Re: preprint

Re: preprint

[Marco Grandis]

[Please do not use this e-address, but only the usual one at dima.unige.it]
Dear Bill,

Thank you very much for all these interesting comments and suggestions, we'll think of them.
We also received others from Ronnie Brown, Marta Bunge and Bob Pare; we  are preparing a revised version.

I am quite convinced of the importance of presheaves on nonempty finite sets. Just for the pleasure of friendly 
discussing, let me add that I like calling them 'symmetric simplicial sets' :-) - a term against which you protested 
sometime ago, in this list if I remember well; at least in a context where their links with simplicial sets and
algebraic topology are relevant.

On the other hand, I think that the classical term of 'simplicial complex' is misleading.
In fact, they sit inside symmetric simplicial sets (not inside simplicial  sets) as the objects that 
'live on their points', according to your terminology. I used for them the term 'combinatorial space',
while I used 'directed combinatorial space' for the (non classical) directed analogue (sitting inside 
simplicial sets), whose objects are defined by privileged finite sequences instead of privileged finite subsets.I think that the opposition between non-directed and directed notions should be kept clear.
Thanks again and best wishes    Marco

On 23/nov/2013, at 02.41, F. William Lawvere wrote:

Dear Ettore and Marco

Much of your very interesting paper is actually taking place 
in an unjustly neglected topos.

The classifying topos for nontrivial Boolean algebras is concretely just 
presheaves on nonempty finite sets. It was one of two examples in my 
1988 Macquarie talk "Toposes generated by codiscrete objects in 
algebraic topology and functional analysis" *

Like many non-localic toposes, this one unites pair of identical 
but adjointly opposite copies of the base topos of abstract sets,
namely a subtopos of codiscretes and its negation, the subcategory of
discretes or constant presheaves. Because in this very special case
the Yoneda inclusion is a restriction of  the codiscrete inclusion, 
it is well justified to consider this topos as "codiscretely generated " 
(wrt colimits).

This topos contains the category of groupoids as a full reflective subcategory.
In fact the reflector preserves finite products, and is a refinement of 
the syntax for presenting groups. It should probably be 
considered as the fundamental combinatorial Poincare functor .

That point of view requires viewing the objects of the topos as 
combinatorial spaces, which  is out of fashion even though a
full coreflective subcategory is that of classical simplicial complexes 
( or "simplicial schemes"according to Godement).  The fear is that
geometric realization will not be exact. But as Joyal and Wraith 
pointed out 30 years ago, one need only replace the usual interval  with the weak
infinite-dimensional sphere as basic parameterizer (of Zitterbewegungen 
instead of ordinary paths ?) in order to obtain  a geometric mo\x10rphism
of the singular/realization kind from any reasonable topological topos.

Actually the groupoids  are already at a low level in the sequence of
essential subtoposes: if the trivial topos is taken as level minus infinity,
the discrete as level zero, and the lowest level whose skeleton functor 
detects connectivity as level one, then the lowest level whose UIAO is
compatible seems to be three.  Explicit consideration of the role of the
groupoids may a  help in calculating the precise Aufhebung  function.

Best wishes
Bill

*the other example was the Gaeta topos on countably  infinite sets, 
closely related to the Kolmogoroff-Mackey bornological generalizations of
Banach spaces, which are  vector space objects in that topos. 		 	   		  

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: preprint

[F. William Lawvere]

Dear Marco
If I understand the history, it is the term "simplicial sets" that was somewhat misleading : Classically the simplicial complexes (based ,from a categorical view, actually on finite families not just finite subsets)
were an important combinatorial structure and indeed still are. Eilenberg and others  recognized the important role in topological calculations for "ordered simplices " and the corresponding toposwas said to consist of "semisimplicial sets". But then the prefixes were dropped !
It does not seem accurate to consider that there is  an "opposition "between the ordered and unordered cases. They are just two subtoposesof the distributive lattice classifier, which consists of presheaves onfinite posets.  The classifying topos point of view is helpful, because it is the choice  of an algebraic structure, of the kind classified, in a topological topos that gives rise to an exact singular/realization pair. For example any topological distributive lattice gives rise to such ;if it happens to be totally ordered, these adjoint functors factor throughthe simplicial subtopos. But a boolean algebra whose operations are continuous represents a realization that depends only on the subtoposthat depends only on the finite trivially ordered sets. The symmetry is concretely the action of boolean negation. The fact that any  distributive lattice space embeds in a boolean algebra space helps to relate these .
There are other subtoposes (i.e. strengthenings of the theory of distributive lattices), so that precise calculation of the Aufhebung andof coHeyting boundary should in particular be relevant to combinatorial topology.
There may be a connection with your earlier work that employed distributive  lattices in the analysis of diagrams. When one represents ageometric category in presheaves on a subcategory P, it is helpful to consider that P parameterizes the basic figure shapes that determine the structureof a general object. In the category of categories, the basic figures arefinite commutative diagrams, are they not ? Thus, if we do not insist on a minimalistic notion of nerve, distributive lattices emerge.
Best wishesBill
> From: grandis43@hotmail.com
> To: wlawvere@hotmail.com; categories@mta.ca
> Subject: categories: Re: preprint
> Date: Sat, 23 Nov 2013 12:03:56 +0100
> 
> [Please do not use this e-address, but only the usual one at dima.unige.it]
> Dear Bill,
> 
> Thank you very much for all these interesting comments and suggestions,  we'll think of them.
> We also received others from Ronnie Brown, Marta Bunge and Bob Pare; we  are preparing a revised version.
> 
> I am quite convinced of the importance of presheaves on nonempty finite sets. Just for the pleasure of friendly 
> discussing, let me add that I like calling them 'symmetric simplicial sets' :-) - a term against which you protested 
> sometime ago, in this list if I remember well; at least in a context where their links with simplicial sets and
> algebraic topology are relevant.
> 
> On the other hand, I think that the classical term of 'simplicial complex' is misleading.
> In fact, they sit inside symmetric simplicial sets (not inside simplicial  sets) as the objects that 
> 'live on their points', according to your terminology. I used for them the term 'combinatorial space',
> while I used 'directed combinatorial space' for the (non classical) directed analogue (sitting inside 
> simplicial sets), whose objects are defined by privileged finite sequences instead of privileged finite subsets.I think that the opposition between non-directed and directed notions should be kept clear.
> Thanks again and best wishes    Marco
> 

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: preprint

[Marco Grandis]

Dear Bill,

The history of simplicial terms is certainly complicated and often
contradictory.
Presently, the term of 'simplicial set/object' seems to be generally
accepted (for a presheaf on positive finite ordinals) and this notion
seems to be mostly considered as the leading one in its framework.
My modest opinion is that simplicial terminology should be organised
taking this term as the leading one.

> It does not seem accurate to consider that there is  an "opposition
> "between the ordered and unordered cases.

Fine, I agree, I should replace 'opposition' with 'distinction'.
Certainly, I would not speak of opposition for the relationships
     set / preordered set
     space / directed space
     classical algebraic topology / directed algebraic topology.

Your comments on distributive lattices are most interesting.

Best wishes    Marco



On 24 Nov 2013, at 01:38, F. William Lawvere wrote:

> Dear Marco
> If I understand the history, it is the term "simplicial sets" that
> was somewhat misleading : Classically the simplicial complexes
> (based ,from a categorical view, actually on finite families not
> just finite subsets)
> were an important combinatorial structure and indeed still are.
> Eilenberg and others  recognized the important role in topological
> calculations for "ordered simplices " and the corresponding
> toposwas said to consist of "semisimplicial sets". But then the
> prefixes were dropped !
> It does not seem accurate to consider that there is  an "opposition
> "between the ordered and unordered cases. They are just two
> subtoposesof the distributive lattice classifier, which consists of
> presheaves onfinite posets.  The classifying topos point of view is
> helpful, because it is the choice  of an algebraic structure, of
> the kind classified, in a topological topos that gives rise to an
> exact singular/realization pair. For example any topological
> distributive lattice gives rise to such ;if it happens to be
> totally ordered, these adjoint functors factor throughthe
> simplicial subtopos. But a boolean algebra whose operations are
> continuous represents a realization that depends only on the
> subtoposthat depends only on the finite trivially ordered sets. The
> symmetry is concretely the action of boolean negation. The fact
> that any  distributive lattice space embeds in a boolean algebra
> space helps to relate these .
> There are other subtoposes (i.e. strengthenings of the theory of
> distributive lattices), so that precise calculation of the
> Aufhebung andof coHeyting boundary should in particular be relevant
> to combinatorial topology.
> There may be a connection with your earlier work that employed
> distributive  lattices in the analysis of diagrams. When one
> represents ageometric category in presheaves on a subcategory P, it
> is helpful to consider that P parameterizes the basic figure shapes
> that determine the structureof a general object. In the category of
> categories, the basic figures arefinite commutative diagrams, are
> they not ? Thus, if we do not insist on a minimalistic notion of
> nerve, distributive lattices emerge.
> Best wishesBill

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: preprint

[Marco Grandis]

Dear Jean,

Presheaves on all finite ordinals are called augmented simplicial sets.

As to 'directed space' I use now this term in a non-technical sense,
as referring to any topological structure suitable for directed
algebraic topology, like:

- preordered topological space,
-  d-space, i.e. space with distinguished paths,
-  c-space, i.e. space with distinguished cubes,
- etc.

See my book on 'Directed Algebraic Topology'. Thanks to Cambridge Un.
Press, it can be freely (and legally) downloaded from:

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


(When I first used the term 'directed space', it was equivalent to d-
space; later I thought it was better to keep it for a more general
meaning.)

Best wishes   Marco


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: preprint

[Jean Bénabou]

Dear Marco,

Many thanks for your explanations, and of course for giving me the possibility to study your book, which I shall do very soon.

I have a notion which I hesitated to call directed category for fear of a conflict in using the word directed. Your non technical use of this word encourage me to use it, but I shall first read your book before I make up my mind.

Best regards, Jean


Le 26 nov. 2013 à 11:11, Marco Grandis a écrit :

> Dear Jean,
> 
> Presheaves on all finite ordinals are called augmented simplicial sets.
> 
> As to 'directed space' I use now this term in a non-technical sense,
> as referring to any topological structure suitable for directed
> algebraic topology, like:
> 
> - preordered topological space,
> -  d-space, i.e. space with distinguished paths,
> -  c-space, i.e. space with distinguished cubes,
> - etc.
> 
> See my book on 'Directed Algebraic Topology'. Thanks to Cambridge Un.
> Press, it can be freely (and legally) downloaded from:
> 
>    http://www.dima.unige.it/~grandis/BkDAT_page.html
> 
> 
> (When I first used the term 'directed space', it was equivalent to d-
> space; later I thought it was better to keep it for a more general
> meaning.)
> 
> Best wishes   Marco


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

preprint

[Marco Grandis]

[-- Warning: decoded text below may be mangled, UTF-8 assumed --]
[-- Attachment #1: Type: text/plain, Size: 1356 bytes --]

The following preprint is available:

-------
E. Carletti - M. Grandis
Fundamental groupoids as generalised pushouts of codiscrete groupoids
Dip. Mat. Univ. Genova, Preprint 603 (2013).
 	http://www.dima.unige.it/~grandis/GpdClm.pdf

Abstract. Every differentiable manifold X has a ‘good cover’, where  
all open sets and their finite intersections are contractible. Using  
a generalised van Kampen theorem for open covers we deduce that the  
fundamental groupoid of X is a ‘generalised pushout’ of codiscrete  
groupoids and inclusions.
This fact motivates the present brief study of generalised pushouts.  
In particular, we show that every groupoid is up to equivalence a  
generalised pushout of codiscrete subgroupoids, and that (in any  
category) finite generalised pushouts amount to ordinary pushouts and  
coequalisers.
-------

Before submitting it, I would like to know if the ‘generalised  
pushouts’ we are using (or similar colimits) have been considered  
elsewhere.
(They are not simply connected colimits, in the sense of Bob Pare,  
and indeed they cannot be constructed with pushouts.)

With best regards to all colleagues and friends. In particular to  
Ronnie Brown and Bob Pare, whose results are used in this preprint.

Marco

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

RE: preprint

[F. William Lawvere]

Dear Ettore and Marco
Much of your very interesting paper is actually taking place in an unjustly  neglected topos.
The classifying topos for nontrivial Boolean algebras is concretely just presheaves on nonempty finite sets. It was one of two examples in my 1988 Macquarie talk "Toposes generated by codiscrete objects in 
algebraic topology and functional analysis" *
Like many non-localic toposes, this one unites pair of identical  but adjointly opposite copies of the base topos of abstract sets,namely a subtopos of codiscretes and its negation, the subcategory ofdiscretes or constant presheaves. Because in this very special casethe Yoneda inclusion is a restriction of  the codiscrete inclusion, it is well justified to consider  this topos as "codiscretely generated " (wrt colimits).
This topos contains the category of groupoids as a full reflective subcategory.In fact the reflector preserves finite products, and is a refinement of the syntax for presenting groups. It should probably be considered as the fundamental combinatorial Poincare functor .
That point of view requires viewing the objects of the topos as combinatorial spaces, which  is out of fashion even though afull coreflective subcategory is that of classical simplicial complexes ( or "simplicial schemes"according to Godement).  The fear is that geometric realization will not be exact. But as Joyal and Wraith pointed out 30 years ago, one need only replace the usual interval  with the weakinfinite-dimensional sphere as basic parameterizer (of Zitterbewegungen instead of ordinary paths ?) in order to  obtain  a geometric mo\x10rphism of the singular/realization kind from any reasonable topological topos.
Actually the groupoids  are already at a low level in the sequence ofessential subtoposes: if the trivial topos is taken as level minus infinity,the  discrete as level zero, and the lowest level whose skeleton functor detects connectivity as level one, then the lowest level whose UIAO is compatible seems to be three.  Explicit consideration of the role of the groupoids may a  help in calculating the precise Aufhebung  function.
Best wishesBill
*the other example was the Gaeta topos on countably  infinite sets, closely related to the Kolmogoroff-Mackey bornological generalizations ofBanach spaces, which are  vector space objects in that topos.






> From: grandis@dima.unige.it
> Subject: categories: preprint
> Date: Mon, 18 Nov 2013 11:51:29 +0100
> To: categories@mta.ca
> 
> The following preprint is available:
> 
> -------
> E. Carletti - M. Grandis
> Fundamental groupoids as generalised pushouts of codiscrete groupoids
> Dip. Mat. Univ. Genova, Preprint 603 (2013).
>  	http://www.dima.unige.it/~grandis/GpdClm.pdf
> 
> Abstract. Every differentiable manifold X has a ‘good cover’, where   
> all open sets and their finite intersections are contractible. Using  
> a generalised van Kampen theorem for open covers we deduce that the  
> fundamental groupoid of X is a ‘generalised pushout’ of codiscrete  
> groupoids and inclusions.
> This fact motivates the present brief study of generalised pushouts.  
> In particular, we show that every groupoid is up to equivalence a  
> generalised pushout of codiscrete subgroupoids, and that (in any  
> category) finite generalised pushouts amount to ordinary pushouts and  
> coequalisers.
> -------
> 
> Before submitting it, I would like to know if the ‘generalised  
> pushouts’ we are using (or similar colimits) have been considered  
> elsewhere.
> (They are not simply connected colimits, in the sense of Bob Pare,  
> and indeed they cannot be constructed with pushouts.)
> 
> With best regards to all colleagues and friends. In particular to  
> Ronnie Brown and Bob Pare, whose results are used in this preprint.
> 
> Marco
> 

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Preprint

[Marco Grandis]

The following preprint is available, in pdf:

M. Grandis
Coherence and distributive lattices in homological algebra
Dip. Mat. Univ. Genova, Preprint 595 (2012)

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

Abstract. Complex systems in homological algebra present problems
of coherence that can be solved by proving the distributivity of the
sublattices of subobjects generated by the system. The main applications
deal with spectral sequences, but the goal of this paper is to convey  
the
importance of distributive lattices (of subobjects) in homological  
algebra,
to researchers outside of this field; a parallel role played by orthodox
semigroups (of endorelations) is referred to but not developed here.

(This article develops part of a conference at CatAlg2011, Gargnano  
(Italy),
in September 2011.)

With best wishes

Marco Grandis

Dipartimento di Matematica
Università di Genova
Via Dodecaneso, 35
16146 Genova
Italy

e-mail: grandis@dima.unige.it
tel: +39 010 353 6805
http://www.dima.unige.it/~grandis/




[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

preprint

[Dusko Pavlovic]

Dear All,

As many of you know, December is the season of two column logic/CS
related preprints. The title of mine is:

    Towards semantics of guarded induction

and it is at the bottom of the page

    http://www.kestrel.edu/HTML/people/pavlovic/

Comments **most** welcome, esp. as I am still a bit in the darkness as
to how to present some parts. This is still an extended abstract, but a
bit more extended and less abstract than the version some of you have
seen before. (Thanks again for the questions that helped me improve it!)

With the very best wishes,
-- Dusko

==============================================================================


    Towards semantics of guarded induction
    by Dusko Pavlovic


    Abstract.

We analyze guarded induction, a coalgebraic method for implementing
abstract data types with infinite elements (e.g. various dynamic
systems, continuous or discrete). It is widely used not just in
computation, but also, tacitly, in many basic constructions of
differential calculus. However, while syntactic characterisations
abound, only the very first steps towards a formal semantics have been
made. A language independent analysis was recently initiated, but just
special cases were covered so far.

In the present paper, we propose a new approach, based on a somewhat
unusual
combination of monads and polynomial categories. The first result is
what appears to be a precise semantic characterisation of guarded
operators on arbitrary final coalgebras.

preprint

[categories]

Date: Thu, 7 Aug 1997 15:43:23 +1000
From: Michael Batanin <mbatanin@mpce.mq.edu.au>

The   preprint

" Finitary monads on globular sets and notions of computad they generate "

is available as
postscript files  at

http://www-math.mpce.mq.edu.au/~mbatanin/papers.html



                               Abstract

Consider a finitary monad on the category of globular sets. We prove
 that the category of its algebras is isomorphic to the category
 of algebras of an appropriate monad on the
special category (of computads) constructed from the data of the
initial monad. In the case of the free $n$-category monad this
definition coincides with R.Street's definition of $n$-computad. In
the case of a monad generated by a higher operad this allows us to
define a pasting operation in a weak $n$-category. It may be also considered
 as the first step toward the proof of equivalence of the different
 definitions of weak $n$-categories.