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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 associ... | 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... | 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,
traditionall... | 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 ... | 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 transformati... | 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 ... | 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 b... | 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... | 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. Fre... | 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 fi... | 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 Cha... | 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 m... | 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 qua... | 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 embeddi... | 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 (m... | 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
observa... | 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 A... | 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 obje... | 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 ... | 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 ... | 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
constructio... | 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 pro... | 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 embeddi... | 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... | 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 exam... | 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)... | 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... | 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 ... | 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 ... | 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 o... | 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, ... | 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 ... | 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 grade... | 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 s... | 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 c... | 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 t... | 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... | 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. Th... | 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... | 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)com... | 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 reg... | 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... | 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 fac... | 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 o... | 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 ... | 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 bot... | 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 ... | 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 th... | 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... | 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, Delig... | 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 categ... | 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 descr... | 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... | 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... | 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 l... | 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 exam... | 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 o... | 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 t... | 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 th... | 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... | 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 ca... | 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 a... | 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 ... | 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 fin... | 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 Karou... | 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 its... | 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... | math |
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