Related papers: Cartesian differential categories revisited
Derivations provide a way of transporting ideas from the calculus of manifolds to algebraic settings where there is no sensible notion of limit. In this paper, we consider derivations in certain monoidal categories, called codifferential…
We present an intrinsic and concrete development of the subdivision of small categories, give some simple examples and derive its fundamental properties. As an application, we deduce an alternative way to compare the homotopy categories of…
Drawing on well-known results from the theory of canonical extensions and the theory of categories enriched over a quantale, we define canonical extensions of quantale-enriched categories and establish their basic properties.
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.
We examine the use of classes to formulate several categorical notions. This leads to two proposals: an explicit structure for working with subobjects, and a hierarchy of $k$-classes. We apply the latter to both ordinary and higher…
Pre-Tannakian categories are a natural class of tensor categories that can be viewed as generalizations of algebraic groups. We define a pre-Tannkian category to be discrete if it is generated by an \'etale commutative algebra; these…
In the category of sets and partial functions, $\mathsf{PAR}$, while the disjoint union $\sqcup$ is the usual categorical coproduct, the Cartesian product $\times$ becomes a restriction categorical analogue of the categorical product: a…
We study and relate categories of modules, comodules and contramodules over a representation of a small category taking values in (co)algebras, in a manner similar to modules over a ringed space. As a result, we obtain a categorical…
Cartesian differential categories provide a categorical framework for multivariable differential calculus and also the categorical semantics of the differential $\lambda$-calculus. Taylor series expansion is an important concept for both…
The notion of Kan extendable subcategories was initially introduced to define the category of compactly generated fibrewise topological spaces over a T1 base space and to establish its cartesian closure. In this paper, we show that the same…
We show that the category of comodules over a coassociative coalgebra in a complete, cocomplete and well-powered category has limits and colimits under additional assumptions.
We show that the derived category of any singularity over a field of characteristic 0 can be embedded fully and faithfully into a smooth triangulated category which has a semiorthogonal decomposition with components equivalent to derived…
We develop a categorical framework for reasoning about abstract properties of differentiation, based on the theory of fibrations. Our work encompasses the first-order fragments of several existing categorical structures for differentiation,…
The notion of a categorical quotient can be generalized since its standard categorical concept does not recover the expected quotients in certain categories. We present a more general formulation in the form of $\mathcal{F}$-quotients in a…
Differential categories were introduced to provide a minimal categorical doctrine for differential linear logic. Here we revisit the formalism and, in particular, examine the two different approaches to defining differentiation which were…
We propose a generalization of Quillen's exact category -- arithmetic exact category and we discuss conditions on such categories under which one can establish the notion of Harder-Narasimhan filtrations and Harder-Narsimhan polygons.…
We prove that the semantics of intuitionistic linear logic in vector spaces which uses cofree coalgebras is also a model of differential linear logic, and that the Cartesian closed category of cofree coalgebras is a model of the…
Restriction categories were introduced to provide an axiomatic setting for the study of partially defined mappings; they are categories equipped with an operation called restriction which assigns to every morphism an endomorphism of its…
We show that the category of categories with pullbacks and pullback preserving functors is cartesian closed.
In this thesis, we introduce Cartesian double categories, motivated by the work of Carboni, Kelly, Walters, and Wood on Cartesian bicategories. Moving from bicategories to the slightly more generalized notion of double categories allows us…