Preprint available

Preprint available

[categories]

Date: Tue, 1 Jul 97 13:38 BST
From: Dr. P.T. Johnstone <P.T.Johnstone@pmms.cam.ac.uk>

The Pure Mathematics Department of Cambridge University has a new electronic
preprint server (accessible via our home page at http://www.pmms.cam.ac.uk).
The first preprint available may be of interest to people on the categories
mailing list: it is

C. Butz and P.T. Johnstone: Classifying toposes for first-order theories

Abstract: By a classifying topos for a first-order theory $\Bbb T$, we mean
a topos $\cal E$ such that, for any topos $\cal F$, models of $\Bbb T$ in
$\cal F$ correspond exactly to open geometric morphisms ${\cal F}
\rightarrow{\cal E}$. We show that not every (infinitary) first-order theory
has a classifying topos in this sense, but we characterize those which do by
an appropriate `smallness condition', and we show that every Grothendieck
topos arises as the classifying topos of such a theory. We also show that
every first-order theory has a conservative extension to one which possesses
a classifying topos, and we obtain a Heyting-valued completeness theorem for
infinitary first-order logic.

For those who would prefer to receive a hard copy of this paper, I shall be
bringing a supply with me to the Vancouver meeting.

Peter Johnstone

preprint available

[claudio pisani]

Dear categorists,

the following preprint is now available at http://arxiv.org/abs/1402.0253 :

"Sequential multicategories"

Abstract:
"We study the monoidal closed category of symmetric multicategories,
especially in relation with its cartesian structure and with sequential multicategories
(whose arrows are sequences of concurrent arrows in a given category).
Then we consider cartesian multicategories in a similar perspective and develop 
some peculiar items such as algebraic products. 
Several classical facts arise as a consequence of this analysis when some of
the multicategories involved are representable."

Among the topics discussed there are:
1) Promonoidal categories as exponentiable multicategories and particular instances of powers of multicategories.
2) Characterization of the sequential multicategories as those of commutative monoids in a symmetric multicategory and the sequential (co)reflection.
3) Characterization of the preadditive ( = cMon-enriched ) categories as those of commutative monoids in a cartesian multicategory and the preadditive coreflection.
4) Algebraic products in cartesian multicategories, generalizing algebraic biproducts in preadditive categories.

Comments are welcome (for instance, maybe there are related works which I am not aware of).

Best regards

Claudio       


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

Preprint available

[Anders Kock]

My preprint "Projective lines as groupoids with projection structure" is now available at arXiv.

A projection structure on a groupoid consists in giving, for each pair
(A,B) of distinct objects in it, a bijection between the set hom(A,B), and
the set of objects distinct from A and B. Groupoids with such structure
occur naturally in projective geometry over a field.  - The present note
investigates when, conversely, an abstractly given groupoid with projection
structure comes about from a field, constructed out of the groupoid.

http://arxiv.org/abs/1310.5357

Anders



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preprint available

[Anders Kock]

My manuscript "Geometric Algebra of Projective Lines" is available at arXiv,
http://arxiv.org/abs/1003.2095

It is a sequel to the preprint on "Abstract Projective Lines" posted in
December 2009.


Abstract: The projective line over a field carries structure of a
groupoid with a certain correspondence between objects and arrows. We
discuss to what extent the field can be reconstructed from the groupoid.

(The *multiplication* of the field comes from the *composition* of the
groupoid; the crux is to reconstruct the *additive* structure.)

Anders Kock





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preprint available

[Anders Kock]

My manuscript "Abstract Projective Lines" is available at arXiv,
http://arxiv.org/abs/0912.0822


Abstract:
     We describe a notion of (abstract) projective line over a field as
a set equipped with a certain first order structure, and a projectivity
between projective lines as a bijection preserving this structure.
     The structure in question is that of a groupoid, with certain
properties.  This leads to a natural notion of bundle of projective
lines, forming a stack.

Anders Kock




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

preprint available

[JONATHON FUNK]

Dear Colleagues,

A preprint, whose abstract follows, is available in compressed .dvi 
form (for DOS and for UNIX) from:

   http://www.emu.edu.tr/academic/facartsc/mathsdep/staffpic/jfunk.htm
   
or if you are browsing the web, click on academics, teaching 
staff, Mathematics, Jonathon Funk, additional information,
after you have reached the EMU homepage http://www.emu.edu.tr

If you would like a copy, but are unable to retrieve the preprint, 
please don't hestitate to contact me, as I would be happy to send 
you the .dvi file personally. 
funk@mozart.emu.edu.tr

----------------------------------------------------------------

``On branched covers in topos theory''

Abstract: We present some new findings conerning branched covers in 
topos theory. Our discussion involves a particular subtopos of a 
given topos that can be described as the smallest subtopos closed 
under small coproducts in the including topos.  We also have some new 
results concerning the general theory of KZ-doctrines, such as the 
the closure under composition of discrete fibrations for a KZ-
doctrine (in the sense of Bunge/Funk, ``On a bicomma object condition 
for KZ-doctrines'').

Regards,
Jonathon Funk



Jonathon Funk
Department of Mathematics
Eastern Mediterranean University
Gazimagusa
Turkish Republic of North Cyprus
via Mersin 10, Turkey

tel: (90) 392 366 6588, Ext: 1227, 1228, 1138
fax: (90) 392 366 1604

preprint available

[zackluo]

Clone Theory: Its Syntax and Semantics, Applications to Universal Algebra, Lambda Calculus and Algebraic Logic

Author: Zhaohua Luo

Abstract: The primary goal of this paper is to present a unified way to transform the syntax of a logic system into certain initial algebraic structure so that it can be studied algebraically. The algebraic structures which one may choose for this purpose are various clones over a full subcategory of a category. We show that the syntax of equational logic, lambda calculus and first order logic can be represented as clones or right algebras of clones over the set of positive integers. The semantics is then represented by structures derived from left algebras of these clones.

URL:
http://www.algebraic.net/cag/ct.pdf

preprint available

[Anders Kock]

Dear all,

This is to announce the availability of a preprint

"Infinitesimal cubical structure, and higher connections"

The preprint can be downloaded from

http://arxiv.org/abs/0705.4406

or from my home page

http://home.imf.au.dk/kock/

In the context of Synthetic Differential Geometry, we describe a
notion of higher connection with values in a cubical groupoid. We do
this by exploiting a certain structure of cubical complex derived
from the first neighbourhood of the diagonal of a manifold. This
cubical complex consists of infinitesimal parallelepipeda.

Yours
Anders

preprint available

[kock]

The preprint
Algebra of Principal Fibre Bundles, and Connections
is available at
ftp://ftp.imf.au.dk/pub/kock/princ4.ps
(102 kb).

The classical relationship between the curvature of a connection,
and the coboundary of its connection form, here comes about from a
pure groupoid calculation.

The preprint updates and expands my 1983/1986 paper,
"Combinatorial notions relating to principal fibre bundles".

Anders Kock
http://www.imf.au.dk/~kock/

Preprint available

[F W Lawvere]

	PREPRINT AVAILABLE

	A transcript of the video of my talk at the September 1997
AMS Meeting in Montreal is now available for downloading in pdf format.
The title is

	TOPOSES OF LAWS OF MOTION

I will be very grateful for comments and suggestions on this paper, 
as well as on the other two papers available:

	http://www.acsu.buffalo.edu/~wlawvere

*******************************************************************************
F. William Lawvere			Mathematics Dept. SUNY 
wlawvere@acsu.buffalo.edu               106 Diefendorf Hall
716-829-2144  ext. 117		        Buffalo, N.Y. 14214, USA

*******************************************************************************
                       

preprint available

[Martin Escardo]

The following preprint is available at

    http://www.dcs.ed.ac.uk/home/mhe/pub/papers/patch-CSLC.ps.gz
&   http://www.dcs.ed.ac.uk/home/mhe/papers.html
&   ftp://ftp.dcs.ed.ac.uk/pub/mhe/patch-CSLC.ps.gz

             On the compact-regular coreflection 
         of a compact stably locally compact locale.

ABSTRACT: The Scott continuous nuclei form a subframe of the frame of
all nuclei. We refer to this subframe as the patch frame. We show that
the patch construction exhibits (i) the category of Stone locales and
continuous maps as a coreflective subcategory of the category of
coherent locales and coherent maps, (ii) the category of compact
regular locales and continuous maps as a coreflective subcategory of
the category of compact stably locally compact locales and perfect
maps, and (iii) the category of regular locally compact locales and
continuous maps as a coreflective subcategory of the category of
stably locally compact locales. We relate our patch construction to
Banaschewski and Brümmer's construction of the dual equivalence of the
category of compact stably locally compact locales and perfect maps
with the category of compact regular biframes and biframe
homomorphisms.

Comments are welcome.
-----------------------------------------------------------------
Martin H. Escardo, LFCS, Computer Science, Edinburgh University
King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, Scotland
office:	2606 (JMCB) fax: +44 131 667 7209 phone: +44 131 650 5135
mailto:mhe@dcs.ed.ac.uk          http://www.dcs.ed.ac.uk/home/mhe
-----------------------------------------------------------------

Preprint available

[Marco Grandis]

The following preprint:

M. Grandis,
"An intrinsic homotopy theory for simplicial complexes
with applications to image processing"

is available at:

ftp://pitagora.dima.unige.it/WWW/FTP/GRANDIS/

as:   Lnk.Nov98.ps

***

Abstract. A simplicial complex is a set equipped with a down-closed family
of distinguished finite subsets; this structure is mostly viewed as
codifying a triangulated space. Here, this structure is used directly to
describe "spaces" of interest in various applications, where the associated
triangulated space would be misleading. An intrinsic homotopy theory, not
based on topological realisation, is introduced.
        The applications considered here are aimed at metric spaces and
digital topology; concretely, at image processing and computer graphics. A
metric space  X  has a structure  t_e(X)  of simplicial complex at each
"resolution"  e > 0;  the resulting n-homotopy group  \pi_n(t_e(X))  detects
those singularities which can be captured by an n-dimensional grid, with
edges bound by  e;  this works equally well for continuous or discrete
regions of euclidean spaces.

***

Comments would be appreciated.

In particular, I am uneasy about a question of terminology.

In my opinion, the term "simplicial complex", quite appropriate when the
structure is viewed as codifying a triangulated space, is unfit when such
objects are treated as "spaces" in themselves (somewhat close to
bornological spaces, which have similar axioms on objects and maps).

In other words, "simplicial complex" should not refer to the category
itself, say  C,  but to its usual embedding in  Top,  the simplicial
realisation. The two aspects may clash, e.g. with respect to initial or
final structures: the coarse C-object on three points (final structure, all
parts are distinguished) is realised as a euclidean triangle; a C-subobject
is sufficient to produce a topological subspace (a regular subobject in
Top), but a C-subspace (a regular subobject in  C)  is a stronger notion.
Moreover, from a more concrete point of view, the simplicial realisation is
quite inappropriate in most of the applications considered in this work.

The opposition  "C-object / simplicial complex" is in part similar to
"sequence / series": the second term refers to a more specific view & use
of the same data; the clashing of the opposition is particularly evident in
the notions of convergence, for a sequence or a series.

That's why I am calling a C-object a "combinatorial space". (The term
"combinatorial complex" has also been used for simplicial complex; and I
wanted a term of the form "attribute + space", to use freely of topological
terms like discrete, coarse, subspace...)
But of course it is embarassing to propose a new term for a classical notion.

Marco Grandis

preprint available

[Susan Niefield]

The following reprint is available at 

	http://www1.union.edu/~niefiels/ESU.ps  
	http://www1.union.edu/~niefiels/ESU.dvi 


EXPONENTIABILITY AND SINGLE UNIVERSES
by Marta BUNGE and Susan NIEFIELD

ABSTRACT - The search for suitable single universes for opposite or dual
pairs of notions (such as those of discrete fibration and discrete
opfibration, or of open and closed inclusions, or of functions and
distributions on a Grothendieck topos) leads naturally to
exponentiability.  Using exponentiability techniques, such as
model-generated categories and glueing, we settle a standing conjecture
and an open problem.  The conjecture, due to F. Lamarche, states that for
a small category B, the category of unique factorization liftings (also
known as discrete Conduche fibrations) over B is a topos.  We also
construct the smallest topos containing the local homeomorphisms
(functions) and the complete spreads (distributions) over any given topos
satisfying a certain condition (true of presheaf toposes).  This solves a
problem posed by F. W. Lawvere.  Along the way, we introduce two new sorts
of geometric morphisms, characterize locally closed inclusions in Cat,
and investigate new features of generalized coverings in topos theory,
such as branched coverings, cuts, and complete spreads. 

preprint available

[Steve Awodey]

Dear Colleagues,

The preprint mentioned below is available from my page on the WWW,

http://www.andrew.cmu.edu/user/awodey/

Please let me know if you have difficulty obtaing or printing it, or if you
would like to have a paper copy sent.

Steve A.


*******************************************************************************

"Topological representation of the lambda-calculus"

S. Awodey

Abstract: The lambda-calculus can be represented  topologically by
assigning certain spaces to the types and certain  continuous maps to the
terms.  Using a recent result from topos theory, the usual calculus of
lambda-conversion is shown to be  deductively complete with respect to such
topological
semantics.  It is also shown to be functionally complete, in  the sense
that there is always a ``minimal'' topological model, in  which every
continuous function is lambda-definable.  These  results subsume earlier
ones using cartesian closed categories, as  well as those employing
so-called Henkin and Kripke lambda-models.

*******************************************************************************

preprint available

[categories]

Date: Mon, 1 Sep 1997 14:47:02 +0200 (MET DST)
From: Koslowski <koslowj@iti.cs.tu-bs.de>

Dear Colleagues,

An updated preprint of my paper "Beyond the Chu-construction", which
I presented in Vancouver in July, is now available on my web-page

	http://www.iti.cs.tu-bs.de/TI-INFO/koslowj/koslowski.html

The abstract follows below.

  From a symmetric monoidal closed (= autonomous) category Po-Hsiang
  Chu originally constructed a *-autonomous one, ie, a self-dual
  autonomous category where the duality is realized by means of a
  dualizing object.  Recently, Michael Barr introduced an extension
  for the non-symmetric, but closed, case that after an initial step
  utilized monads and modules between them.  Since these tools are
  well-understood in a bicategorical setting, we introduce a notion
  of local *-autonomy for closed bicategories that turns out to
  be inherited by the bicategories of monads and the bicategory of
  interpolads.  Since the first step of Barr's construction carries
  over directly to the bicategorical setting, we recover his main
  result as an easy corollary.  Furthermore, the Chu-construction at
  this level may be viewed as a procedure to turn the endo-1-cells of
  a bicategory into the objects of a new bicategory, and hence is
  conceptually close to the constructions of bicategories of monads
  and of interpolads.

Best regards,

-- J"urgen

-- 
J"urgen Koslowski       % If I don't see you no more in this world
ITI                     % I meet you in the next world
TU Braunschweig         % and don't be late!
koslowj@iti.cs.tu-bs.de %              Jimi Hendrix (Voodoo Child)

Preprint available

[categories]

Date: Wed, 30 Jul 1997 15:51:32 +0200 (MET DST)
From: Carsten Butz <butz@daimi.aau.dk>

Dear Colleagues,

the ps-file of the following preprint is available at the homepage
http://www.brics.dk/~butz :

Topological Completeness for Higher-Order Logic

by Steve Awodey (awodey@cmu.edu),
   Carsten Butz (butz@brics.dk).

Abstract: Using recent results in topos theory, two systems of
higher-order logic are shown to be complete with respect to sheaf
models over topological spaces---so-called ``topological semantics''.
The first is classical higher-order logic, with relational
quantification of finitely high type; the second system is a
predicative fragment thereof with quantification over functions
between types, but not over arbitrary relations.  The second theorem
applies to intuitionistic as well as classical logic.

Best regards,

Steve Awodey and Carsten Butz

preprint available

[categories]

Date: Tue, 22 Apr 1997 21:26:04 +0200 (METDST)
From: Anders Kock <kock@mi.aau.dk>

The article:
"Geometric Construction of the Levi-Civita Parallelism"
by Anders Kock
is available  from
ftp://ftp.mi.aau.dk/pub/kock/parallel.ps
(about 150 kb).
(The Levi-Civita Parallellism is also called the Riemannian Connection; it
is the unique symmetric affine connection compatible with a given
Riemannian metric. We present a geometric construction of it, using
variational principles and synthetic differential geometry.)

Preprint available.

[categories]

Date: Thu, 10 Apr 1997 17:04:46 +0100
From: Marcelo Fiore <mf@dcs.ed.ac.uk>

The following preprint is available at

                    http://www.dcs.ed.ac.uk/home/mf/ADT/

as cub.dvi and cub.ps.  Best, Marcelo.


               Complete Cuboidal Sets in Axiomatic Domain Theory


      Marcelo Fiore            Gordon Plotkin               John Power
    <mf@dcs.ed.ac.uk>        <gdp@dcs.ed.ac.uk>          <ajp@dcs.ed.ac.uk>

                       Department of Computer Science 
                Laboratory for Foundations of Computer Science 
                University of Edinburgh, The King's Buildings 
                        Edinburgh EH9 3JZ, Scotland 


                                 Synopsis

We study the enrichment of models of axiomatic domain theory.  To this end, we
introduce a new and broader notion of domain, viz. that of complete cuboidal
set, that complies with the axiomatic requirements.  We show that the category
of complete cuboidal sets provides a general notion of enrichment for a wide 
class of axiomatic domain-theoretic structures. 

 Cuboidal sets play a role similar to that played by posets in the traditional
 setting.  They are the analogue of simplicial sets but with the simplicial 
 category enlarged to the cuboidal category of cuboids, i.e. of finite 
 products  O_n1 x ... x O_ni  of finite ordinals.  These cuboids are the 
 possible shapes of paths.  A cuboidal set  P  has a set  P(C)  of paths of 
 every shape  C = n1 x ... x ni;  indeed, it is a (rooted) presheaf over the 
 cuboidal category.  The set of points of  P  is  P(O_1).  The set of 
 (one-dimensional) paths of length  n  is  P(O_n+1);  they can be thought of 
 as (linear) computations conditional on the occurrence of  n  linearly 
 ordered events  e_1 < ... < e_n.  Evidently,  O_n  is the partial order 
 associated to this simple linear event structure, and can be considered as a 
 sequential process of length  n.  At higher dimensions,  P(O_n1 x ... x O_ni)
 can be thought of as the set of computations conditional on the occurrence of 
 n_1 + ... + n_i  events ordered by  e_1,1 < ... < e_1,n1 ; ... ; 
 e_i,1 < ... < e_i,ni.  This is the event structure which can be considered 
 as  i  sequential processes, of respective lengths  O_n1, ..., O_ni,  running 
 concurrently.

 Complete cuboidal sets are cuboidal sets equipped with a formal-lub operator 
 satisfying three algebraic laws, which are exactly those needed of the lub 
 operator in order to prove the fixed-point theorem.  Computationally, the 
 passage from cuboidal sets to complete cuboidal sets corresponds to allowing 
 infinite processes.  In fact, the formal-lub operator assigns paths of shape 
 C  to `paths of shape  C x omega',  for every  C.  Here the set of paths of 
 shape  C x omega  is the colimit of the paths of shape  C x O_n;  such paths 
 can be thought of as the higher-dimensional analogue of the increasing 
 sequences of traditional domain theory.