to my message of September 22 addressed to Michael Barr
> and all
> of you (concerning 'Directed Algebraic Topology'/quasi-uniform
> spaces) I am
> forwarding a message from Guillaume Brummer:
>
> "...Thank you for copying this interesting material to me, and for
> mentioning quasi-uniform spaces to Michael Barr. Serious early work
> in this
> field was by Leopoldo Nachbin of Rio de Janeiro, published in CRASP
> 226
> (1948) 774-775, -- just 11 years after Andre' Weil's monograph on
> uniform
> spaces. Then came Nachbin's monograph Topologia e ordem (Chicago
> 1950), of
> which an English translation Topology and order was published by Van
> Nostrand (1964). Meantime the book by A'. Csa'sza'r, Fondements de la
> topologie ge'ne'rale, had appeared in Paris (1960)..."
>
> Guillaume Brummer also says:
>
> "...Hans-Peter K"unzi has published several surveys of this field
> since
> 1993, with lots of bibliography..."
>
> However, I still see no connection (which does not mean that it
> will never
> occur!) between "directed/asymmetric algebraic topology" and
> "asymmetric
> general topology". Surely Marco Grandis is the right person to ask
> about
> this. Well, since Marco mentioned preordered topological spaces,
> one could
> think of bitopological spaces and then quasi-uniform spaces might
> occur
> naturally...
>
> George Janelidze
>
>



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Re: 'Directed Algebraic Topology'

[George Janelidze]

Dear Colleagues,

In addition to my message of September 22 addressed to Michael Barr and all
of you (concerning 'Directed Algebraic Topology'/quasi-uniform spaces) I am
forwarding a message from Guillaume Brummer:

"...Thank you for copying this interesting material to me, and for
mentioning quasi-uniform spaces to Michael Barr. Serious early work in this
field was by Leopoldo Nachbin of Rio de Janeiro, published in CRASP 226
(1948) 774-775, -- just 11 years after Andre' Weil's monograph on uniform
spaces. Then came Nachbin's monograph Topologia e ordem (Chicago 1950), of
which an English translation Topology and order was published by Van
Nostrand (1964). Meantime the book by A'. Csa'sza'r, Fondements de la
topologie ge'ne'rale, had appeared in Paris (1960)..."

Guillaume Brummer also says:

"...Hans-Peter K"unzi has published several surveys of this field since
1993, with lots of bibliography..."

However, I still see no connection (which does not mean that it will never
occur!) between "directed/asymmetric algebraic topology" and "asymmetric
general topology". Surely Marco Grandis is the right person to ask about
this. Well, since Marco mentioned preordered topological spaces, one could
think of bitopological spaces and then quasi-uniform spaces might occur
naturally...

George Janelidze



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Re: 'Directed Algebraic Topology'

[Martin Escardo]

"Asymmetric spaces" doesn't give relevant hits in google, but
"asymmetric topology does". People have considered "quasi-uniform
spaces" (losing the symmetry in the same way as metric spaces with
Lavwere's work), and much more. An interesting example, which has shown
up in topos theory via the work of Johnstone, are stably locally compact
spaces/locales, where the patch modification makes (coreflectively) such
spaces symmetric. Some people use two topologies on the same set, or two
subframes of a frame that generate, or variations of this idea.
Symmetric spaces give "positive and negative information", and
asymmetric ones give one of the two only. Some people argue that it is
good to keep the topologies of positive and of negative information
separate. Anyway, I just wanted to say that there is a large body of
work on this. Whether it is relevant for the original question I don't
know, but it is relevant for Barr's subquestion. MHE.

Michael Barr wrote:
> One obvious thing that comes to mind are asymmetric spaces--a metric
> without the symmetry axiom.  This can obviously be extended to uniform
> spaces, although I am not aware anyone has.  As for topological spaces,
> I know of nothing there.
>
> Michael
>
> On Mon, 21 Sep 2009, Urs Schreiber wrote:
>
>> Marco Grandis wrote:
>>
>>> My book
>>>
>>>    'Directed Algebraic Topology'
>>>    Models of non-reversible worlds
>>>
>>> has appeared, at Cambridge University Press.
>>
>> In that context I am wondering about the following:
>>
>> it would be nice to have a notion of directed topological space that
>> would extend the relation between (nice) topological spaces and
>> oo-groupoids to one between (nice) directed topological spaces and
>> (oo,1)-categories.
>>
>> More generally, it would be nice to have a notion of "r-directed
>> topological space" for r in N that would extend the relation between
>> (nice) topological spaces and oo-groupoids to one of (nice)
>> "r-directed spaces" and (oo,r)-cateories.
>>
>> (Probably such a notion of directed spaces can't be supporrted by
>> plain topological spaces with direction information, but requires
>> filtered directed spaces or the like. )
>>
>> Has anything like this been considered?
>
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
>


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Re: 'Directed Algebraic Topology'

[Gaucher Philippe]

Le mardi 22 septembre 2009 10:37:13, Marco Grandis a écrit :
> Dear Urs,
>
> There are various directed topological structures, which have
> directed homotopies
> and fundamental (higher) categories, like:
>
> - preordered topological spaces (simple but poor);
> - locally preordered topological spaces (in a suitable sense);
> - d-spaces = topological spaces equipped with distinguished paths;
> - spaces equipped with distinguished cubes;
> - cubical sets (in the combinatorial world);
> - generalised metric spaces (in the sense of Lawvere);
> - 'inequilogical spaces';
> - etc.
>
> I prefer d-spaces, which have also been studied by other authors.

Dear categorists,

Some remarks about topological and categorical models of directed algebraic 
topology.

Indeed, d-spaces have a lot of interesting features as topological model since 
i also work in a recent preprint with multipointed d-spaces (1) and i prove 
that from a homotopical point of view, they are the topological version (in 
the sense of concrete topological functors) of the category of flows (Or on the 
contrary, the flows are a categorical version of multipointed d-spaces). I had 
introduced them for studying branching and merging homology theories, which 
require a specific feature of the topological model that d-spaces do not have. 
This study can also be done with multipointed d-spaces in theory, but the 
theory remains to be written (I am working on that...). I also have a model 
structure on multipointed d-spaces preserving the homotopy type of path 
spaces. I'd like to mention in this mail that I do not know how to prove that 
it is left-proper and any idea would be really welcome.

Concerning cubical sets now. The usual degeneracy maps have no interest in 
computer science. But they do not "disturb". On the contrary, there is a new 
kind of degeneracy map that I call transverse degeneracy (2) which are of 
interest in computer science. The symmetric transverse precubical sets  are 
the only kind of precubical set such that the 1-dimensional coskeleton functor 
is well-behaved from a computer science point of view (see (2)). The base 
category \widehat{\square} appears also in the study of topological models of 
concurrency. Roughly speaking, the space of morphisms from a topological m-
cube to a topological n-cube preserving the labelling will be always homotopy 
equivalent to \widehat{\square}([m],[n]), if the m+n labels are the same (if 
they are not the same, it will be homotopy equivalent to the subset 
corresponding to maps preserving the labelling of course).

Precubical sets are enough to model all process algebras (3) but they are too 
poor for a mathematical treatment, even in the combinatorial world. For 
example, in the category of precubical sets, the labelled square corresponding 
to the concurrent execution of a and b is not isomorphic to the labelled 
square corresponding to the concurrent execution of b and a if a<>b ! 
Symmetric precubical sets are better (or less bad) because this drawback 
disappears. Transverse symmetric precubical sets are even better but they are 
more complicated to understand. I have an explicit combinatorial description 
of the symmetric precubical set of labels. I do not have such a description 
for the transverse symmetric precubical set of labels for example.
Symmetric precubical sets are also related to higher dimensional transition 
systems in a non-trivial way: the latter can be identified to a full reflective 
subcategory of the (labelled) former (4).

pg. http://www.pps.jussieu.fr/~gaucher/

(1) Homotopical interpretation of globular complex by multipointed d-space 
(preprint)
(2) Combinatorics of labelling in higher dimensional automata (preprint)
(3) Towards a homotopy theory of process algebra (HHA)
(4) Directed algebraic topology and higher dimensional transition system 
(preprint)











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Re: 'Directed Algebraic Topology'

[Peter Bubenik]

On Tue, Sep 22, 2009 at 5:00 AM, Urs Schreiber
<urs.schreiber@googlemail.com> wrote:

> Meanwhile probably Peter Bubenik's message to the mailing list will
> have appeared, where he says that with David Spivak he is in the
> process of investigating the connection between directed topological
> spaces and (oo,1)-categories. I am wondering what model of directed
> spaces they are using and to which extent they find an equivalence.

We use a variant of Marco Grandis' d-spaces (topological spaces with a
distinguished set of paths).

Peter


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Re: 'Directed Algebraic Topology'

[Urs Schreiber]

On Tue, Sep 22, 2009 at 10:37 AM, Marco Grandis <grandis@dima.unige.it> wrote:

> In my web page you can find references to many papers of mine on this
> domain, and such
> papers have many references to other authors.

[...]

> The latter does not cover higher fundamental categories, which - in
> dimension 2 - can be found
> in:
>
> -, Modelling fundamental 2-categories for directed homotopy, Homology
> Homotopy Appl. 8 (2006), 31-70.
>
>
> -, Lax 2-categories and directed homotopy, Cah. Topol. Géom. Différ. Catég.
> 47 (2006), 107-128.
>
> -, Absolute lax 2-categories, Appl. Categ. Struct. 14 (2006), 191-214.


Thanks for these references. While I haven't read all of them in
detail, I am aware of many of them, I think. In fact, the question I
asked arose in discussion of nLab entries on directed space

 http://ncatlab.org/nlab/show/directed+space

and directed homotopy theory

 http://ncatlab.org/nlab/show/directed+homotopy+theory

(which still are greatly in need of improvement)

that list some of these.

My question revolves around the issue whether and to which degree
forming the fundamental category or 2-category or ... or
(oo,n)-catgory of a directed space -- for instance a d-space --
establishes an equivalence, in a suitable sense, between directed
spaces and these categorical structures that is analogous to the
(Quillen) equivalence between (nice) topological spaces and
oo-groupoids (modeled as Kan complexes) that is given by forming the
fundamental oo-groupoid Pi(X) = S(X) given by the singular simplicial
complex.

It would seem that in order to have the formation of the "fundamental
(oo,1)-category" (if any) of a directed space be a suitable
equivalence of sorts, one would need something like filtered or
stratified directed spaces.

Do you know if this has been considered?

Meanwhile probably Peter Bubenik's message to the mailing list will
have appeared, where he says that with David Spivak he is in the
process of investigating the connection between directed topological
spaces and (oo,1)-categories. I am wondering what model of directed
spaces they are using and to which extent they find an equivalence.


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Re: 'Directed Algebraic Topology'

[Marco Grandis]

Dear Urs,

There are various directed topological structures, which have  
directed homotopies
and fundamental (higher) categories, like:

- preordered topological spaces (simple but poor);
- locally preordered topological spaces (in a suitable sense);
- d-spaces = topological spaces equipped with distinguished paths;
- spaces equipped with distinguished cubes;
- cubical sets (in the combinatorial world);
- generalised metric spaces (in the sense of Lawvere);
- 'inequilogical spaces';
- etc.

I prefer d-spaces, which have also been studied by other authors.  
(Notice that the one-dimensional
information which is added to a topological space has effects in all  
dimension.) However,
directed homology works much better for cubical sets, or spaces with  
distinguished cubes.

In my web page you can find references to many papers of mine on this  
domain, and such
papers have many references to other authors. You could begin by:

- M. Grandis, Directed homotopy theory, I. The fundamental category,  
Cah. Topol. Géom. Différ. Catég. 44 (2003), 281-316.

-, The shape of a category up to directed homotopy, Theory Appl.  
Categ. 15 (2005/06), No. 4, 95-146.


A more complete study can be found in my book.
The latter does not cover higher fundamental categories, which - in  
dimension 2 - can be found
in:

-, Modelling fundamental 2-categories for directed homotopy, Homology  
Homotopy Appl. 8 (2006), 31-70.


-, Lax 2-categories and directed homotopy, Cah. Topol. Géom. Différ.  
Catég. 47 (2006), 107-128.

-, Absolute lax 2-categories, Appl. Categ. Struct. 14 (2006), 191-214.



Marco Grandis

http://www.dima.unige.it/~grandis/


On 21 Sep 2009, at 11:44, Urs Schreiber wrote:

> Marco Grandis wrote:
>
>> My book
>>
>>    'Directed Algebraic Topology'
>>    Models of non-reversible worlds
>>
>> has appeared, at Cambridge University Press.
>
> In that context I am wondering about the following:
>
> it would be nice to have a notion of directed topological space that
> would extend the relation between (nice) topological spaces and
> oo-groupoids to one between (nice) directed topological spaces and
> (oo,1)-categories.
>
> ...
> Has anything like this been considered?


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Re: 'Directed Algebraic Topology'

[George Janelidze]

Dear Michael,

The asymmetry you are mentioning is very different.

One of your old theorems, expressed in a language recently used by Maria
Manuel Clementino, Dirk Hofmann, Walter Tholen and others, says that the
category of topological spaces can be identified with the category of
(T,V)-categories (=lax (T,V)-algebras) for V = 2 = {0,1} considered as a
symmetric monoidal category, and T being the ultrafilter monad on the
category of sets.

If T is the identity monad on the category of sets, a (T,V)-category becomes
just a V-category (=a category enriched in V), and - at least when V is a
quantale - it makes sense to call a V-category X "symmetric" if X(a,b) =
X(b,a) for all objects a and b in X (where X(a,b) denotes the internal hom
object).

Example 1 (the old observation of Bill Lawvere - as you know of course):
When V = R+ (suitable monoidal category of nonnegative real numbers), a
V-category is exactly an asymmetric metric space, while a symmetric
V-category is an ordinary metric space (well, ignoring the axiom d(x,y) = 0
=> x = y of course).

Example 2 (obvious): When V = 2, a V-category is a preorder (=reflexive and
transitive binary relation on a set), while a symmetric V-category is an
equivalence relation (=symmetric preorder).

But when T is not an identity monad as in your old theorem, the symmetry
does not even make sense - simply because, say, a relation between T(X) and
X cannot be symmetric. That is, the ordinary topological spaces are already
"much more asymmetric" than asymmetric metric spaces!

And what I said about (asymmetric) metric spaces can be repeated for
(quasi)uniform spaces with "pro" involved: see [M. M. Clementino, D.
Hofmann, W. Tholen, One setting for all: Metric, Topology, Uniformity,
Approach Structure, Applied Categorical Structures 12, 2004, 127-154]. So
again, "the ordinary topological spaces are much more asymmetric than
asymmetric uniform spaces (=quasiuniform spaces)".

But yes, quasiuniform spaces have been and are studied seriously -
particularly categorically and particularly by Guillaume Brummer and
Hans-Peter Kunzi in Cape Town.

George

----- Original Message -----
From: "Michael Barr" <barr@math.mcgill.ca>
To: "Urs Schreiber" <urs.schreiber@googlemail.com>; <categories@mta.ca>
Sent: Monday, September 21, 2009 5:56 PM
Subject: categories: Re: 'Directed Algebraic Topology'


One obvious thing that comes to mind are asymmetric spaces--a metric
without the symmetry axiom.  This can obviously be extended to uniform
spaces, although I am not aware anyone has.  As for topological spaces, I
know of nothing there.

Michael

On Mon, 21 Sep 2009, Urs Schreiber wrote:

> Marco Grandis wrote:
>
>> My book
>>
>> 'Directed Algebraic Topology'
>> Models of non-reversible worlds
>>
>> has appeared, at Cambridge University Press.
>
> In that context I am wondering about the following:
>
> it would be nice to have a notion of directed topological space that
> would extend the relation between (nice) topological spaces and
> oo-groupoids to one between (nice) directed topological spaces and
> (oo,1)-categories.
>
> More generally, it would be nice to have a notion of "r-directed
> topological space" for r in N that would extend the relation between
> (nice) topological spaces and oo-groupoids to one of (nice)
> "r-directed spaces" and (oo,r)-cateories.
>
> (Probably such a notion of directed spaces can't be supporrted by
> plain topological spaces with direction information, but requires
> filtered directed spaces or the like. )
>
> Has anything like this been considered?



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Re: 'Directed Algebraic Topology'

[Michael Barr]

One obvious thing that comes to mind are asymmetric spaces--a metric 
without the symmetry axiom.  This can obviously be extended to uniform 
spaces, although I am not aware anyone has.  As for topological spaces, I 
know of nothing there.

Michael

On Mon, 21 Sep 2009, Urs Schreiber wrote:

> Marco Grandis wrote:
>
>> My book
>>
>>    'Directed Algebraic Topology'
>>    Models of non-reversible worlds
>>
>> has appeared, at Cambridge University Press.
>
> In that context I am wondering about the following:
>
> it would be nice to have a notion of directed topological space that
> would extend the relation between (nice) topological spaces and
> oo-groupoids to one between (nice) directed topological spaces and
> (oo,1)-categories.
>
> More generally, it would be nice to have a notion of "r-directed
> topological space" for r in N that would extend the relation between
> (nice) topological spaces and oo-groupoids to one of (nice)
> "r-directed spaces" and (oo,r)-cateories.
>
> (Probably such a notion of directed spaces can't be supporrted by
> plain topological spaces with direction information, but requires
> filtered directed spaces or the like. )
>
> Has anything like this been considered?


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'Directed Algebraic Topology'

[Marco Grandis]

Dear categorists,

My book

    'Directed Algebraic Topology'
    Models of non-reversible worlds

has appeared, at Cambridge University Press. Its aims are mentioned
below.

It is likely well known that the policy of Cambridge UP, with respect
to publication,
is open and liberal.

But I must say I was pleased and surprised, during the preparation of
this volume,
by their way of handling things, which was at the same time effective
and informal,
precise and very flexible.

For the many people in this list that are concerned with the problems
of our libraries,
because of the high prices of scientific books and journals, I will
add that royalties for
this volume have been converted into CUP books for the library of my
Departement.

Marco Grandis
________

FROM THE BEGINNING OF THE INTRODUCTION
Aims

Directed Algebraic Topology is a recent subject which arose in the
1990's, on the one hand in abstract settings for homotopy theory,
and on the other hand in investigations in the theory of concurrent
processes.
Its general aim should be stated as `modelling non-reversible
phenomena'.
The subject has a deep relationship with category theory.

	The domain of Directed Algebraic Topology should be distinguished
from the domain of classical Algebraic Topology by the principle that
{\it directed spaces
have privileged directions and directed paths therein need not be
reversible}.
While the classical domain of Topology and Algebraic Topology is a
reversible world,
where a path in a space can always be travelled backwards, the study
of non-reversible
phenomena requires broader worlds, where a directed space can have
non-reversible paths.

The homotopical tools of Directed Algebraic Topology, corresponding
in the classical case to
ordinary homotopies, the fundamental group and fundamental $n$-
groupoids, should be similarly
`non-reversible': {\it directed homotopies}, the {\it fundamental
monoid} and {\it fundamental
$n$-categories}.
Similarly, its homological theories will take values in `directed'
algebraic structures, like {\it
preordered} abelian groups or abelian {\it monoids}. Homotopy
constructions like mapping cone,
cone and suspension, occur here in a directed version; this gives
rise to new `shapes', like (lower and
upper) directed cones and directed spheres, whose elegance is
strengthened by the fact that such
constructions are determined by universal properties.

Applications will deal with domains where privileged directions
appear, such as concurrent
processes, rewrite systems, traffic networks, space-time models,
biological systems, etc.
At the time of writing, the most developed ones are concerned with
concurrency.


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