We prove a general theorem which includes most notions of "exact completion".
The theorem is that "k-ary exact categories" are a reflective sub-2-category of
"k-ary sites", for any regular cardinal k. A k-ary exact category is an exact
category with disjoint and universal k-small coproducts, and a k-ary site is a
site whose covering sieves are generated by k-small families and which
satisfies a weak size condition.

This is a report on aspects of the theory and use of monoidal categories. The
first section introduces the main concepts through the example of the category
of vector spaces. String notation is explained and shown to lead naturally to a
link between knot theory and monoidal categories. The second section reviews
the light thrown on aspects of representation theory by the machinery of
monoidal category theory, such as braidings and convolution. The category
theory of Mackey functors is reviewed in the third section.

We call a finitely complete category diexact if every Mal'cev relation admits
a pushout which is stable under pullback and itself a pullback.

Given a horizontal monoid M in a duoidal category F, we examine the
relationship between bimonoid structures on M and monoidal structures on the
category of right M-modules which lift the vertical monoidal structure of F. We
obtain our result using a variant of the Tannaka adjunction. The approach taken
utilizes hom-enriched categories rather than categories on which a monoidal
category acts ("actegories"). The requirement of enrichment in F itself demands
the existence of some internal homs, leading to the consideration of
convolution for duoidal categories.

Two fundamental contributions to categorical quantum mechanics are presented.
First, we generalize the CP-construction, that turns any dagger compact
category into one with completely positive maps, to arbitrary dimension.
Second, we axiomatize when a given category is the result of this construction.

We apply the representation theory of left-handed skew Boolean algebras by
sections of their dual \'{e}tale spaces, given in \cite{K}, to construct a
series of dual adjunctions between the categories of locally compact Boolean
spaces and left-handed skew Boolean algebras by means of extensions of certain
enriched $\Hom$-set functors induced by objects sitting in two categories. The
constructed adjunctions are "deformations" of Stone duality obtained by the
replacement in the latter of the category of Boolean algebras by the category
of left-handed skew Boolean algebras.

We present a purely category-theoretic characterization of retracts of
Fra\"iss\'e limits. For this aim, we consider a natural version of injectivity
with respect to a pair of categories (a category and its subcategory). It turns
out that retracts of Fra\"iss\'e limits are precisely the objects that are
injective relatively to such a pair. One of the applications is a
characterization of non-expansive retracts of Urysohn's universal metric space.

The notion of Ann-categories is a categorification of the ring structure.
Regular Ann-categories were classified by Shukla algebraic cohomology. In this
article, we state and prove the precise theorem on classification for the
general case due to Mac Lane cohomology for rings. And an application for
classification problem of ring extensions is also introduced.

We prove the $L^2$-Euler characteristic has the invariance under the
barycentric subdivision only for finite acyclic categories. And we extend the
definition of $L^2$-Euler characteristic and prove the extended $L^2$-Euler
characteristic has the invariance under the barycentric subdivision for more
wide class of finite categories.

In this note we show that the known relation between double groupoids and
matched pairs of groups may be extended, or seems to extend, to the triple
case. The references give some other occurrences of double groupoids.

We prove for a large family of rings R that their lambda-pure global
dimension is greater than one for each infinite regular cardinal lambda. This
answers in negative a problem posed by Rosicky. The derived categories of such
rings then do not satisfy the Adams lambda-representability for morphisms for
any lambda. Equivalently, they are examples of well generated triangulated
categories whose lambda-abelianization in the sense of Neeman is not a full
functor for any lambda.

We show that every internal biequivalence in a tricategory T is part of a
biadjoint biequivalence. We give two applications of this result, one for
transporting monoidal structures and one for equipping a monoidal bicategory
with invertible objects with a coherent choice of those inverses.

Category computation theory deals with a web-based systemic processing that
underlies the morphic webs, which constitute the basis of categorial logical
calculus. It is proven that, for these structures, algorithmically
incompressible binary patterns can be morphically compressed, with respect to
the local connectivities, in a binary morphic program. From the local
connectivites, there emerges a global morphic connection that can be
characterized by a low length binary string, leading to the identification of
chaotic categorial dynamics, underlying the algorithmically random pattern.

The purpose of this survey article is to introduce the reader to a connection
between Logic, Geometry, and Algebra which has recently come to light in the
form of an interpretation of the constructive type theory of Martin-L\"of into
homotopy theory, resulting in new examples of higher-dimensional categories.

Dual monoidal category $\mathcal C^\ast$ of a monoidal functor $F:\mathcal
C\to \mathcal V$ has been constructed by S. Majid. In this paper, we extend the
construction of dual structures for an Ann-functor $F:\mathcal B\to \mathcal
A$. In particular, when $F=id_{\mathcal A}$, then the dual category $\mathcal
A^{\ast}$ is indeed the center of $\mathcal A$ and this is a braided
Ann-category.

Let $F, G: \mathcal{I} \to \mathcal{C}$ be strong monoidal functors from a
skeletally small monoidal category $\mathcal{I}$ to a tensor category
$\mathcal{C}$ over an algebraically closed field $k$. The set $Nat(F, G)$ of
natural transformations $F \to G$ is naturally a vector space over $k$. We show
that the set $Nat_\otimes(F, G)$ of monoidal natural transformations $F \to G$
is linearly independent as a subset of $Nat(F, G)$.

Similar to $k$-bilinear forms, $k$-bilinear maps determine a duality between
two complete lattices. Unlike forms, minimal intervals in the lattice are not
only copies of $k$ -- they can be arbitrary division $k$-bilinear maps. The
category of $k$-bilinear maps with adjoint-morphisms is constructed to probe
this structure. It is shown that this category is equivalent to the category of
$k$-modules and comes with a natural contra-variant involution.

In this paper, we will prove that the 2-category (2-SGp) of symmetric
2-groups and 2-category ($\cR$-2-Mod) of $\cR$-2-modules(\cite{5}) have enough
projective objects, respectively.

For any locally cartesian closed category E, we prove that a local fibered
right adjoint between slices of E is given by a polynomial. The slices in
question are taken in a well known fibered sense.

We extend Barr's well-known characterization of the final coalgebra of a
$Set$-endofunctor as the completion of its initial algebra to the
Eilenberg-Moore category of algebras for a $Set$-monad $\mathbf{M}$ for
functors arising as liftings. As an application we introduce the notion of
commuting pair of endofunctors with respect to the monad $\mathbf{M}$ and show
that under reasonable assumptions, the final coalgebra of one of the
endofunctors involved can be obtained as the free algebra generated by the
initial algebra of the other endofunctor.

Our work is a fundamental study of the notion of approximation in
V-categories and in (U,V)-categories, for a quantale V and the ultrafilter
monad U. We introduce auxiliary, approximating and Scott-continuous
distributors, the way-below distributor, and continuity of V- and
(U,V)-categories. We fully characterize continuous V-categories (resp.
(U,V)-categories) among all cocomplete V-categories (resp. (U,V)-categories) in
the same ways as continuous domains are characterized among all dcpos.

The additivity of trace in certain tensor triangulated categories for an
endomorphisms of finite order of a distinguished triangles is investigated. For
identity endomorphism this has been fully established by J. P. May ("The
additivity of traces in triangulated categories", Adv. Math., 2001, 163,
34-73). By imposing extra conditions on the coefficients we show how May's
result implies a stronger additivity.

Let A, B, S be categories, let F:A-->S and G:B-->S be functors. We assume
that for "many" objects a in A, there exists an object b in B such that F(a) is
isomorphic to G(b). We establish a general framework under which it is possible
to transfer this statement to diagrams of A. These diagrams are all indexed by
posets in which every principal ideal is a join-semilattice and the set of all
upper bounds of any finite subset is a finitely generated upper subset.

Let $G$ and $A$ be objects of a finitely cocomplete homological category
$\mathbb C$. We define a notion of an (internal) action of $G$ of $A$ which is
functorially equivalent with a point in $\mathbb C$ over $G$, i.e. a split
extension in $\mathbb C$ with kernel $A$ and cokernel $G$. This notion and its
study are based on a preliminary investigation of cross-effects of functors in
a general categorical context. These also allow us to define higher categorical
commutators.

We show that the relative Auslander-Buchweitz context on a triangulated
category $\T$ coincides with the notion of co-$t$-structure on certain
triangulated subcategory of $\T$ (see the Theorem \ref{M2}). In the
Krull-Schmidt case, we stablish a bijective correspondence between
co-$t$-structures and cosuspended, precovering subcategories (see the Theorem
\ref{correspond}). We also give a description of the bounded non-degenerated
co-$t$-structures on $\Kb$ (see the Theorem \ref{Msc}).

We introduce and study the analogous of the Auslander-Buchweitz approximation
theory (see \cite{AB}) for triangulated categories $\mathcal{T}.$ We also
relate different kinds of relative homological dimensions by using suitable
subcategories of $\mathcal{T}.$ Moreover, we establish the existence of
preenvelopes (and precovers) in certain triangulated subcategories of
$\mathcal{T}.$

We first compare several algebraic notions of normality, from a categorical
viewpoint. Then we introduce an intrinsic description of Higgins' commutator
for ideal-determined categories, and we define a new notion of normality in
terms of this commutator. Our main result is to extend to any semi-abelian
category the following well-known characterization of normal subgroups: a
subobject $K$ is normal in $A$ if, and only if, $[A,K]\leq K$.

The complex numbers are an important part of quantum theory, but are
difficult to motivate from a theoretical perspective. We describe a simple
formal framework for theories of physics, and show that if a theory of physics
presented in this manner satisfies certain completeness properties, then it
necessarily includes the complex numbers as a mathematical ingredient. Central
to our approach are the techniques of category theory, and we introduce a new
category-theoretical tool, called the dagger-limit, which governs the way in
which systems can be combined to form larger systems.

This is the first draft of a book about higher categories approached by
iterating Segal's method, as in Tamsamani's definition of $n$-nerve and
Pelissier's thesis. If $M$ is a tractable left proper cartesian model category,
we construct a tractable left proper cartesian model structure on the category
of $M$-precategories. The procedure can then be iterated, leading to model
categories of $(\infty, n)$-categories.

A Quillen model structure on the category Gray-Cat of Gray-categories is
described, for which the weak equivalences are the triequivalences. It is shown
to restrict to the full subcategory Gray-Gpd of Gray-groupoids. This is used to
provide a functorial and model-theoretic proof of the unpublished theorem of
Joyal and Tierney that Gray-groupoids model homotopy 3-types. The model
structure on Gray-Cat is conjectured to be Quillen equivalent to a model
structure on the category Tricat of tricategories and strict homomorphisms of
tricategories.

The focus of this essay is a rigorous treatment of infinite games. An
infinite game is defined as a play consisting of a fixed number of players
whose sequence of moves is repeated, or iterated ad infinitum. Each sequence
corresponds to a single iteration of the play, where there are an infinite
amount of iterations. There are two distinct concepts within this broad
definition which encompass all infinite games: the strong infinite game and the
weak infinite game. Both differ in terms of imputations.

The goal of this paper is to prove coherence results with respect to
relational graphs for monoidal endofunctors, i.e. endofunctors of a monoidal
category that preserve 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.

We make explicit some conditions on a semi-abelian category $\mathbb D$ such
that the cohomology group homomorphisms $j^n_A:H^n_{M(\mathbb D/Y)}(A)\to
H^n_{\mathbb D/Y}(A)$, induced by the inclusion $j: Ab\mathbb D\totail \mathbb
D$ of the abelian objects of $\mathbb D$, are actually isomorphisms. These
conditions hold when $\mathbb D$ is the category $Gp$ of groups, and this
allows us to give a new insight on the Eilenberg-Mac Lane cohomology of groups.
They hold also when $\mathbb D$ is the category $\mathbb D=\mathbb K$-$Lie$ of
Lie-algebras.

The homotopy coherent nerve from simplicial categories to simplicial sets and
its left adjoint C are important to the study of (infinity,1)-categories
because they provide a means for comparing two models of their homotopy theory,
giving a Quillen equivalence between the model structures for quasi-categories
and simplicial categories. However, the hom-spaces of the simplicial category
CX arising from a quasi-category X are comparatively poorly understood. We show
that when X is a quasi-category, all 2,1-horns in the hom-spaces of its
simplicial category can be filled.

This paper presents non-commutative and structural notions of torsor. The two
are related by the machinery of Tannaka-Krein duality.

We give two examples of categorical axioms asserting that a canonically
defined natural transformation is invertible where the invertibility of any
natural transformation implies that the canonical one is invertible. The first
example is distributive categories, the second (semi-)additive ones. We show
that each follows from a general result about monoidal functors.

We consider Frobenius algebras and their bimodules in certain abelian
monoidal categories. In particular we study the Picard group of the category of
bimodules over a Frobenius algebra, i.e. the group of isomorphism classes of
invertible bimodules. The Rosenberg-Zelinsky sequence describes a homomorphism
from the group of algebra automorphisms to the Picard group, which however is
typically not surjective. We investigate under which conditions there exists a
Morita equivalent Frobenius algebra for which the corresponding homomorphism is
surjective.

The notion of Grothendieck topos may be considered as a generalisation of
that of topological space, one in which the points of the space may have
non-trivial automorphisms. However, the analogy is not precise, since in a
topological space, it is the points which have conceptual priority over the
open sets, whereas in a topos it is the other way around. Hence a topos is more
correctly regarded as a generalised locale, than as a generalised space.

We explore the relationship between polynomial functors and (rooted) trees.
In the first part we use polynomial functors to derive a new convenient
formalism for trees, and obtain a natural and conceptual construction of the
category $\Omega$ of Moerdijk and Weiss; its main properties are described in
terms of some factorisation systems. Although the constructions are motivated
and explained in terms of polynomial functors, they all amount to elementary
manipulations with finite sets.

In this paper we call generalized lax epimorphism a functor defined on a ring
with several objects, with values in an abelian AB5 category, for which the
associated restriction functor is fully faithful. We characterize such a
functor with the help of a conditioned right cancellation of another,
constructed in a canonical way from the initial one.

In the context of quantaloid-enriched categories, we explain how each
saturated class of weights defines, and is defined by, an essentially unique
full sub-KZ-doctrine of the free cocompletion KZ-doctrine. The KZ-doctrines
which arise as full sub-KZ-doctrines of the free cocompletion, are
characterised by two simple "fully faithfulness" conditions. Conical weights
form a saturated class, and the corresponding KZ-doctrine is precisely (the
generalisation to quantaloid-enriched categories of) the Hausdorff doctrine of
[Akhvlediani et al., 2009].

It is well known that "Fukaya category" is in fact an
$A_{\infty}$-pre-category in sense of Kontsevich and Soibelman \cite{KS}. The
reason is that in general the morphism spaces are defined only for transversal
pairs of Lagrangians, and higher products are defined only for transversal
sequences of Lagrangians. In \cite{KS} it is conjectured that for any graded
commutative ring $k,$ quasi-equivalence classes of $A_{\infty}$-pre-categories
over $k$ are in bijection with quasi-equivalence classes of
$A_{\infty}$-categories over $k$ with strict (or weak) identity morphisms.

We generalize the notion of identities among relations, well known for
presentations of groups, to presentations of n-categories by polygraphs. To
each polygraph, we associate a track n-category, generalizing the notion of
crossed module for groups, in order to define the natural system of identities
among relations. We relate the facts that this natural system is finitely
generated and that the polygraph has finite derivation type.

Bicategories of spans are characterized as cartesian bicategories in which
every comonad has an Eilenberg-Moore ob ject and every left adjoint arrow is
comonadic.

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 the preceding part (I) of this paper, we showed that for any torsion pair
(i.e., $t$-structure without the shift-closedness) in a triangulated category,
there is an associated abelian category, which we call the heart. Two extremal
cases of torsion pairs are $t$-structures and cluster tilting subcategories. If
the torsion pair comes from a $t$-structure, then its heart is nothing other
than the heart of this $t$-structure. In this case, as is well known, by
composing certain adjoint functors, we obtain a cohomological functor from the
triangulated category to the heart.

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 offer two proofs that categories weakly enriched over symmetric monoidal
categories can be strictified to categories enriched in permutative categories.
This is a "many 0-cells" version of the strictification of bimonoidal
categories to strict ones.

One of the open problems in higher category theory is the systematic
construction of the higher dimensional analogues of the Gray tensor product of
2-categories. In this paper we continue the developments of [3] and [2] by
understanding the natural generalisations of Gray's little brother, the funny
tensor product of categories. In fact we exhibit for any higher categorical
structure definable by an n-operad in the sense of Batanin [1], an analogous
tensor product which forms a symmetric monoidal closed structure on the
category of algebras of the operad.

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.

We develop further the algebra of cospans and spans of graphs introduced by
Katis, Sabadini and Walters for the sequential and parallel composition of
processes, adding here data types.

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.

For an associative ring $R$, let $P$ be an $R$-module with $S=\End_R(P)$. C.\
Menini and A. Orsatti posed the question of when the related functor
$\Hom_R(P,-)$ (with left adjoint $P\ot_S-$) induces an equivalence between a
subcategory of $_R\M$ closed under factor modules and a subcategory of $_S\M$
closed under submodules. They observed that this is precisely the case if the
unit of the adjunction is an epimorphism and the counit is a monomorphism. A
module $P$ inducing these properties is called a $\star$-module.

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.

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 $n$.

We generalize Barr's embedding theorem for regular categories to the context
of enriched categories.

This article is intended as a reference guide to various notions of monoidal
categories and their associated string diagrams. It is hoped that this will be
useful not just to mathematicians, but also to physicists, computer scientists,
and others who use diagrammatic reasoning.