Let V be a cofibrantly generated closed symmetric monoidal model category and
M a model V-category. We say that a weighted colimit W*D of a diagram D
weighted by W is a homotopy weighted colimit if the diagram D is pointwise
cofibrant and the weight W is cofibrant in the projective model structure on
[C^op,V]. We then proceed to describe such homotopy weighted colimits through
homotopy tensors and ordinary (conical) homotopy colimits. This is a homotopy
version of the well known isomorphism W*D=\int^C(W\tensor D).

We introduce the notion of pie algebra for a 2-monad, these bearing the same
relationship to the flexible and semiflexible algebras as pie limits do to
flexible and semiflexible ones. We see that in many cases, the pie algebras are
precisely those "free at the level of objects" in a suitable sense; so that,
for instance, a strict monoidal category is pie just when its underlying monoid
of objects is free.

We consider a notion of exact sequences in any -not necessarily exact-
pointed category relative to a given (E;M)-factorization structure. We apply
this notion to introduce and investigate a new notion of exact sequences of
semimodules over semirings relative to the canonical image factorization.
Several homological results are proved using the new notion of exactness
including some restricted versions of the Short Five Lemma and the Snake Lemma
opening the door for introducing and investigating homology objects in such
categories.

We investigate when the categories of all rational $A$-modules and of finite
dimensional rational modules are closed under extensions inside the category of
$C^*$-modules, where $C^*$ is the cofinite topological completion of $A$. We
give a complete characterization of these two properties, in terms of a
topological and a homological condition. We also give connections to other
important notions in coalgebra theory such as coreflexive coalgebras.

We summarize our axioms for higher categories, and describe the blob complex.
Fixing an n-category C, the blob complex associates a chain complex B_*(W;C)$
to any n-manifold W. The 0-th homology of this chain complex recovers the usual
topological quantum field theory invariants of W. The higher homology groups
should be viewed as generalizations of Hochschild homology (indeed, when W=S^1
they coincide). The blob complex has a very natural definition in terms of
homotopy colimits along decompositions of the manifold W.

We classify extensions of a group $G$ by a braided 2-group $\mathcal{B}$ as
defined by Drinfeld, Gelaki, Nikshych, and Ostrik. We describe such extensions
as homotopy classes of maps from the classifying space of $G$ to the
classifying space of the 3-group of braided $\mathcal{B}$-bitorsors. The
Postnikov system of the latter space contains a generalization of the classical
Pontryagin square to the setting of local coefficients, which has been
previously discussed by Baues and more recently, in a setting close to ours, by
Etingof, Nikshych, and Ostrik.

In this paper, we state and prove precise theorems on the classification of
the category of (braided) categorical groups and their (braided) monoidal
functors, and some applications obtained from the basic studies on monoidal
functors between categorical groups.

In this paper we show that the category of frames, and, thus, the cate- gory
of locales is 'rigid'. This means that every endo-equivalence on them is
isomorphic to the identity functor. To reach this result we prove new results
concerning the number of automorphisms between frames and new results
concerning the order preserving properties of endo-equivalences.

By studying NIM-representations we show that the Fibonacci category and its
tensor powers are completely anisotropic; that is, they do not have any
non-trivial separable commutative ribbon algebras. As an application we deduce
that a chiral algebra with the representation category equivalent to a product
of Fibonacci categories is maximal; that is, it is not a proper subalgebra of
another chiral algebra. In particular the chiral algebras of the Yang-Lee
model, the WZW models of G2 and F4 at level 1, as well as their tensor powers,
are maximal.

We prove a coherence theorem for braided monoidal bicategories and relate it
to the coherence theorem for monoidal bicategories. We show how coherence for
these structures can be interpretted topologically using up-to-homotopy operad
actions and the algebraic classification of surface braids.

In this paper we ?rst de?ne the category of fuzzy hyper BCK- algebras. After
that we show that the category of hyper BCK-algebras has equalizers,
coequalizers, products. It is a consequence that this category is complete and
hence has pullbacks.

A relative derived category for the category of modules over a presheaf of
algebras is constructed to identify the relative Yoneda and Hochschild
cohomologies with its homomorphism groups. The properties of a functor between
this category and the relative derived category of modules over the algebra
associated to the presheaf are studied. We obtain a generalization of the
$Special$ $ Cohomology$ $Comparison$ $Theorem$ of M. Gerstenhaber and S. D.
Schack.

The basic notions of category theory, such as limit, adjunction, and
orthogonality, all involve assertions of the existence and uniqueness of
certain arrows. Weak notions arise when one drops the uniqueness requirement
and asks only for existence. The enriched versions of the usual notions involve
certain morphisms between hom-objects being invertible; here we introduce
enriched versions of the weak notions by asking that the morphisms between
hom-objects belong to a chosen class of "surjections".

This reports on the fundamental objects revealed by Ross Street, which he
called `orientals'. Street's work was in part inspired by Robert's attempts to
use N-category ideas to construct nets of C*-algebras in Minkowski space for
applications to relativistic quantum field theory: Roberts' additional
challenge was that `no amount of staring at the low dimensional cocycle
conditions would reveal the pattern for higher dimensions'.

We discuss cyclic star-autonomous categories; that is, unbraided star-
autonomous categories in which the left and right duals of every object p are
linked by coherent natural isomorphism. We settle coherence questions which
have arisen concerning such cyclicity isomorphisms, and we show that such
cyclic structures are the natural setting in which to consider enriched
profunctors. Specifically, if V is a cyclic star-autonomous category, then the
collection of V-enriched profunctors carries a canonical cyclic structure.

We classify the matrices M which correspond to finite categories

We observe that the notion of two sets being equal up to finitely many
elements is a homotopy equivalence relation in a model category, and suggest a
homotopy-invariant variant of Generalised Continuum Hypothesis about which more
can be proven within ZFC and which first appeared in PCF theory. The formalism
allows to draw analogies between notions of set theory and those of homotopy
theory, and we indeed observe a similarity between homotopy theory
ideology/yoga and that of PCF theory. We also briefly discuss conjectural
connections with model theory and arithmetics and geometry.

In an earlier paper we introduced a notion of Markov automaton, together with
parallel operations which permit the compositional description of Markov
processes. We illustrated by showing how to describe a system of n dining
philosophers, and we observed that Perron-Frobenius theory yields a proof that
the probability of reaching deadlock tends to one as the number of steps goes
to infinity. In this paper we add sequential operations to the algebra (and the
necessary structure to support them).

The aim of this paper is twofold. One is to give a definition of the Euler
characteristic of infinite acyclic categories with filtrations and the other is
to prove the invariance of the Euler characteristic under the subdivision of
finite categories.

We produce a cofibrantly generated simplicial symmetric monoidal model
structure for the category of (small unital) C*-categories, whose weak
equivalences are the unitary equivalences. The closed monoidal structure
consists of the maximal tensor product, which generalizes that of C*-algebras,
with the Ghez-Lima-Roberts' C*-categories of *-functors, C*(A,B), providing the
internal Hom's.

We prove that a monomorphic functor $F:Comp\to Comp$ with finite supports is
epimorphic, continuous, and its maximal $\emptyset$-modification $F^\circ$
preserves intersections. This implies that a monomorphic functor $F:Comp\to
Comp$ of finite degree $deg F\le n$ preserves (finite-dimensional) compact
ANR's if the spaces $F\emptyset$, $F^\circ\emptyset$, and $Fn$ are
finite-dimensional ANR's. This improves a known result of Basmanov.

In this paper we define 3-crossed modules for commutative (Lie) algebras and
investigate the relation between this construction and the simplicial algebras.
Also we define the projective 3-crossed resolution for investigate a higher
dimensional homological information and show the existence of this resolution
for an arbitrary $\mathbf{k}$-algebra.

We generalize the concept of Sato Grassmannians of locally linearly compact
topological vector spaces (Tate spaces) to the category limA of the "locally
compact objects" of an exact category A, and study some of their properties.
This allows us to generalize the Kapranov dimensional torsor Dim(X) and
determinantal gerbe Det(X) for the objects of limA. We then introduce a class
of exact categories, that we call quasiabelian exact, and prove that if A is
quasiabelian exact, Dim(X) and Det(X) are multiplicative in admissible short
exact sequences.

We define a weak bimonad as a monad T on a monoidal category M with the
property that the Eilenberg-Moore category M^T is monoidal and the forgetful
functor from M^T to M is separable Frobenius. Whenever M is also Cauchy
complete, a simple set of axioms is provided, that characterizes the monoidal
structure of M^T as a weak lifting of the monoidal structure of M . The
relation to bimonads, and the relation to weak bimonoids in a braided monoidal
category are revealed. We also discuss antipodes, obtaining the notion of weak
Hopf monad.

A combinatorial category Disks was introduced by Andr\'e Joyal to play a role
in his definition of weak omega-category. He defined the category Theta to be
dual to Disks. In the ensuing literature, a more concrete description of Theta
was provided. In this paper we provide another proof of the dual equivalence
and introduce various categories equivalent to Disks or Theta, each providing a
helpful viewpoint.

Alain Bruguieres, in his talk [1], announced his work [2] with Alexis
Virelizier and the second author which dealt with lifting closed structure on a
monoidal category to the category of Eilenberg-Moore algebras for an opmonoidal
monad. Our purpose here is to generalize that work to the context internal to
an autonomous monoidal bicategory. The result then applies to quantum
categories and bialgebroids.

Given an algebraic theory which can be described by a (possibly symmetric)
operad $P$, we propose a definition of the \emph{weakening} (or
\emph{categorification}) of the theory, in which equations that hold strictly
for $P$-algebras hold only up to coherent isomorphism. This generalizes the
theories of monoidal categories and symmetric monoidal categories, and several
related notions defined in the literature.

This note informally describes a way to build certain cubical n-categories by
iterating a process of taking models of certain finite limits theories. We base
this discussion on a construction of "double bicategories" as bicategories
internal to Bicat, and see how to extend this to n-tuple bicategories (and
similarly for tricategories etc.) We briefly consider how to reproduce
"simpler" definitions of weak cubical n-category from these.

We provide a framework to deal with "diagrammatic" operadic actions in Cat,
i.e. actions given by compositions of diagrams, rather than strings of objects.
We achieve this by introducing a monoidal structure on the category of small
diagrams in Cat, which generalizes simultaneously the composition product of
collections in the theory of operads, and the semi-direct product of groups.
Familial operads are given then as monoids with respect to this monoidal
structure, and algebras are defined as categories, carrying actions of such
monoids.

We define the thin fundamental Gray 3-groupoid $S_3(M)$ of a smooth manifold
$M$ and define (by using differential geometric data) 3-dimensional holonomies,
to be smooth strict Gray 3-groupoid maps $S_3(M) \to C(H)$, where $H$ is a
2-crossed module of Lie groups and $C(H)$ is the Gray 3-groupoid naturally
constructed from $H$. As an application, we define Wilson 3-sphere observables.

The goal of this paper is to prove coherence results with respect to
relational graphs for monoidal monads and comonads, i.e. monads and comonads in
a monoidal category such that the endofunctor of the monad or comonad is a
monoidal functor (this means that it preserves the monoidal structure up to a
natural transformation that need not be an isomorphism). These results are
proved first in the absence of symmetry in the monoidal structure, and then
with this symmetry. The monoidal structure is also allowed to be given with
finite products or finite coproducts.

Given a continuous monadic functor T in the category of Tychonov spaces for
each discrete topological semigroup X we extend the semigroup operation of X to
a right-topological semigroup operation on TX whose topological center contains
the dense subsemigroup of all elements of TX that have finite support.

We give a characterisation of those local not necessary commutative rings,
for which the category of projective modules admits a triangulation with the
identity as translation functor. By "admits a triangulation" we mean that the
category can be given the structure of a triangulated category that satisfies
the standard set of axioms including the octahedral axiom.

We introduce the notion of a lax monoidal fibration and we show how it can be
conveniently used to deal with various algebraic structures that play an
important role in some definitions of the opetopic sets (Baez-Dolan,
Hermida-Makkai-Power). We present the 'standard' such structures, the
exponential fibrations of basic fibrations and three areas of applications.
First area is related to the T-categories of A. Burroni. The monoids in the
Burroni lax monoidal fibrations form the fibration of T-categories.

Given a thick subcategory of a triangulated category, we define a
colocalisation and a natural long exact sequence that involves the original
category and its localisation and colocalisation at the subcategory. Similarly,
we construct a natural long exact sequence containing the canonical map between
a homological functor and its total derived functor with respect to a thick
subcategory.

We establish a criterion for deciding whether a class of structures is the
class of models of a geometric theory inside Grothendieck toposes; then we
specialize this result to obtain a characterization of the infinitary
first-order theories which are geometric in terms of their models in
Grothendieck toposes, solving a problem posed by Ieke Moerdijk in 1989.

The metric jets, introduced in the first chapter, generalize the jets (at
order one) of Charles Ehresmann. In short, for a "good" map $f$ (said to be
"tangentiable" at $a$), we define its metric jet tangent at $a$ (composed of
all the maps which are locally lipschitzian at $a$ and tangent to $f$ at $a$)
called the "tangential" of $f$ at $a$, and denoted T$f_a$ (the domain and
codomain of $f$ being metric spaces).

We briefly relate the existence of a middle-four interchange map in a
category with two monoidal structures, to the standard Cockett and Seely notion
of a weakly distributive category.

A construction of Kleisli objects in 2-categories of noncartesian internal
categories or categories internal to monoidal categories is presented.

A classical result of Tannaka duality is the fact that a coalgebra over a
field can be reconstructed from its category of finite dimensional
representations by using the forgetful functor which sends a representation to
its underlying vector space. There is also a corresponding recognition result,
which characterizes those categories equipped with a functor to finite
dimensional vector spaces which are equivalent to the category of finite
dimensional representations of a coalgebra.

For any abelian category \calC satsifying (AB5) over a separated,
quasi-compact scheme S, we construct a stack of 2-groups \GL(\calC) over the
flat site of S. We will give a concrete description of \GL(\calC) when \calC is
the category of quasi-coherent sheaves on a separated, quasi-compact scheme X
over S. We will show that the tangent space \gl(\calC) of \GL(\calC) at the
origin has a structure as a Lie 2-algebra.

Given a pair of adjoint functors between two arbitrary categories it induces
mutually inverse equivalences between the full subcategories of the initial
ones, consisting of objects for which the arrows of adjunction are
isomorphisms. We investigate some cases in which these subcategories may be
better characterized. One application is the construction of cellular
approximations. Other is the definition and the characterization of (weak)
*-objects in the non additive case.

Categorical universal algebra can be developed either using Lawvere theories
(single-sorted finite product theories) or using monads, and the category of
Lawvere theories is equivalent to the category of finitary monads on Set. We
show how this equivalence, and the basic results of universal algebra, can be
generalized in three ways: replacing Set by another category, working in an
enriched setting, and by working with another class of limits than finite
products.

Natural weak factorization systems are an algebraization of weak
factorization systems, which are familiar components of model categories. By a
modified version of Quillen's small object argument due to Richard Garner,
cofibrantly generated natural weak factorization systems can be constructed,
allowing many applications. We define a new notion of a "natural model
structure" and prove "natural" analogs of some classical results about model
categories.

In this paper we examine on changing the base which induces a pair of
functors for a subcategory of a category of crossed modules over commutative
algebras. We give some examples and results on induced crossed modules.

We describe the 2-category of quantum categories.

A completeness conjecture is advanced concerning the free small-colimit
completion P(A) of a (possibly large) category A. The conjecture is based on
the existence of a small generating-cogenerating set of objects in A. We sketch
how the validity of the result would lead to the existence of an Isbell-Lambek
bicompletion C(A) of such an A, without a "change-of-universe" procedure being
necessary to describe or discuss the bicompletion.

One of the open problems in higher category theory is the systematic
construction of the higher dimensional analogues of the Gray tensor product. In
this paper we continue the work of [7] to adapt the machinery of globular
operads [4] to this task. The resulting theory includes the Gray tensor product
of 2-categories and the Crans tensor product [12] of Gray categories.

This paper contains some contributions to the study of the relationship
between 2-categories and the homotopy types of their classifying spaces.
Mainly, generalizations are given of both Quillen's Theorem B and Thomason's
Homotopy Colimit Theorem to 2-functors.

A question of Jack Morava is answered by generalising the notion of Moore
paths to that of Moore hyperrectangles, so obtaining a strict cubical
omega-category. This also has the structure of connections in the sense of
Brown and Higgins, but cancellation of connections does not hold.

The Eilenberg-Moore constructions and a Beck-type theorem for pairs of monads
are described. More specifically, a notion of a {\em Morita context} comprising
of two monads, two bialgebra functors and two connecting maps is introduced. It
is shown that in many cases equivalences between categories of algebras are
induced by such Morita contexts. The Eilenberg-Moore category of
representations of a Morita context is constructed.

A semisimple algebraic tensor category over an algebraically closed field k
of characteristic zero is the representation category of all finite dimensional
twisted super representations of an affine reductive supergroup G over k. Such
a supergroup is reductive if and only if its connected component is reductive.
The connected component is reductive if and only if the Lie superalgebra
divided by its center is a product of simple Lie algebras of classical type and
Lie superalgebras spo(1,2r) of the orthosymplectic types BC_r.

A braided Ann-category $\A$ is an Ann-category $\A$ together with the
braiding $c$ such that $(\A, \otimes, a, c, (I,l,r))$ is a braided tensor
category, and $c$ is compatible with the distributivity constraints. The paper
shows the dependence of the left (or right) distributivity constraint on other
axioms. Hence, the paper shows the relation to the concepts of {\it
distributivity category} due to M. L. Laplaza and {\it ring-like category} due
to A. Frohlich and C.T.C Wall.

We develop the basic constructions of homological algebra in the
(appropriately defined) unbounded derived categories of modules over algebras
over coalgebras over noncommutative rings (which we call semialgebras over
corings). We define double-sided derived functors SemiTor and SemiExt of the
functors of semitensor product and semihomomorphisms, and construct an
equivalence between the exotic derived categories of semimodules and
semicontramodules.

As Koenig and Zhu showed, quotient of a triangulated category by a maximal
1-orthogonal subcategory becomes an abelian category. In this paper, we
generalize this result to a maximal $n$-orthogonal subcategory for an arbitrary
positive integer $n$.

Finding coherent relations to define non Abelian cohomology is a thriller
which entertains the mathematical community since fifty one years. The purpose
of this paper is to simplify the attempt to beat it defined by the author which
used the notion of sequences of fibred categories and to apply the resulting
theory to higher divisors and Chow theory.