preprints available

preprints available

[categories]

Date: Wed, 12 Nov 1997 14:16:30 +0100 (MET)
From: Anders Kock <kock@mi.aau.dk>

The following preprints are available:

Differential Forms as Infinitesimal Cochains

This is essentially my contribution at the Vancvouver Category Theory
Meeting in July. It proves that the simplicial complex given by the first
neighbourhood of the diagonal of a manifold (in a well adapted model for
SDG) has de Rham cohomology of the manifold as its R-dual.


Extension Theory for Local Groupoids

We relate Extension Theory for (non-abelian) groups (a la Eilenberg-Mac
Lane) with the theory of Connections (a la Ehresmann), via a notion of
local groupoid. In particular, we give in this setting a kind of converse
to the statement "the curvature 2-form of a connection satisfies Bianchi
identity".


Both these preprints are accessible via my home page:
http://www.mi.aau.dk/~kock/
or directly at
ftp://ftp.mi.aau.dk/pub/kock/Cochains.ps
(respectively ../locg.ps)

Anders Kock

preprints available

[Ronnie]

The following are available from my preprint page, by me unless stated otherwise: 

http://www.bangor.ac.uk/~mas010/brownpr.html

1) 08.04 Exact sequences of fibrations of crossed complexes, homotopy classification of maps, and nonabelian extensions of groups 
ABSTRACT:  The classifying space of a crossed complex generalises the construction of Eilenberg-Mac Lane spaces. We show how the theory of fibrations of crossed complexes allows the analysis of homotopy classes of maps from a free crossed complex to such a classifying space. This gives results on the homotopy classification of maps from a CW-complex to the classifying space of a crossed module and also, more generally, of a crossed complex whose homotopy groups vanish in dimensions between 1 and n. The results are analogous to those for the obstruction to an abstract kernel in group extension theory. 

2) 06.04 R. Brown, I. Morris, J. Shrimpton and C.D. Wensley 

Graphs of morphisms of graphs 

 ABSTRACT: This is an account for the combinatorially minded reader of various categories of directed and undirected graphs, and their analogies with the category of sets. As an application, the endomorphisms of a graph are in this context not only composable, giving a monoid structure, but also have a notion of adjacency, so that the set of endomorphisms is both a monoid and a graph. We extend Shrimpton's (unpublished) investigations on the morphism digraphs of reflexive digraphs to the undirected case by using an equivalence between a category of reflexive, undirected graphs and the category of reflexive, directed graphs with reversal. In so doing, we emphasise a picture of the elements of an undirected graph, as involving two types of edges with a single vertex, namely `bands' and `loops'. Such edges are distinguished by the behaviour of morphisms with respect to these elements.

3) Possible connections between whiskered categories and groupoids, many object Lie algebras, automorphism structures and local-to-global questions 

ABSTRACT: We define the notion of whiskered categories and groupoids and discuss potential applications and extensions, for example to a many object Lie theory, and to resolutions of monoids. This paper is more an outline of a possible programme or programmes than giving conclusive results. 

4) A new higher homotopy groupoid: the fundamental  globular $\omega$-groupoid  of a filtered space 
MSC Classification:18D10, 18G30, 18G50, 20L05, 55N10, 55N25.
KEY WORDS: filtered space, higher homotopy van Kampen theorem, cubical singular complex, free globular  groupoid
ABSTRACT: We show that the graded set of filter homotopy classes rel vertices of maps from the $n$-globe to a filtered space may be given the structure of globular $\omega$--groupoid. The proofs use an analogous fundamental cubical $\omega$--groupoid due to the author and Philip Higgins. This method also relates the construction to the fundamental crossed complex of a filtered space, and this relation allows the proof that the crossed complex associated to the free globular $\omega$-groupoid on one element of dimension $n$ is the fundamental crossed complex of the $n$-globe. 

Ronnie 

preprints available

[Anders Kock]

Dear all,

This is to announce the availability of two preprints.

1)
"Group valued differential forms revisited"

We study the relationship between combinatorial group valued
differential forms and classical differential forms with values in
the corresponding Lie algebra. In particular, we compare simplicial
coboundary and exterior derivative. The results represent
strengthening of results I obtained in 1982.

This preprint can be downloaded from
http://www.imf.au.dk/publs?id=636

or from my home page

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

2)
"Some matrices with nilpotent entries, and their determinants"

We study algebraic properties of matrices whose rows are mutual
neighbours, and are also neigbours of 0 (neighbour in the sense of a
certain nilpotency condition). The intended application is in synthetic
differential geometry. For a square matrix of this kind, the product of
the diagonal entries equals the determinant, modulo a factor n!

This preprint can be downloaded from
http://arxiv.org/abs/math.RA/0612435

or from my home page (address as above).


Yours
Anders

preprints available

[jvoosten]

The following preprints are available from the WWW:

Jaap van Oosten
Realizability: A Historical Essay
http://www.math.uu.nl/publications/preprints/1131.ps.gz

Abstract: historical survey on Realizability. Focuses on
notions of realizability used in the study of metamathematics of
arithmetical theories, and topos-theoretic developments.
Bibliography contains 96 items.24 pages

Lars Birkedal and Jaap van Oosten
Relative and Modified Relative Realizability
http://www.math.uu.nl/publications/preprints/1146.ps.gz

Abstract: We approach `relative realizability' from an abstract point
of view, studying internal partial combinatory algebras in an
arbitrary topos E. Let RT(E,A) denote the standard realizability
topos over E w.r.t. A.
We define the notion of `elementary subobject'
in a topos; if, for two internal pca's A and B in E, there is
an embedding which maps A as elementary subobject into B, there
is a local geometric morphism from RT(E,B) to RT(E,A).
Next we study the situation where an internal topology j is given;
we have a tripos over E using only the j-closed subobjects of A,
giving a topos RTj(E,A). RTj(E,A) is a subtopos of RT(E,A) and we
have a pullback diagram of toposes:

  Sh_j(E)--->RTj(E,A)
    |           |
    |           |
    V           V
    E   ---->  RT(E,A)

If A--> B is an embedding with A elementary subobject of B, the local
geometric morphism restricts to a local geometric morphism:
RTj(E,B)-->RTj(E,A)
Moreover if A --> B is a j-dense embedding, there is a logical functor
(a filter-quotient situation): RTj(E,A) --> RTj(E,B).
If j is an open topology, the inclusion RTj(E,A)-->RT(E,A) is open
too, and it makes sense to consider its closed complement; we call this
the modified relative realizability topos Mj(E,A) w.r.t. A and j.
We have an automatic pullback diagram

  Sh_k(E)---->Mj(E,A)
    |           |
    |           |
    V           V
    E  ---->  RT(E,A)

where k denotes the closed complement of j.
In a section `Examples' we show that our treatment generalizes former
definitions of relative realizability (in Awodey, Birkedal, Scott) and
modified realizability.
16 pages.

preprints available

[categories]

Date: Thu, 4 Dec 1997 11:54:52 +0100
From: Marco Grandis <grandis@dima.unige.it>

The following preprints are now accessible as ps-files, via web of ftp:

http://www.dima.unige.it/STAFF/GRANDIS/

ftp://www.dima.unige.it/pub/STAFF/GRANDIS


(1). "Limits in double categories",  by Marco Grandis and Robert Pare
Dbl.Dec97.ps

(2). "Weak subobjects and weak limits in categories and homotopy
categories", by M.G.
Var1.Aug97.ps

(3). "Weak subobjects and the epi-monic completion of a category", by M.G.
Var2.Dec97.ps

***

The first was announced on this mailing list, on 13 Nov 1997.
(With respect to the printed preprint, this is a slightly revised version,
containing a more detailed comparison with Bastiani-Ehresmann's "limits
relative to double categories".)

The second and third form an expanded version of a printed preprint
("Variables and weak limits in categories and homotopy categories", Dec
1996), announced on this list on 13 Dec 1996.
Abstracts for (2) and (3) are given below.

***

(2). Abstract. We introduce the notion of "variation", or  "weak
subobject", in a category, as an extension of the notion of subobject. The
dual notion is called a covariation, or weak quotient.
    Variations are important in homotopy categories, where they are well
linked to weak limits, much in the same way as, in "ordinary" categories,
subobjects are linked to limits. Thus, "homotopy variations" for a space
S,  with respect to the homotopy category  HoTop,  form a lattice  Fib(S)
of "types of fibration" over  S.
    Nevertheless, the study of weak subobjects in ordinary categories, like
abelian groups or groups, is interesting in itself and relevant to classify
variations in homotopy categories of spaces, by means of homology and
homotopy functors.   (To appear in: Cahiers Top. Geom. Diff. Categ.)

(3). Abstract. Formal properties of weak subobjects are considered. The
variations in a category  X  can be identified with the (distinguished)
subobjects in the epi-monic completion of  X,  or Freyd completion  FrX,
the free category with epi-monic factorisation system over  X,  which
extends the Freyd embedding of the stable homotopy category of spaces in an
abelian category (P. Freyd, Stable homotopy, La Jolla 1965).
    If  X  has products and weak equalisers, as  HoTop  and various other
homotopy categories,  FrX  is complete. If  X  has zero-object, weak
kernels and weak cokernels, as the homotopy category of pointed spaces,
then  FrX  is a "homological" category. Finally, if  X  is triangulated,
FrX  is abelian and the embedding  X --> FrX  is the universal homological
functor on  X,  as in the original case. These facts have consequences on
the ordered sets of variations.


Marco Grandis

Dipartimento di Matematica
Universita' di Genova
via Dodecaneso 35
16146 GENOVA, Italy
e-mail: grandis@dima.unige.it
tel: +39.10.353 6805   fax: +39.10.353 6752
http://www.dima.unige.it/STAFF/GRANDIS/

Preprints available

[categories]

Date: Tue, 5 Aug 1997 17:19:51 -0400
From: Walter Tholen <tholen@mathstat.yorku.ca>

The following two preprints (joint work with George Janelidze) are available as
postscript files from my home page at
http://www.math.yorku.ca/Who/Faculty/Tholen/menu.html
For titles and abstracts, see below.

Walter Tholen

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

"Functorial Factorization, Well-pointedness and Separabilty"

Abstract: A functorial treatment of factorization structures is presented,
under extensive use of well-pointed endofunctors. Actually, so-called weak
factorization systems are interpreted as pointed lax indexed endofunctors,
and this sheds new light on the correspondence between reflective subcategories
and factorization systems. The second part of the paper presents two improtant
factorization structures in the context of pointed endofunctors:
concordant-dissonant and inseparable-sepaprable.


"Extended Galois Theory And Dissonant Morphisms"

Abstract: For a given Galois structure on a category C and an effective descent
morphism p: E --> B in C we describe the category of so-called weakly split
objects over (E,p) in terms of internal actions of the Galois (pre)groupoid of
(E,p) with an additional structure. We explain that this generates various
known results in categorical Galois theory and in particular two results of M.
Barr and R. Diaconescu. We also give an elaborate list of examples and
applications.