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600 |
Nuclear and Trace Ideals in Tensored *-Categories
|
math.CT
|
We generalize the notion of nuclear maps from functional analysis by defining
nuclear ideals in tensored *-categories. The motivation for this study came
from attempts to generalize the structure of the category of relations to
handle what might be called ``probabilistic relations''. The compact closed
structure associated with the category of relations does not generalize
directly, instead one obtains nuclear ideals. We introduce the notion of
nuclear ideal to analyze these classes of morphisms. In compact closed
categories, we see that all morphisms are nuclear, and in the category of
Hilbert spaces, the nuclear morphisms are the Hilbert-Schmidt maps.
We also introduce two new examples of tensored *-categories, in which
integration plays the role of composition. In the first, morphisms are a
special class of distributions, which we call tame distributions. We also
introduce a category of probabilistic relations which was the original
motivating example.
Finally, we extend the recent work of Joyal, Street and Verity on traced
monoidal categories to this setting by introducing the notion of a trace ideal.
For a given symmetric monoidal category, it is not generally the case that
arbitrary endomorphisms can be assigned a trace. However, we can find ideals in
the category on which a trace can be defined satisfying equations analogous to
those of Joyal, Street and Verity. We establish a close correspondence between
nuclear ideals and trace ideals in a tensored *-category, suggested by the
correspondence between Hilbert-Schmidt operators and trace operators on a
Hilbert space.
|
math
|
601 |
Basic Bicategories
|
math.CT
|
A concise guide to very basic bicategory theory, from the definition of a
bicategory to the coherence theorem.
|
math
|
602 |
Applications of Rewriting Systems and Groebner Bases to Computing Kan Extensions and Identities Among Relations
|
math.CT
|
This thesis concentrates on the development and application of rewriting and
Groebner basis methods to a range of combinatorial problems.
Chapter Two contains the most important result, which is the application of
Knuth-Bendix procedures to Kan extensions, showing how rewriting provides a
useful method for attempting to solve a variety of combinatorial problems which
can be phrased in terms of Kan extensions.
Chapter Three shows that the standard Knuth-Bendix algorithm is step-for-step
a special case of Buchberger's algorithm. The one-sided cases and higher
dimensions are considered.
Chapter Four relates rewrite systems, Groebner bases and automata. Automata
which only accept irreducibles, and automata which output reduced forms are
discussed for presentations of Kan extensions. Reduction machines for rewrite
systems are identified with standard output automata and the reduction machines
devised for algebras are expressed as Petri nets.
Chapter Five uses the completion of a group rewriting system to
algorithmically determine a contracting homotopy necessary in order to compute
the set of generators for the module of identities among relations using the
covering groupoid methods devised by Brown and Razak Salleh. Reducing the
resulting set of submodule generators is identified as a Groebner basis
problem. Algorithms are implemented in GAP3.
|
math
|
603 |
K-Theory for Triangulated Categories III(A): The Theorem of the Heart
|
math.CT
|
This is the fourth installment of a series. The main point of the entire
series is the following: given a triangulated category T, it is possible to
attach to it a K-theory space.
|
math
|
604 |
fc-multicategories
|
math.CT
|
fc-multicategories are a very general kind of two-dimensional structure,
encompassing bicategories, monoidal categories, double categories and ordinary
multicategories. We define them and explain how they provide a natural setting
for two familiar categorical ideas. The first is the bimodules construction,
traditionally carried out on suitably cocomplete bicategories but perhaps more
naturally carried out on fc-multicategories. The second is enrichment: there is
a theory of categories enriched in an fc-multicategory, extending the usual
theory of enrichment in a monoidal category. We finish by indicating how this
work is just the simplest case of a much larger phenomenon.
|
math
|
605 |
On Ideals and Homology in Additive Categories
|
math.CT
|
Ideals are used to define homological functors for additive categories. In
abelian categories the ideals corresponding to the usual universal objects are
principal, and the construction reduces, in a choice dependent way, to homology
groups.
Applications are considered: derived categories and functors.
|
math
|
606 |
Grothendieck Categories
|
math.CT
|
The general theory of Grothendieck categories is presented. We systemize the
principle methods and results of the theory, showing how these results can be
used for studying rings and modules.
|
math
|
607 |
Algebraic duality for partially ordered sets
|
math.CT
|
For an arbitrary partially ordered set $P$ its {\em dual} $P^*$ is built as
the collection of all monotone mappings $P\to\2$ where $\2=\{0,1\}$ with $0<1$.
The set of mappings $P^*$ is proved to be a complete lattice with respect to
the pointwise partial order. The {\em second dual} $P^{**}$ is built as the
collection of all morphisms of complete lattices $P^*\to\2$ preserving
universal bounds. Then it is proved that the partially ordered sets $P$ and
$P^{**}$ are isomorphic.
|
math
|
608 |
Coherence in Substructural Categories
|
math.CT
|
It is proved that MacLane's coherence results for monoidal and symmetric
monoidal categories can be extended to some other categories with
multiplication; namely, to relevant, affine and cartesian categories. All
results are formulated in terms of natural transformations equipped with
``graphs'' (g-natural transformations), and corresponding morphism theorems are
given as consequences. Using these results, some basic relations between the
free categories of these classes are obtained.
|
math
|
609 |
From Coherent Structures to Universal Properties
|
math.CT
|
Given a 2-category $\twocat{K}$ admitting a calculus of bimodules, and a
2-monad T on it compatible with such calculus, we construct a 2-category
$\twocat{L}$ with a 2-monad S on it such that: (1)S has the
adjoint-pseudo-algebra property. (2)The 2-categories of pseudo-algebras of S
and T are equivalent. Thus, coherent structures (pseudo-T-algebras) are
transformed into universally characterised ones (adjoint-pseudo-S-algebras).
The 2-category $\twocat{L}$ consists of lax algebras for the pseudo-monad
induced by T on the bicategory of bimodules of $\twocat{K}$. We give an
intrinsic characterisation of pseudo-S-algebras in terms of representability.
Two major consequences of the above transformation are the classifications of
lax and strong morphisms, with the attendant coherence result for
pseudo-algebras. We apply the theory in the context of internal categories and
examine monoidal and monoidal globular categories (including their monoid
classifiers) as well as pseudo-functors into $\Cat$.
|
math
|
610 |
On the Galois Theory of Grothendieck
|
math.CT
|
In this paper we deal with Grothendieck's interpretation of Artin's
interpretation of Galois's Galois Theory (and its natural relation with the
fundamental group and the theory of coverings) as he developed it in Expose V,
section 4, ``Conditions axiomatiques d'une theorie de Galois'' in the SGA1
1960/61.
This is a beautiful piece of mathematics very rich in categorical concepts,
and goes much beyond the original Galois's scope (just as Galois went much
further than the non resubility of the quintic equation). We show explicitly
how Grothendieck's abstraction corresponds to Galois work.
We introduce some axioms and prove a theorem of characterization of the
category (topos) of actions of a discrete group. This theorem corresponds
exactly to Galois fundamental result. The theorem of Grothendieck characterizes
the category (topos) of continuous actions of a profinite topological group. We
develop a proof of this result as a "passage into the limit'' (in an inverse
limit of topoi) of our theorem of characterization of the topos of actions of a
discrete group. We deal with the inverse limit of topoi just working with an
ordinary filtered colimit (or union) of the small categories which are their
(respective) sites of definition.
We do not consider generalizations of Grothendieck's work, except by
commenting briefly in the last section how to deal with the prodiscrete (not
profinite) case. We also mention the work of Joyal-Tierney, which falls
naturally in our discussion.
There is no need of advanced knowledge of category theory to read this paper,
exept for the comments in the last section.
|
math
|
611 |
A relative Yoneda Lemma (manuscript)
|
math.CT
|
We construct set-valued right Kan-extensions via a relative Yoneda Lemma.
|
math
|
612 |
Localic Galois Theory
|
math.CT
|
In Proposition I of "Memoire sur les conditions de resolubilite des equations
par radicaux", Galois established that any intermediate extension of the
splitting field of a polynomial with rational coefficients is the fixed field
of its galois group.
We first state and prove the (dual) categorical interpretation of of this
statement, which is a theorem about atomic sites with a representable point. In
the general case, the point determines a proobject and it becomes
(tautologically) prorepresentable. We state and prove the, mutatus mutatis,
prorepresentable version of Galois theorem. In this case the classical group of
automorphisms has to be replaced by the localic group of automorphisms. These
developments form the content of a theory that we call "Localic Galois Theory".
An straightforward corollary of this theory is the theorem: "A topos with a
point is connected atomic if and only if it is the classifying topos of a
localic group, and this group can be taken to be the locale of automorphisms of
the point". This theorem was first proved in print in Joyal A, Tierney M. "An
extension of the Galois Theory of Grothendieck", Mem. AMS 151, Theorem 1,
Section 3, Chapter VIII. Our proof is completely independent of descent theory
and of any other result in that paper.
|
math
|
613 |
On the monad of proper factorisation systems in categories
|
math.CT
|
It is known that factorisation systems in categories can be viewed as unitary
pseudo algebras for the "squaring" monad in Cat.
We show in this note that an analogous fact holds for proper (i.e., epi-mono)
factorisation systems and a suitable quotient of the former monad, deriving
from a construct introduced by P. Freyd for stable homotopy.
Structural similarities of the previous monad with the path endofunctor of
topological spaces are considered.
|
math
|
614 |
On Ext in the Category of Functors to Preabelian Category
|
math.CT
|
The work is devoted to the extension groups in the category of functors from
a small category to an additive category with an Abelian structure in the sense
of Heller. It is constructed a spectral sequence which converges to the
extension group. Example for diagrams of locally convex spaces is given.
|
math
|
615 |
n-Categories Admissible by n-graph
|
math.CT
|
The concept of n-categories and related subject is considered. An n-category
is described as an n-graph with a composition. A new definition of operad is
presented. Some illustrative examples are given.
|
math
|
616 |
Calculating limits and colimits in pro-categories
|
math.CT
|
We present some constructions of limits and colimits in pro-categories. These
are critical tools in several applications. In particular, certain technical
arguments concerning strict pro-maps are essential for a theorem about \'etale
homotopy types. Also, we show that cofiltered limits in pro-categories commute
with finite colimits.
|
math
|
617 |
Flows in Graphs and Homology of Free Categories
|
math.CT
|
We introduce the notion of a generalized flow on a graph with coefficients in
a R-representation and show that the module of flows is isomorphic to the first
derived functor of the colimit. We generalize Kirchhoff's laws and build an
exact sequence for calculating the module of flows on the union of graphs.
|
math
|
618 |
Cohomologie non abelienne d'ordre superieur et applications
|
math.CT
|
In this paper we propose a higher non abelian cohomology theory without using
the notion of n-category. We use this to study compositions series of affine
manifolds and cohomology of manifolds.
|
math
|
619 |
Structures in higher-dimensional category theory
|
math.CT
|
This paper, written in 1998, aims to clarify various higher categorical
structures, mostly through the theory of generalized operads and
multicategories. Chapters I and II, which cover this theory and its application
to give a definition of weak n-category, are largely superseded by my thesis
(math.CT/0011106), but Chapters III and IV have not appeared elsewhere. The
main result of Chapter III is that small Gray-categories can be characterized
as the sub-tricategories of the tricategory of 2-categories, homomorphisms,
strong transformations and modifications; there is also a conjecture on
coherence in higher dimensions. Chapter IV defines opetopes and a category of
n-pasting diagrams for each n, which in the case n=2 is a definition of the
category of trees.
|
math
|
620 |
Some properties of the theory of n-categories
|
math.CT
|
Let $L_n$ denote the Dwyer-Kan localization of the category of weak
n-categories divided by the n-equivalences. We propose a list of properties
that this simplicial category is likely to have, and conjecture that these
properties characterize $L_n$ up to equivalence. We show, using these
properties, how to obtain the morphism $n-1$-categories between two points in
an object of $L_n$ and how to obtain the composition map between the morphism
objects.
|
math
|
621 |
On the Structure of Modular Categories
|
math.CT
|
For a braided tensor category C and a subcategory K there is a notion of
centralizer C_C(K), which is a full tensor subcategory of C. A pre-modular
tensor category is known to be modular in the sense of Turaev iff the center
Z_2(C):=C_C(C) (not to be confused with the center Z_1 of a tensor category,
related to the quantum double) is trivial, i.e. equivalent to Vect, and
dim(C)<>0. Here dim(C)=sum_i d(X_i)^2, the X_i being the simple objects. We
prove the following double centralizer theorem: Let C be a modular category and
K a full tensor subcategory closed w.r.t. direct sums, subobjects and duals.
Then C_C(C_C(K))=K and dim(K)dim(C_C(K))=dim(C). We give several applications,
the most important being the following. If C is modular and K is a full modular
subcategory, then also L=C_C(K) is modular and C is equivalent as a ribbon
category to the direct product of K and L. Thus every modular category
factorizes (non-uniquely, in general) into prime ones. We study the prime
factorizations of the categories D(G)-Mod, where G is a finite abelian group.
|
math
|
622 |
Pushout stability of embeddings, injectivity and categories of algebras
|
math.CT
|
In several familiar subcategories of the category ${\mathbb T}$ of
topological spaces and continuous maps, embeddings are not pushout-stable. But,
an interesting feature, capturable in many categories, namely in categories
$\mathcal{B}$ of topological spaces, is the following: For $\mathcal{M}$ the
class of all embeddings, the subclass of all pushout-stable
$\mathcal{M}$-morphisms (that is, of those $\mathcal{M}$-morphisms whose
pushout along an arbitrary morphism always belongs to $\mathcal{M}$) is of the
form $A^{Inj}$ for some space $A$, where $A^{Inj}$ consists of all morphisms
$m:X \to Y$ such that the map $Hom(m,A): Hom(Y,A) \to Hom(X,A)$ is surjective.
We study this phenomenon. We show that, under mild assumptions, the reflective
hull of such a space $A$ is the smallest $\mathcal{M}$-reflective subcategory
of $\mathcal{B}$; furthermore, the opposite category of this reflective hull is
equivalent to a reflective subcategory of the Eilenberg-Moore category
$Set^{\mathbb T}, where ${\mathbb T}$ is the monad induced by the right adjoint
$Hom(-,A): {\mathbb T}^{op} \to Set$. We also find conditions on a category
$\mathcal{B}$ under which the pushout-stable $\mathcal{M}$-morphisms are of the
form $\mathcal{A}^{Inj}$ for some category $\mathcal{A}$.
|
math
|
623 |
Generalized enrichment of categories
|
math.CT
|
We define the phrase `category enriched in an fc-multicategory' and explore
some examples. An fc-multicategory is a very general kind of 2-dimensional
structure, special cases of which are double categories, bicategories, monoidal
categories and ordinary multicategories. Enrichment in an fc-multicategory
extends the (more or less well-known) theories of enrichment in a monoidal
category, in a bicategory, and in a multicategory. Moreover, fc-multicategories
provide a natural setting for the bimodules construction, traditionally
performed on suitably cocomplete bicategories. Although this paper is
elementary and self-contained, we also explain why, from one point of view,
fc-multicategories are the natural structures in which to enrich categories.
|
math
|
624 |
On Regular Closure Operators and Cowellpowered Subcategories
|
math.CT
|
Many Properties of a category X, as for instance the existence of an adjoint
or a factorization system, are a consequence of the cowellpoweredness of X. In
the absence of cowellpoweredness, for general results, fairly strong assumption
on the category are needed. This paper provides a number of novel and useful
observations to tackle the cowellpoweredness problem of subcategories by means
of regular closure operators. Our exposition focusses on the question when two
subcategories A and B induce the same regular closure operators, then
information about (non)-cowellpoweredness of A may be gained from corresponding
property of B, and vice versa.
|
math
|
625 |
The omega-Categories Associated With Products of Infinite-Dimensional Globes
|
math.CT
|
This thesis studies the omega-categories associated with products of
infinite-dimensional globes.
|
math
|
626 |
On the representation theory of Galois and Atomic Topoi
|
math.CT
|
We elaborate on the representation theorems of topoi as topoi of discrete
actions of various kinds of localic groups and groupoids. We introduce the
concept of "proessential point" and use it to give a new characterization of
pointed Galois topoi. We establish a hierarchy of connected topoi:
[1. essentially pointed Atomic = locally simply connected],
[2. proessentially pointed Atomic = pointed Galois],
[3. pointed Atomic].
These topoi are the classifying topos of, respectively: 1. discrete groups,
2. prodiscrete localic groups, and 3. general localic groups.
We analyze also the unpoited version, and show that for a Galois topos, may
be pointless, the corresponding groupoid can also be considered, in a sense,
the groupoid of "points". In the unpointed theories, these topoi classify,
respectively: 1. connected discrete groupoids, 2. connected (may be pointless)
prodiscrete localic groupoids, and 3. connected groupoids with discrete space
of objects and general localic spaces of hom-sets, when the topos has points
(we do not know the class of localic groupoids that correspond to pointless
connected atomic topoi).
We comment and develop on Grothendieck's galois theory and its generalization
by Joyal-Tierney, and work by other authors on these theories.
|
math
|
627 |
Computads and slices of operads
|
math.CT
|
For a given $\omega$-operad $A$ on globular sets we introduce a sequence of
symmetric operads on $Set$ called slices of $A$ and show how the connected
limit preserving properties of slices are related to the property of the
category of $n$-computads of $A$ being a presheaf topos.
|
math
|
628 |
Galois extensions of braided tensor categories and braided crossed G-categories
|
math.CT
|
We show that the author's notion of Galois extensions of braided tensor
categories [22], see also [3], gives rise to braided crossed G-categories,
recently introduced for the purposes of 3-manifold topology [31]. The Galois
extensions C \rtimes S are studied in detail, and we determine for which g in G
non-trivial objects of grade g exist in C \rtimes S.
|
math
|
629 |
Group Objects and Internal Categories
|
math.CT
|
Algebraic structures such as monoids, groups, and categories can be
formulated within a category using commutative diagrams. In many common
categories these reduce to familiar cases. In particular, group objects in Grp
are abelian groups, while internal categories in Grp are equivalent both to
group objects in Cat and to crossed modules of groups. In this exposition we
give an elementary introduction to some of the key concepts in this area.
|
math
|
630 |
Remarks on 2-Groups
|
math.CT
|
A 2-group is a `categorified' version of a group, in which the underlying set
G has been replaced by a category and the multiplication map m: G x G -> G has
been replaced by a functor. A number of precise definitions of this notion have
already been explored, but a full treatment of their relationships is difficult
to extract from the literature. Here we describe the relation between two of
the most important versions of this notion, which we call `weak' and `coherent'
2-groups. A weak 2-group is a weak monoidal category in which every morphism
has an inverse and every object x has a `weak inverse': an object y such that x
tensor y and y tensor x are isomorphic to 1. A coherent 2-group is a weak
2-group in which every object x is equipped with a specified weak inverse x*
and isomorphisms i_x: 1 -> x tensor x*, e_x: x* tensor x -> 1 forming an
adjunction. We define 2-categories of weak and coherent 2-groups and construct
an `improvement' 2-functor which turns weak 2-groups into coherent ones; using
this one can show that these 2-categories are biequivalent. We also internalize
the concept of a coherent 2-group. This gives a way of defining topological
2-groups, Lie 2-groups, and the like.
|
math
|
631 |
Non abelian cohomology: the point of view of gerbed tower
|
math.CT
|
In this paper we define a notion of gerbed tower, and use this notion to give
a geometric representation of cohomological classes.
|
math
|
632 |
Paracategories I: internal parategories and saturated partial algebras
|
math.CT
|
Based on the monoid classifier, we give an alternative axiomatization of
Freyd's paracategories, which can be interpreted in any bicategory of partial
maps. Assuming furthermore a free-monoid monad T in our ambient category, and
coequalisers satisfying some exactness conditions, we give an abstract envelope
construction, putting paramonoids (and paracategories) in the more general
context of partial algebras. We introduce for the latter the crucial notion of
saturation, which characterises those partial algebras which are isomorphic to
the ones obtained from their enveloping algebras. We also set up a
factorisation system for partial algebras, via epimorphisms and (monic) Kleene
morphisms and relate the latter to saturation.
|
math
|
633 |
Some calculus with extensive quantities: wave equation
|
math.CT
|
We take some first steps in providing a synthetic theory of distributions. In
particular, we are interested in the use of distribution theory as foundation,
not just as tool, in the study of the wave equation.
|
math
|
634 |
The monoidal centre as a limit
|
math.CT
|
The centre of a monoidal category is a braided monoidal category. Monoidal
categories are monoidal objects (or pseudomonoids) in the monoidal bicategory
of categories. This paper provides a universal construction in a braided
monoidal bicategory that produces a braided monoidal object from any monoidal
object. Some properties and sufficient conditions for existence of the
construction are examined.
|
math
|
635 |
Weak n-categories: opetopic and multitopic foundations
|
math.CT
|
We generalise the concepts introduced by Baez and Dolan to define opetopes
constructed from symmetric operads with a category, rather than a set, of
objects. We describe the category of 1-level generalised multicategories, a
special case of the concept introduced by Hermida, Makkai and Power, and
exhibit a full embedding of this category in the category of symmetric operads
with a category of objects. As an analogy to the Baez-Dolan slice construction,
we exhibit a certain multicategory of function replacement as a slice
construction in the multitopic setting, and use it to construct multitopes. We
give an explicit description of the relationship between opetopes and
multitopes.
|
math
|
636 |
Weak n-categories: comparing opetopic foundations
|
math.CT
|
We define the category of tidy symmetric multicategories. We construct for
each tidy symmetric multicategory Q a cartesian monad (E_Q,T_Q) and extend this
assignation to a functor. We exhibit a relationship between the slice
construction on symmetric multicategories, and the `free operad' monad
construction on suitable monads. We use this to give an explicit description of
the relationship between Baez-Dolan and Leinster opetopes.
|
math
|
637 |
The category of opetopes and the category of opetopic sets
|
math.CT
|
We give an explicit construction of the category Opetope of opetopes. We
prove that the category of opetopic sets is equivalent to the category of
presheaves over Opetope.
|
math
|
638 |
Opetopic bicategories: comparison with the classical theory
|
math.CT
|
We continue our previous modifications of the Baez-Dolan theory of opetopes
to modify the Baez-Dolan definition of universality, and thereby the category
of opetopic n-categories and lax functors. For the case n=2 we exhibit an
equivalence between this category and the category of bicategories and lax
functors. We examine notions of strictness in the opetopic theory.
|
math
|
639 |
An alternative characterisation of universal cells in opetopic n-categories
|
math.CT
|
We address the fact that composition in an opetopic weak n-category is in
general not unique and hence is not a well-defined operation. We define
composition with a given k-cell in an n-category by a span of (n-k)-categories.
We characterise such a cell as universal if its composition span gives an
equivalence of (n-k)-categories.
|
math
|
640 |
A relationship between trees and Kelly-Mac Lane graphs
|
math.CT
|
We give a precise description of combed trees in terms of Kelly-Mac Lane
graphs. We show that any combed tree is uniquely expressed as an allowable
Kelly-Mac Lane graph of a certain shape. Conversely, we show that any such
Kelly-Mac Lane graph uniquely defines a combed tree.
|
math
|
641 |
The theory of opetopes via Kelly-Mac Lane graphs
|
math.CT
|
This paper follows from two earlier works. In the first we gave an explicit
construction of opetopes, the underlying cell shapes in the theory of opetopic
n-categories; at the heart of this construction is the use of certain trees. In
the second we gave a description of trees using Kelly-Mac Lane graphs. In the
present paper we apply the latter to the former, to give a construction of
opetopes using Kelly-Mac Lane graphs.
|
math
|
642 |
Category Theory and Higher Dimensional Algebra: potential descriptive tools in neuroscience
|
math.CT
|
We explain the notion of colimit in category theory as a potential tool for
describing structures and their communication, and the notion of higher
dimensional algebra as a potential yoga for dealing with processes and
processes of processes.
|
math
|
643 |
A Guided Tour in the Topos of Graphs
|
math.CT
|
In this paper we survey the fundamental constructions of a presheaf topos in
the case of the elementary topos of graphs. We prove that the transition graphs
of nondeterministic automata (a.k.a. labelled transition systems) are the
separated presheaves for the double negation topology, and obtain as an
application that their category is a quasitopos.
|
math
|
644 |
Strengthening track theories
|
math.CT
|
Using cohomology of categories with coefficients in natural systems it is
proved that a groupoid enrichad category with pseudoproducts is
pseudoequivalent to one with strict products.
|
math
|
645 |
A generalization and a new proof of Plotkin's reduction theorem
|
math.CT
|
It is known that Plotkin's reduction theorem is very important for his theory
of universal algebraic geometry [arXiv:math. GM/0210187], [arXiv:math.
GM/0210194]. It turns out that this theorem can be generalized to arbitrary
categories containing two special objects and in this case its proof becomes
considerable more simple. This new proof and applications are the subject of
the present paper.
|
math
|
646 |
Flatness, preorders and general metric spaces
|
math.CT
|
This paper studies a general notion of flatness in the enriched context:
P-flatness where the parameter P stands for a class of presheaves. One obtains
a completion of a category A by considering the category Flat_P(A) of P-flat
presheaves over A. This completion is related to the free cocompletion of A
under a class of colimits defined by Kelly. For a category A, for P = P0 the
class of all presheaves, Flat_P0(A) is the Cauchy-completion of A. Two classes
P1 and P2 of general interest for general metric spaces are considered. The P1-
and P2-flatness are investigated and the associated completions are
characterized for general metric spaces (enrichemnts over R+) and preorders
(enrichments over Bool). We get this way two non-symmetric completions for
metric spaces and retrieve the ideal completion for preorders.
|
math
|
647 |
Grothendieck categories and support conditions
|
math.CT
|
We give examples of pairs (G1,G2) where G1 is a Grothendieck category and G2
a full Grothendieck subcategory of G1, the inclusion G2 --> G1 being denoted i,
for which R^+i : D^+G2 --> D^+G1 (or even Ri : DG2 --> DG1) is a full
embedding. This yields generalizations of some results of Bernstein and Lunts,
and of Cline, Parshall and Scott.
|
math
|
648 |
Monad interleaving: a construction of the operad for Leinster's weak $ω$-categories
|
math.CT
|
We show how to "interleave" the monad for operads and the monad for
contractions on the category \coll of collections, to construct the monad for
the operads-with-contraction of Leinster. We first decompose the adjunction for
operads and the adjunction for contractions into a chain of adjunctions each of
which acts on only one dimension of the underlying globular sets at a time. We
then exhibit mutual stability conditions that enable us to alternate the
dimension-by-dimension free functors. Hence we give an explicit construction of
a left adjoint for the forgetful functor $\owc \lra \coll$, from the category
of operads-with-contraction to the category of collections. By applying this to
the initial (empty) collection, we obtain explicitly an initial
operad-with-contraction, whose algebras are by definition the weak
$\omega$-categories of Leinster.
|
math
|
649 |
Free ${A}_\infty$-categories
|
math.CT
|
For a differential graded k-quiver Q we define the free A-infinity-category
FQ generated by Q. The main result is that for an arbitrary A-infinity-category
A the restriction A-infinity-functor A_\infty(FQ,A) -> A_1(Q,A) is an
equivalence, where objects of the last A-infinity-category are morphisms of
differential graded k-quivers Q -> A.
|
math
|
650 |
Homotopical structures in categories
|
math.CT
|
In this paper is presented a new approach to the axiomatic homotopy theory in
categories, which offers a simpler and more useful answer to this old question:
how two objects in a category (without any topological feature) can be deformed
each in other?
|
math
|
651 |
A duality Hopf algebra for holomorphic N=1 special geometries
|
math.CT
|
We find a self-dual noncommutative and noncocommutative Hopf algebra acting
as a universal symmetry on the modules over inner Frobenius algebras of modular
categories (as used in two dimensional boundary conformal field theory) similar
to the Grothendieck-Teichmueller group GT as introduced by Drinfeld as a
universal symmetry of quasitriangular quasi-Hopf algebras. We discuss the
relationship to a similar self-dual, noncommutative, and noncocommutative Hopf
algebra, previously found as the universal symmetry of trialgebras and three
dimensional extended topological quantum field theories. As an application of
our result, we get a transitive action of a sub-Hopf algebra of the latter
universal symmetry algebra on the relative period matrices of holomorphic N=1
special geometries.
|
math
|
652 |
Flatness, accessibility and metric spaces
|
math.CT
|
This paper studies a notion of parameterized flatness in the enriched
context: p-flatness where the parameter p stands for a class of presheaves. One
obtains a completion of a category A by considering the category F_p(A) of
p-flat presheaves over A. The completion is related to the free cocompletion
under a class of colimits defined by Kelly. We define a notion of Q-accessible
categories where Q is the class of p-flat indexes. For a category A, for p = P0
the class of all presheaves, F_P0(A) is the Cauchy-completion of A. Two classes
P1 and P2 of interest for general metric spaces and prorders are considered.
The F_P1- and F_P2- flatess are characterized yielding non-symmetric
completions of metric spaces a la Cauchy involving non-symmetric filters.
|
math
|
653 |
Omega-categories and chain complexes
|
math.CT
|
There are several ways to construct omega-categories from combinatorial
objects such as pasting schemes or parity complexes. We make these
constructions into a functor on a category of chain complexes with additional
structure, which we call augmented directed complexes. This functor from
augmented directed complexes to omega-categories has a left adjoint, and the
adjunction restricts to an equivalence on a category of augmented directed
complexes with good bases. The omega-categories equivalent to augmented
directed complexes with good bases include the omega-categories associated to
globes, simplexes and cubes; thus the morphisms between these omega-categories
are determined by morphisms between chain complexes. It follows that the entire
theory of omega-categories can be expressed in terms of chain complexes; in
particular we describe the biclosed monoidal structure on omega-categories and
calculate some internal homomorphism objects.
|
math
|
654 |
Les groupements
|
math.CT
|
Neocategories, semicategories, precategories are well-known generalizations
of categories. But they all suppose that sources and targets of morphisms
fulfilled identity conditions. Here we intend to suppress those conditions. In
doing this we get at the construction of a simple framework which seems
appropiate to study Moore surfaces and their possible extensions in higher
dimensions.
|
math
|
655 |
Tours de torseurs, geometrie differentielle des suites de fibres principaux, et theorie des cordes
|
math.CT
|
In this paper we interpret cohomological class using the notion of tower of
torsors, we apply our construction to string theory.
|
math
|
656 |
The Chu construction for complete atomistic coatomistic lattices
|
math.CT
|
The Chu construction is used to define a *-autonomous structure on a category
of complete atomistic coatomistic lattices. This construction leads to a new
tensor product that is compared with a certain number of other existing tensor
products.
|
math
|
657 |
A strict totally coordinatized version of Kapranov and Voevodsky's 2-category {\bf 2Vect}
|
math.CT
|
We give a concrete description of a strict totally coordinatized version of
Kapranov and Voevodsky's 2-category of finite dimensional 2-vector spaces. In
particular, we give explicit formulas for composition of 1-morphisms and the
two compositions between 2-morphisms
|
math
|
658 |
A Full and faithful Nerve for 2-categories
|
math.CT
|
The notion of geometric nerve of a 2-category (Street, \cite{refstreet})
provides a full and faithful functor if regarded as defined on the category of
2-categories and lax 2-functors. Furthermore, lax 2-natural transformations
between lax 2-functors give rise to homotopies between the corresponding
simplicial maps. These facts allow us to prove a representation theorem of the
general non abelian cohomology of groupoids (classifying non abelian extensions
of groupoids) by means of homotopy classes of simplicial maps.
|
math
|
659 |
State monads and their algebras
|
math.CT
|
State monads in cartesian closed categories are those defined by the familiar
adjunction between product and exponential. We investigate the structure of
their algebras, and show that the exponential functor is monadic provided the
base category is sufficiently regular, and the exponent is a non-empty object.
|
math
|
660 |
Enlargements of Categories
|
math.CT
|
In order to apply nonstandard methods to modern algebraic geometry, as a
first step in this paper we study the applications of nonstandard constructions
to category theory. It turns out that many categorial properties are well
behaved under enlargements.
|
math
|
661 |
Non-well-founded trees in categories
|
math.CT
|
Non-well-founded trees are used in mathematics and computer science, for
modelling non-well-founded sets, as well as non-terminating processes or
infinite data-structures. Categorically, they arise as final coalgebras for
polynomial endofunctors, which we call M-types. In order to reason about trees,
we need the notion of path, which can be formalised in the internal logic of
any locally cartesian closed pretopos with a natural number object. In such
categories, we derive existence results about M-types, leading to stability of
locally cartesian closed pretoposes with a natural number object and M-types
under slicing, formation of coalgebras (for a cartesian comonad), and sheaves
for an internal site.
|
math
|
662 |
Categorical structures enriched in a quantaloid: categories, distributors and functors
|
math.CT
|
We thoroughly treat several familiar and less familiar definitions and
results concerning categories, functors and distributors enriched in a base
quantaloid Q. In analogy with V-category theory we discuss such things as
adjoint functors, (pointwise) left Kan extensions, weighted (co)limits,
presheaves and free (co)completion, Cauchy completion and Morita equivalence.
With an appendix on the universality of the quantaloid Dist(Q) of Q-enriched
categories and distributors.
|
math
|
663 |
Categorical structures enriched in a quantaloid: regular presheaves, regular semicategories
|
math.CT
|
We study presheaves on semicategories enriched in a quantaloid: this gives
rise to the notion of regular presheaf. A semicategory is regular when its
representable presheaves are regular, and its regular presheaves then
constitute an essential (co)localization of the category of all of its
presheaves. The notion of regular semidistributor allows to establish the
Morita equivalence of regular semicategories. Continuous orders and Omega-sets
provide examples.
|
math
|
664 |
Categorical structures enriched in a quantaloid: orders and ideals over a base quantaloid
|
math.CT
|
Applying (enriched) categorical structures we define the notion of ordered
sheaf on a quantaloid Q, which we call `Q-order'. This requires a theory of
semicategories enriched in the quantaloid Q, that admit a suitable Cauchy
completion. There is a quantaloid Idl(Q) of Q-orders and ideal relations, and a
locally ordered category Ord(Q) of Q-orders and monotone maps; actually,
Ord(Q)=Map(Idl(Q)). In particular is Ord(Omega), with Omega a locale, the
category of ordered objects in the topos of sheaves on Omega. In general
Q-orders can equivalently be described as Cauchy complete categories enriched
in the split-idempotent completion of Q. Applied to a locale Omega this
generalizes and unifies previous treatments of (ordered) sheaves on Omega in
terms of Omega-enriched structures.
|
math
|
665 |
Towards an axiomatization of the theory of higher categories
|
math.CT
|
We define a notion of "theory of (1,infty)-categories", and we prove that
such a theory is unique up to equivalence.
|
math
|
666 |
Categorical non abelian cohomology, and the Schreier theory of groupoids
|
math.CT
|
By regarding the classical non abelian cohomology of groups from a
2-dimensional categorical viewpoint, we are led to a non abelian cohomology of
groupoids which continues to satisfy classification, interpretation and
representation theorems generalizing the classical ones. This categorical
approach is based on the fact that if groups are regarded as categories, then,
on the one hand, crossed modules are 2-groupoids and, cocycles are lax
2-functors and the cocycle conditions are precisely the coherence axioms for
lax 2-functors, and, on the other hand group extensions are fibrations of
categories. Furthermore, $n$-simplices in the nerve of a 2-category are lax
2-functors.
|
math
|
667 |
Higher gauge theory I: 2-Bundles
|
math.CT
|
I categorify the definition of fibre bundle, replacing smooth manifolds with
differentiable categories, Lie groups with coherent Lie 2-groups, and bundles
with a suitable notion of 2-bundle. To link this with previous work, I show
that certain 2-categories of principal 2-bundles are equivalent to certain
2-categories of (nonabelian) gerbes. This relationship can be (and has been)
extended to connections on 2-bundles and gerbes.
The main theorem, from a perspective internal to this paper, is that the
2-category of 2-bundles over a given 2-space under a given 2-group is (up to
equivalence) independent of the fibre and can be expressed in terms of
cohomological data (called 2-transitions). From the perspective of linking to
previous work on gerbes, the main theorem is that when the 2-space is the
2-space corresponding to a given space and the 2-group is the automorphism
2-group of a given group, then this 2-category is equivalent to the 2-category
of gerbes over that space under that group (being described by the same
cohomological data).
|
math
|
668 |
Categorical structures enriched in a quantaloid: tensored and cotensored categories
|
math.CT
|
Our subject is that of categories, functors and distributors enriched in a
base quantaloid Q. We show how cocomplete Q-categories are precisely those
which are tensored and conically cocomplete, or alternatively, those which are
tensored, cotensored and order-cocomplete. Bearing this in mind, we analyze how
Sup-valued homomorphisms on Q are related to Q-categories. With an appendix on
action, representation and variation.
|
math
|
669 |
Covering groupoids
|
math.CT
|
Topos properties of the category of covering groupoids over a fixed groupoid
are discussed. A classification result for connected covering groupoids over a
fixed groupoid analogous to the fundamental theorem of Galois theory is given.
|
math
|
670 |
On the $\mathbb{Z} D_\infty$-category
|
math.CT
|
In this paper we give a direct proof of the properties of the $\ZZ D_\infty$
category which was introduced in the classification of noetherian, hereditary
categories with Serre duality by Idun Reiten and the author.
|
math
|
671 |
Notes on enriched categories with colimits of some class
|
math.CT
|
Given a class Phi of weights, we study the following classes: Phi^+ of
Phi-flat weights which are the psi for which psi-colimits commute in the base V
with limits with weights in Phi; and Phi^-, dually defined, of weights psi for
which psi-limits commute in the base V with colimits with weights in Phi. We
show that both these classes are saturated (i.e. closed under the terminology
of Albert-Kelly or Betti's coverings). We prove that for the class P of all
weights P^+ = P^-. For any small B, we defined an enriched adjunction a` la
Isbell [B,V]^op -> [B^op,V] and show how it restricts to an equivalence
(P^-(B^op))^op ~ P^-(B) between subcategories of small projectives.
|
math
|
672 |
Towards "dynamic domains": totally continuous cocomplete Q-categories
|
math.CT
|
It is common practice in both theoretical computer science and theoretical
physics to describe the (static) logic of a system by means of a complete
lattice. When formalizing the dynamics of such a system, the updates of that
system organize themselves quite naturally in a quantale, or more generally, a
quantaloid. In fact, we are lead to consider cocomplete quantaloid-enriched
categories as fundamental mathematical structure for a dynamic logic common to
both computer science and physics. Here we explain the theory of totally
continuous cocomplete categories as generalization of the well-known theory of
totally continuous suplattices. That is to say, we undertake some first steps
towards a theory of "dynamic domains''.
|
math
|
673 |
Cohomology of the Grothendieck construction
|
math.CT
|
We consider cohomology of small categories with coefficients in a natural
system in the sense of Baues and Wirsching. For any funtor L: K -> CAT, we
construct a spectral sequence abutting to the cohomology of the Grothendieck
construction of L in terms of the cohomology of K and of L(k), for k an object
in K.
|
math
|
674 |
Quadratic categories, Koszul resolutions and operads
|
math.CT
|
A quadratic algebra is a homogeneous algebra generated by its elements of
degree 1. Manin has endowed the category of quadratic algebras with two tensor
products. These structures have been adapted to operads by Ginsburg and
Kapranov. Berger has defined such tensor products for n-homogeneous algebras.
The purpose of this paper is to define the notion of quadratic category, which
is a category endowed with two tensor products. The Manin and Ginsburg-Kapranov
constructions are examples of quadratic categories. We define also a Koszul
complex, n-homogeneous operads and show how this notion can be applied to study
coherence relations for n-categories.
|
math
|
675 |
Generalized Brown representability in homotopy categories
|
math.CT
|
We show that the homotopy category of a combinatorial stable model category
$\ck$ is well generated. It means that each object $K$ of $\Ho(\ck)$ is an
iterated weak colimit of $\lambda$-compact objects for some cardinal $\lambda$.
A natural question is whether each $K$ is a weak colimit of $\lambda$-compact
objects. We show that this is related to (generalized) Brown representability
of $\Ho(\mathcal K)$.
|
math
|
676 |
Files for Gabriel-Zisman localization
|
math.CT
|
This preprint contains the Coq proof files for Gabriel-Zisman localization,
bundled with the source. The text of this preprint consists of the definitions
and lemma statements of the main files, with proofs removed. See the other
preprint ``Explaining GZ localization to the computer'' for explanation and
discussion.
|
math
|
677 |
On the non additivity of the trace in derived categories
|
math.CT
|
In this note we provide an example of an endomorphism of a short exact
sequence of perfect complexes, with the trace of the middle map not equal to
the sum of the traces of the two other ones. The point is that the squares
involved are commutative only up to homotopy. In view of this example I have
found in 1968, Deligne immediately created his "categories spectrales", and
soon afterwards Illusie introduced the "filtered derived categories" where a
satisfactory kind of additivity is restored for the trace. This paper, written
in French, ends up with a brief chronological comment.
|
math
|
678 |
Weak identity arrows in higher categories
|
math.CT
|
There are a dozen definitions of weak higher categories, all of which loosen
the notion of composition of arrows. A new approach is presented here, where
instead the notion of identity arrow is weakened -- these are tentatively
called fair categories. The approach is simplicial in spirit, but the usual
simplicial category $\Delta$ is replaced by a certain `fat' delta of `coloured
ordinals', where the degeneracy maps are only up to homotopy. The first part of
this exposition is aimed at a broad mathematical readership and contains also a
brief introduction to simplicial viewpoints on higher categories in general. It
is explained how the definition of fair $n$-category is almost forced upon us
by three standard ideas.
The second part states some basic results about fair categories, and give
examples. The category of fair 2-categories is shown to be equivalent to the
category of bicategories with strict composition law. Fair 3-categories
correspond to tricategories with strict composition laws. The main motivation
for the theory is Simpson's weak-unit conjecture according to which
$n$-groupoids with strict composition laws and weak units should model all
homotopy $n$-types. A proof of a version of this conjecture in dimension 3 is
announced, obtained in joint work with A. Joyal. Technical details and a fuller
treatment of the applications will appear elsewhere.
|
math
|
679 |
Elementary remarks on units in monoidal categories
|
math.CT
|
We explore an alternative definition of unit in a monoidal category
originally due to Saavedra: a Saavedra unit is a cancellative idempotent (in a
1-categorical sense). This notion is more economical than the usual notion in
terms of left-right constraints, and is motivated by higher category theory. To
start, we describe the semi-monoidal category of all possible unit structures
on a given semi-monoidal category and observe that it is contractible (if
nonempty). Then we prove that the two notions of units are equivalent in a
strong functorial sense. Next, it is shown that the unit compatibility
condition for a (strong) monoidal functor is precisely the condition for the
functor to lift to the categories of units, and it is explained how the notion
of Saavedra unit naturally leads to the equivalent non-algebraic notion of fair
monoidal category, where the contractible multitude of units is considered as a
whole instead of choosing one unit. To finish, the lax version of the unit
comparison is considered. The paper is self-contained. All arguments are
elementary, some of them of a certain beauty.
|
math
|
680 |
On lifting stable diagrams in Frobenius categories
|
math.CT
|
Suppose given a Frobenius category E, i.e. an exact category with a big
enough subcategory B of bijectives. Let_E_ := E/B denote its classical stable
category. For example, we may take E to be the category of complexes C(A) with
entries in an additive category A, in which case_E_ is the homotopy category of
complexes K(A). Suppose given a finite poset D that satisfies the combinatorial
condition of being ind-flat. Then, given a diagram of shape D with values in_E_
(i.e. commutative up to homotopy), there exists a diagram consisting of pure
monomorphisms with values in E (i.e. commutative) that is isomorphic, as a
diagram with values in_E_, to the given diagram.
|
math
|
681 |
Notes on enriched categories with colimits of some class (completed version)
|
math.CT
|
The paper is in essence a survey of categories having $\phi$-weighted
colimits for all the weights $\phi$ in some class $\Phi$. We introduce the
class $\Phi^+$ of {\em $\Phi$-flat} weights which are those $\psi$ for which
$\psi$-colimits commute in the base $\V$ with limits having weights in $\Phi$;
and the class $\Phi^-$ of {\em $\Phi$-atomic} weights, which are those $\psi$
for which $\psi$-limits commute in the base $\V$ with colimits having weights
in $\Phi$. We show that both these classes are {\em saturated} (that is, what
was called {\em closed} in the terminology of \cite{AK88}). We prove that for
the class $\p$ of {\em all} weights, the classes $\p^+$ and $\p^-$ both
coincide with the class $\Q$ of {\em absolute} weights. For any class $\Phi$
and any category $\A$, we have the free $\Phi$-cocompletion $\Phi(\A)$ of $\A$;
and we recognize $\Q(\A)$ as the Cauchy-completion of $\A$. We study the
equivalence between ${(\Q(\A^{op}))}^{op}$ and $\Q(\A)$, which we exhibit as
the restriction of the Isbell adjunction between ${[\A,\V]}^{op}$ and
$[\A^{op},\V]$ when $\A$ is small; and we give a new Morita theorem for any
class $\Phi$ containing $\Q$. We end with the study of $\Phi$-continuous
weights and their relation to the $\Phi$-flat weights.
|
math
|
682 |
Lambda-presentable morphisms, injectivity and (weak) factorization systems
|
math.CT
|
We show that in a locally lambda-presentable category, every
lambda(m)-injectivity class (i.e., the class of all the objects injective with
respect to some class of lambda-presentable morphisms) is a weakly reflective
subcategory determined by a functorial weak factorization system cofibrantly
generated by a class of lambda-presentable morphisms. This was known for
small-injectivity classes, and referred to as the "small object argument". An
analogous result is obtained for orthogonality classes and factorization
systems, where lambda-filtered colimits play the role of the transfinite
compositions in the injectivity case. Lambda-presentable morphisms are also
used to organize and clarify some related results (and their proofs), in
particular on the existence of enough injectives (resp. pure-injectives).
|
math
|
683 |
Identity and Categorification
|
math.CT
|
In the paper I check approaches to identity in mathematics by Plato, Frege,
and Geach against Category theory.
|
math
|
684 |
Cryptography and Encryption
|
math.CT
|
In cryptography, encryption is the process of obscuring information to make
it unreadable without special knowledge. This is usually done for secrecy, and
typically for confidential communications. Encryption can also be used for
authentication, digital signatures, digital cash e.t.c. In this paper we are
going to examine and analyse all these topics in detail.
|
math
|
685 |
Commutation Structures
|
math.CT
|
For a fixed object X in a monoidal category, an X-commutation structure on an
object A is just a map from XA to AX. We study aspects of such structure in
case A has a dual.
|
math
|
686 |
Notes on 2-groupoids, 2-groups and crossed-modules
|
math.CT
|
This paper contains some basic results on 2-groupoids, with special emphasis
on computing derived mapping 2-groupoids between 2-groupoids and proving their
invariance under strictification. Some of the results proven here are
presumably folklore (but do not appear in the literature to the author's
knowledge) and some of the results seem to be new. The main technical tool used
throughout the paper is the Quillen model structure on the category of
2-groupoids introduced by Moerdijk and Svensson.
|
math
|
687 |
Bipolar spaces
|
math.CT
|
Some basic features of the simultaneous inclusion of discrete fibrations and
discrete opfibrations on a category A in the category of categories over A are
studied; in particular, the reflections and the coreflections of the latter in
the former are considered, along with a negation-complement operator which,
applied to a discrete fibration, gives a functor with values in discrete
opfibrations (and vice versa) and which turns out to be classical, in that the
strong contraposition law holds. Such an analysis is developed in an
appropriate conceptual frame that encompasses similar "bipolar" situations and
in which a key role is played by "cofigures", that is components of products;
e.g. the classicity of the negation-complement operator corresponds to the fact
that discrete opfibrations (or in general "closed parts") are properly analyzed
by cofigures with shape in discrete fibrations ("open parts"), that is, that
the latter are "coadequate" for the former, and vice versa. In this context, a
very natural definition of "atom" is proposed and it is shown that, in the
above situation, the category of atoms reflections is the Cauchy completion of
A.
|
math
|
688 |
Quadratic categories and Koszul resolutions
|
math.CT
|
In this paper we define quadratic categories and their representations.
|
math
|
689 |
Orientals
|
math.CT
|
The orientals or oriented simplexes are a family of strict omega-categories
constructed by Ross Street. We show that the category of orientals is
isomorphic to a subcategory of the category of chain complexes. This leads to a
very simple combinatorial description of the morphisms between orientals. We
also show that the category of orientals is the closure of the category of
simplexes under certain filler operations which represent complicial
operations.
|
math
|
690 |
Thin fillers in the cubical nerves of omega-categories
|
math.CT
|
It is shown that the cubical nerve of a strict omega-category is a sequence
of sets with cubical face operations and distinguished subclasses of thin
elements satisfying certain thin filler conditions. It is also shown that a
sequence of this type is the cubical nerve of a strict omega-category unique up
to isomorphism; the cubical nerve functor is therefore an equivalence of
categories. The sequences of sets involved are in effect the analogues of
cubical T-complexes appropriate for strict omega-categories. Degeneracies are
not required in the definition of these sequences, but can in fact be
constructed as thin fillers. The proof of the thin filler conditions uses chain
complexes and chain homotopies.
|
math
|
691 |
Categories, norms and weights
|
math.CT
|
The well-known Lawvere category R of extended real positive numbers comes
with a monoidal closed structure where the tensor product is the sum. But R has
another such structure, given by multiplication, which is *-autonomous.
Normed sets, with a norm in R, inherit thus two symmetric monoidal closed
structures, and categories enriched on one of them have a 'subadditive' or
'submultiplicative' norm, respectively. Typically, the first case occurs when
the norm expresses a cost, the second with Lipschitz norms.
This paper is a preparation for a sequel, devoted to 'weighted algebraic
topology', an enrichment of directed algebraic topology. The structure of R,
and its extension to the complex projective line, might be a first step in
abstracting a notion of algebra of weights, linked with physical measures.
|
math
|
692 |
Universal coefficient theorem in triangulated categories
|
math.CT
|
Let T be a triangulated category, A a graded abelian category and h: T -> A a
homology theory on T with values in A. If the functor h reflects isomorphisms,
is full and is such that for any object x in A there is an object X in T with
an isomorphism between h(X) and x, we prove that A is a hereditary abelian
category and the ideal Ker(h) is a square zero ideal which as a bifunctor on T
is isomorphic to Ext^1_A(h(-)[1], h(-)).
|
math
|
693 |
Finite Products are Biproducts in a Compact Closed Category
|
math.CT
|
If a compact closed category has finite products or finite coproducts then it
in fact has finite biproducts, and so is semi-additive.
|
math
|
694 |
Exponentiable functors between quantaloid-enriched categories
|
math.CT
|
Exponentiable functors between quantaloid-enriched categories are
characterized in elementary terms. The proof goes as follows: the elementary
conditions on a given functor translate into existence statements for certain
adjoints that obey some lax commutativity; this, in turn, is precisely what is
needed to prove the existence of partial products with that functor; so that
the functor's exponentiability follows from the works of Niefield [1980] and
Dyckhoff and Tholen [1987].
|
math
|
695 |
Strict 2-toposes
|
math.CT
|
A 2-categorical generalisation of elementary topos is provided and some of
the properties of the yoneda structure it generates are explored. Examples
relevant to the globular approach to higher category theory are discussed. This
paper also contains some expository material on the theory of fibrations
internal to a finitely complete 2-category as well as a self-contained
development of the necessary background on yoneda structures.
|
math
|
696 |
Generalized 2-vector spaces and general linear 2-groups
|
math.CT
|
In this paper a notion of {\it generalized 2-vector space} is introduced
which includes Kapranov and Voevodsky 2-vector spaces. Various kinds of
generalized 2-vector spaces are considered and examples are given. The
existence of non free generalized 2-vector spaces and of generalized 2-vector
spaces which are non Karoubian (hence, non abelian) categories is discussed,
and it is shown how any generalized 2-vector space can be identified with a
full subcategory of an (abelian) functor category with values in the category
${\bf VECT}_K$ of (possibly infinite dimensional) vector spaces. The
corresponding general linear 2-groups $\mathbb{G}\mathbb{L}({\bf
Vect}_K[\mathcal{C}])$ are considered. Specifically, it is shown that
$\mathbb{G}\mathbb{L}({\bf Vect}_K[\mathcal{C}])$ always contains as a (non
full) sub-2-group the 2-group ${\sf Equiv}_{Cat}(\mathcal{C})$ (hence, for
finite categories $\mathcal{C}$, they contain {\sl Weyl sub-2-groups} analogous
to usual Weyl subgroups of the general linear groups), and
$\mathbb{G}\mathbb{L}({\bf Vect}_K[\mathcal{C}])$ is explicitly computed (up to
equivalence) in a special case of generalized 2-vector spaces which include
those of Kapranov and Voevodsky. Finally, other important drawbacks of the
notion of generalized 2-vector space, besides the fact that it is in general a
non Karoubian category, are also mentioned at the end of the paper.
|
math
|
697 |
Double Clubs
|
math.CT
|
We develop a theory of double clubs which extends Kelly's theory of clubs to
the pseudo double categories of Pare and Grandis. We then show that the club
for symmetric strict monoidal categories on Cat extends to a `double club' on
the pseudo double category of `categories, functors, profunctors and
transformations'.
|
math
|
698 |
Polycategories via pseudo-distributive laws
|
math.CT
|
In this paper, we give a novel abstract description of Szabo's
polycategories. We use the theory of double clubs -- a generalisation of
Kelly's theory of clubs to `pseudo' (or `weak') double categories -- to
construct a pseudo-distributive law of the free symmetric strict monoidal
category pseudocomonad on Mod over itself qua pseudomonad, and show that monads
in the `two-sided Kleisli bicategory' of this pseudo-distributive law are
precisely symmetric polycategories.
|
math
|
699 |
2-nerves for bicategories
|
math.CT
|
We describe a Cat-valued nerve of bicategories, which associates to every
bicategory a simplicial object in Cat, called the 2-nerve. We define a
2-category NHom whose objects are bicategories and whose 1-cells are normal
homomorphisms of bicategories, in such a way that the 2-nerve construction
becomes a full embedding of NHom in the 2-category of simplicial objects in
Cat. This embedding has a left biadjoint, and we characterize its image. The
2-nerve of a bicategory is always a weak 2-category in the sense of Tamsamani,
and we show that NHom is biequivalent to a certain 2-category whose objects are
Tamsamani weak 2-categories.
|
math
|
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