Related papers: Derivations as Algebras
Higher-dimensional category theory is the study of n-categories, operads, braided monoidal categories, and other such exotic structures. It draws its inspiration from areas as diverse as topology, quantum algebra, mathematical physics,…
A differential modality is a comonad on an additive symmetric monoidal category $(\mathsf{C},\otimes,I)$, whose underlying functor we denote $!\colon\mathsf{C} \rightarrow \mathsf{C}$, together with some additional structure including a…
The principle behind algebraic language theory for various kinds of structures, such as words or trees, is to use a compositional function from the structures into a finite set. To talk about compositionality, one needs some way of…
We study $N$-differential graded ($NDG$) categories and their the derived categories. First, we introduce $N$-differential modules over an $NDG$ category $\mathcal{A}$. Then we show that the category $\mathsf{C}_{Ndg}(\mathcal{A})$ of…
We continue the investigation of tabular algebras with trace (a certain class of associative ${\Bbb Z}[v, v^{-1}]$-algebras equipped with distinguished bases) by determining the extent to which the tabular structure may be recovered from a…
In general, all constructions of algebraic topology are functorial; the notions of category, functor and natural transformation originated here. The arrow categories are more simple forms of the \emph{comma} categories and were introduced…
We study thick subcategories of derived categories of gentle algebras. Any thick subcategory of a derived category of a gentle algebra is generated by a set of string objects or a set of band objects. We show the thick subcategories…
We present a common framework to study varieties in great generality from a categorical point of view. The main application of this study is in the setting of algebraic categories, where we introduce Birkhoff varieties which are essentially…
This is the second in a series of papers laying the foundations for a differential graded approach to derived differential geometry (and other geometries in characteristic zero). In this paper, we extend the classical notion of a dg-algebra…
We view difference algebra as the study of algebraic objects in the topos of difference sets. The methods of topos theory and categorical logic enable us to develop difference homological algebra, identify a solid foundation for difference…
A lifting of a semilattice S is an algebra A such that the semilattice of compact (=finitely generated) congruences of A is isomorphic to S. The aim of this work is to give a categorical theory of partial algebras endowed with a partial…
Tangent categories are categories equipped with a tangent functor: an endofunctor with certain natural transformations which make it behave like the tangent bundle functor on the category of smooth manifolds. They provide an abstract…
The theories of strings and $D$-branes have motivated the development of non Abelian cohomology techniques in differential geometry, on the purpose to find a geometric interpretation of characteristic classes. The spaces studied here, like…
We study "higher-dimensional" generalizations of differential forms. Just as differential forms can be defined as the universal commutative differential algebra containing C^\infty(M), we can define differential gorms as the universal…
We develop further the theory of operads and analytic functors. In particular, we introduce a bicategory that has operads as 0-cells, operad bimodules as 1-cells and operad bimodule maps as 2-cells, and prove that this bicategory is…
Operads often arise from geometry. The standard $A_\infty$ operad can be derived from the cellular chains on the Stasheff associahedra, and an $A_\infty$ algebra is an algebra over this operad. The notion of an $\mathbf{fc}$-multicategory,…
Having in mind applications to particle physics we develop the differential calculus over Jordan algebras and the theory of connections on Jordan modules. In particular we focus on differential calculus over the exceptional Jordan algebra…
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…
Category theory unifies mathematical concepts, aiding comparisons across structures by incorporating objects and morphisms, which capture their interactions. It has influenced areas of computer science such as automata theory, functional…
We pose a new algebraic formalism for studying differential calculus in vector bundles. This is achieved by studying various functors of differential calculus over arbitrary graded commutative algebras (DCGCA) and applying this language to…