Related papers: Categories for Me, and You?
In this study, internal categories in the category of the crossed modules are characterized and it has been shown that there is a natural equivalence between the category of the crossed modules over crossed modules, i.e. crossed squares,…
Recall that the definition of the $K$-theory of an object C (e.g., a ring or a space) has the following pattern. One first associates to the object C a category A_C that has a suitable structure (exact, Waldhausen, symmetric monoidal, ...).…
We give a presentation of Feynman categories from a representation--theoretical viewpoint. Feynman categories are a special type of monoidal categories and their representations are monoidal functors. They can be viewed as a far reaching…
A relevant category is a symmetric monoidal closed category with a diagonal natural transformation that satisfies some coherence conditions. Every cartesian closed category is a relevant category in this sense. The denomination 'relevant'…
Categorification is the process of finding category-theoretic analogs of set-theoretic concepts by replacing sets with categories, functions with functors, and equations between functions by natural isomorphisms between functors, which in…
This survey article is intended as an introduction to the recent categorical classification theorems of the three authors, restricting to the special case of the category of modules for a finite group.
We study, in an abstract axiomatic setting, the notion of sectional category of a morphism. From this, we unify and generalize known results about this invariant in different settings as well as we deduce new applications.
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…
We establish a DK-equivalence between the relative category of $\pi$-tribes and the relative category of locally cartesian closed quasicategories. From this follows one of the internal languages conjecture: Martin-L\"of type theory with…
We introduce a theory for encoding and manipulating algebraic data on categories via $\textit{concentration structures}$, which are equivalence relations on morphisms that satisfy certain axioms. For any category with a concentration…
The monumental treatise "\'El\'ements de math\'ematique" of N. Bourbaki is based on the notion of structure and on the theory of sets. On the other hand, the theory of categories is based on the notions of morphism and functor. An…
This article introduces Hilbert $*$-categories: an abstraction of categories with similar algebraic and analytic properties to the categories of real, complex, and quaternionic Hilbert spaces and bounded linear maps. Other examples include…
Categories are coreflectively embedded in multicategories via the "discrete cocone" construction, the right adjoint being given by the monoid construction. Furthermore, the adjunction lifts to the "cartesian level": preadditive categories…
The paper is devoted to a categorical study of the category of probabilistic metric spaces. The study is based on an isomorphic description of the category of probabilistic metric spaces. The isomorphic description was obtained in [3] and…
In [1] we introduced the concept of structured space, which is a topological space that locally resembles some algebraic structures. In [2] we proceeded the study of these spaces, developing two cohomology theories. The aim of this paper is…
This dissertation has two main parts. The first part deals with questions relating to Haghverdi and Scott's notion of partially traced categories. The main result is a representation theorem for such categories: we prove that every…
The representation theory for categorical groups is constructed. Each categorical group determines a monoidal bicategory of representations. Typically, these categories contain representations which are indecomposable but not irreducible. A…
We present here definitions and constructions basic for the theory of monoidal and tensor categories. We provide references to the original sources, whenever possible. Group-theoretical categories are used as examples
Algebraic theories with dependency between sorts form the structural core of Martin-L\"of type theory and similar systems. Their denotational semantics are typically studied using categorical techniques; many different categorical…
We discuss what it means for a symmetric monoidal category to be a module over a commutative semiring category. Each of the categories of (1) cartesian monoidal categories, (2) semiadditive categories, and (3) connective spectra can be…