TAC: Abstracts from Volume 3, 1997

TAC: Abstracts from Volume 3, 1997

[categories]

Date: Mon, 19 Jan 1998 17:31:10 -0400 (AST)
From: Bob Rosebrugh <rrosebru@mta.ca>

Following are a table of contents and abstracts of articles in Volume 3 of
Theory and Applications of Categories. They are accessible on the Web from

www.tac.mta.ca/tac/

or by ftp from 

ftp.tac.mta.ca/pub/tac/html/volumes/1997

Submission of articles to any of the Editors (who are listed after the
abstracts) is invited. Please consult the Information for Authors at the
Web or ftp sites. For subscription write to tac@mta.ca including a postal
address. 

Robert Rosebrugh, Managing Editor                 http://www.tac.mta.ca/tac
Theory and Applications of Categories                            tac@mta.ca
Department of Mathematics and Computer Science       
67 York Street                     ********NEW STREET ADDRESS********
Sackville, NB E4L 1E6              ********NEW POSTAL CODE***********
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+1-506-364-2530                                   (fax)+1-506-364-2645



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

                                                        ISSN 1201-561X

                  THEORY AND APPLICATIONS OF CATEGORIES

                             Volume 3 - 1997 


Higher dimensional Peiffer elements in simplicial commutative algebras, 
     Z. Arvasi and T. Porter,                                             1

Doctrines whose structure forms a fully faithful adjoint string,
     F. Marmolejo,                                                       23

Note on a theorem of Putnam's, 
     Michael Barr,                                                       45

Lax operad actions and coherence for monoidal n-categories, A_{\infty}
rings and modules,
     Gerald Dunn,                                                        50

Proof theory for full intuitionistic linear logic, bilinear logic, and
MIX categories,
     J.R.B. Cockett and R.A.G. Seely,                                    85

The reflectiveness of covering morphisms in algebra and geometry, 
     George Janelidze and Max Kelly,                                    132

Crossed squares and 2-crossed modules of commutative algebras,
     Zekeriya Arvasi,                                                   160

Monads and interpolads in bicategories,
     J"urgen Koslowski,                                                 182

On property-like structures,
     G. M. Kelly and Stephen Lack,                                      213

Closed model categories for [n,m]-types,
     J. Ignacio Extremiana Aldana, Luis J. Hernandez Paricio, and M.
     Teresa Rivas Rodriguez,                                            251

Multilinearity of Sketches,
     David B. Benson,                                                   269



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

ABSTRACTS:


Higher Dimensional Peiffer Elements in Simplicial Commutative
Algebras 

Z. Arvasi and T. Porter 

Let E be a simplicial commutative algebra such that E_n is generated by
degenerate elements. It is shown that in this case the n^th term of the
Moore complex of E is generated by images of certain pairings from lower
dimensions. This is then used to give a description of the boundaries in
dimension n-1 for n = 2, 3, and 4. 

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

Doctrines whose structure forms a fully faithful adjoint string 

F. Marmolejo 

We pursue the definition of a KZ-doctrine in terms of a fully faithful
adjoint string Dd -| m -| dD. We give the definition in any Gray-category.
The concept of algebra is given as an adjunction with invertible counit.
We show that these doctrines are instances of more general pseudomonads.
The algebras for a pseudomonad are defined in more familiar terms and
shown to be the same as the ones defined as adjunctions when we start with
a KZ-doctrine. 


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

Note on a theorem of Putnam's 

Michael Barr 

In a 1981 book, H. Putnam claimed that in a pure relational language
without equality, for any model of a relation that was neither empty nor
full, there was another model that satisfies the same first order
sentences. Ed Keenan observed that this was false for finite models since
equality is a definable predicate in such cases. This note shows that
Putnam's claim is true for infinite models, although it requires a more
sophisticated proof than the one outlined by Putnam. 

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

Lax Operad Actions and Coherence for Monoidal n-Categories,
A_{\infty} Rings and Modules 

Gerald Dunn 

We establish a general coherence theorem for lax operad actions on an
n-category which implies that an n-category with such an action is lax
equivalent to one with a strict action. This includes familiar coherence
results (e.g. for symmetric monoidal categories) and many new ones. In
particular, any braided monoidal n-category is lax equivalent to a strict
braided monoidal n-category. We also obtain coherence theorems for
A_{\infty} and E_{\infty} rings and for lax modules over such rings. 
Using these results we give an extension of Morita equivalence to
A_{\infty} rings and some applications to infinite loop spaces and
algebraic K-theory. 

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

Proof theory for full intuitionistic linear logic, bilinear logic, and
MIX categories 

J.R.B. Cockett and R.A.G. Seely 

This note applies techniques we have developed to study coherence in
monoidal categories with two tensors, corresponding to the tensor-par
fragment of linear logic, to several new situations, including Hyland and
de Paiva's Full Intuitionistic Linear Logic (FILL), and Lambek's Bilinear
Logic (BILL). Note that the latter is a noncommutative logic; we also
consider the noncommutative version of FILL. The essential difference
between FILL and BILL lies in requiring that a certain tensorial strength
be an isomorphism. In any FILL category, it is possible to isolate a full
subcategory of objects (the ``nucleus'') for which this transformation is
an isomorphism. In addition, we define and study the appropriate
categorical structure underlying the MIX rule. For all these structures,
we do not restrict consideration to the ``pure'' logic as we allow
non-logical axioms. We define the appropriate notion of proof nets for
these logics, and use them to describe coherence results for the
corresponding categorical structures. 

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

The reflectiveness of covering morphisms in algebra and
geometry 

G. Janelidze and G. M. Kelly 

Each full reflective subcategory X of a finitely-complete category C gives
rise to a factorization system (E, M) on C, where E consists of the
morphisms of C inverted by the reflexion I : C --> X. Under a simplifying
assumption which is satisfied in many practical examples, a morphism f : A
--> B lies in M precisely when it is the pullback along the unit \etaB : B
--> IB of its reflexion If : IA --> IB;  whereupon f is said to be a
trivial covering of B. Finally, the morphism f : A --> B is said to be a
covering of B if, for some effective descent morphism p : E --> B, the
pullback p^*f of f along p is a trivial covering of E. This is the
absolute notion of covering; there is also a more general relative one,
where some class \Theta of morphisms of C is given, and the class Cov(B)
of coverings of B is a subclass -- or rather a subcategory -- of the
category C \downarrow B \subset C/B whose objects are those f : A --> B
with f in \Theta. Many questions in mathematics can be reduced to asking
whether Cov(B) is reflective in C \downarrow B; and we give a number of
disparate conditions, each sufficient for this to be so. In this way we
recapture old results and establish new ones on the reflexion of local
homeomorphisms into coverings, on the Galois theory of commutative rings,
and on generalized central extensions of universal algebras. 

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

Crossed squares and 2-crossed modules of commutative
algebras 

Zekeriya Arvasi 

In this paper, we construct a neat description of the passage from crossed
squares of commutative algebras to 2-crossed modules analogous to that
given by Conduche in the group case.  We also give an analogue, for
commutative algebra, of T. Porter's simplicial groups to n-cubes of groups
which implies an inverse functor to Conduche's one. 

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

Monads and interpolads in bicategories 

Jurgen Koslowski 

Given a bicategory, 2, with stable local coequalizers, we construct a
bicategory of monads Y-mnd by using lax functors from the generic 0-cell,
1-cell and 2-cell, respectively, into Y. Any lax functor into Y factors
through Y-mnd and the 1-cells turn out to be the familiar bimodules. The
locally ordered bicategory rel and its bicategory of monads both fail to
be Cauchy-complete, but have a well-known Cauchy-completion in common.
This prompts us to formulate a concept of Cauchy-completeness for
bicategories that are not locally ordered and suggests a weakening of the
notion of monad. For this purpose, we develop a calculus of general
modules between unstructured endo-1-cells. These behave well with respect
to composition, but in general fail to have identities. To overcome this
problem, we do not need to impose the full structure of a monad on
endo-1-cells. We show that associative coequalizing multiplications
suffice and call the resulting structures interpolads. Together with
structure-preserving i-modules these form a bicategory Y-int that is
indeed Cauchy-complete, in our sense, and contains the bicategory of
monads as a not necessarily full sub-bicategory. Interpolads over rel are
idempotent relations, over the suspension of set they correspond to
interpolative semi-groups, and over spn they lead to a notion of
``category without identities'' also known as ``taxonomy''. If Y locally
has equalizers, then modules in general, and the bicategories Y-mnd and
Y-int in particular, inherit the property of being closed with respect to
1-cell composition. 

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

On property-like structures 

G. M. Kelly and Stephen Lack 

A category may bear many monoidal structures, but (to within a unique
isomorphism) only one structure of `category with finite products'. To
capture such distinctions, we consider on a 2-category those 2-monads for
which algebra structure is essentially unique if it exists, giving a
precise mathematical definition of `essentially unique' and investigating
its consequences. We call such 2-monads property-like. We further consider
the more restricted class of fully property-like 2-monads, consisting of
those property-like 2-monads for which all 2-cells between (even lax)
algebra morphisms are algebra 2-cells. The consideration of lax morphisms
leads us to a new characterization of those monads, studied by Kock and
Zoberlein, for which `structure is adjoint to unit', and which we now call
lax-idempotent 2-monads: both these and their colax-idempotent duals are
fully property-like. We end by showing that (at least for finitary
2-monads) the classes of property-likes, fully property-likes, and
lax-idempotents are each coreflective among all 2-monads. 

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

Closed model categories for [n,m] types 

J. Ignacio Extremiana Aldana, Luis J. Hernandez Paricio, Maria T. Rivas
Rodriguez 

For m >= n > 0, a map f between pointed spaces is said to be a weak
[n,m]-equivalence if f induces isomorphisms of the homotopy groups \pi_k
for n <= k <= m~. Associated with this notion we give two different closed
model category structures to the category of pointed spaces. Both
structures have the same class of weak equivalences but different classes
of fibrations and therefore of cofibrations. Using one of these
structures, one obtains that the localized category is equivalent to the
category of n-reduced CW-complexes with dimension less than or equal to
m+1 and m-homotopy classes of cellular pointed maps. Using the other
structure we see that the localized category is also equivalent to the
homotopy category of (n-1)-connected (m+1)-coconnected CW-complexes. 

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

Multilinearity of Sketches 

David B. Benson 

We give a precise characterization for when the models of the tensor
product of sketches are structurally isomorphic to the models of either
sketch in the models of the other. For each base category K call the just
mentioned property (sketch) K-multilinearity. Say that two sketches are
K-compatible with respect to base category K just in case in each K-model,
the limits for each limit specification in each sketch commute with the
colimits for each colimit specification in the other sketch and all limits
and colimits are pointwise. Two sketches are K-multilinear if and only if
the two sketches are K-compatible. This property then extends to strong
Colimits of sketches. 

We shall use the technically useful property of limited completeness and
completeness of every category of models of sketches. That is, categories
of sketch models have all limits commuting with the sketched colimits and
and all colimits commuting with the sketched limits. Often used
implicitly, the precise statement of this property and its proof appears
here. 


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

Editors of Theory and Applications of Categories

John Baez, University of California Riverside 
     baez@math.ucr.edu 
Michael Barr, McGill University 
     barr@math.mcgill.ca 
Lawrence Breen, Universite de Paris 13 
     breen@math.univ-paris13.fr 
Ronald Brown , University of North Wales 
     r.brown@bangor.ac.uk 
Jean-Luc Brylinski, Pennsylvania State University 
     jlb@math.psu.edu 
Aurelio Carboni, Universita della Calabria
     carboni@unical.it 
Peter T. Johnstone, University of Cambridge 
     ptj@pmms.cam.ac.uk 
G. Max Kelly, University of Sydney 
     kelly_m@maths.usyd.edu.au 
Anders Kock, University of Aarhus 
     kock@mi.aau.dk 
F. William Lawvere, State University of New York at Buffalo 
     wlawvere@acsu.buffalo.edu 
Jean-Louis Loday, Universite Louis Pasteur et CNRS, Strasbourg 
     loday@math.u-strasbg.fr 
Ieke Moerdijk, University of Utrecht 
     moerdijk@math.ruu.nl 
Susan Niefield , Union College 
     niefiels@union.edu 
Robert Pare, Dalhousie University 
     pare@mscs.dal.ca 
Andrew Pitts , University of Cambridge 
     ap@cl.cam.ac.uk 
Robert Rosebrugh , Mount Allison University 
     rrosebrugh@mta.ca 
Jiri Rosicky, Masaryk University 
     rosicky@math.muni.cz 
James Stasheff , University of North Carolina 
     jds@charlie.math.unc.edu 
Ross Street , Macquarie University 
     street@macadam.mpce.mq.edu.au 
Walter Tholen , York University 
     tholen@mathstat.yorku.ca 
Myles Tierney, Rutgers University 
     tierney@math.rutgers.edu 
Robert F. C. Walters , University of Sydney 
     walters_b@maths.usyd.edu.au 
R. J. Wood, Dalhousie University 
     rjwood@mscs.dal.ca