Related papers: Directed Containers as Categories
Lenses, optics and dependent lenses (or equivalently morphisms of containers, or equivalently natural transformations of polynomial functors) are all widely used in applied category theory as models of bidirectional processes. From the…
We give an alternate conception of string diagrams as labeled 1-dimensional oriented cobordisms, the operad of which we denote by Cob/O, where O is the set of string labels. The axioms of traced (symmetric monoidal) categories are fully…
The category of flows is not cartesian closed. We construct a closed symmetric monoidal structure which has moreover a satisfactory behavior from the computer scientific viewpoint.
Every directed set is Tukey equivalent to (a) the family of all compact subsets, ordered by inclusion, of a (locally compact) space, to (b) a neighborhood filter, ordered by reverse inclusion, of a point (of a compact space, and of a…
The structure of the category of matroids and strong maps is investigated: it has coproducts and equalizers, but not products or coequalizers; there are functors from the categories of graphs and vector spaces, the latter being faithful;…
A mechanism responsible for the directed transport and molecular separation in a symmetric channel is proposed. We found that under the action of spatial harmonic oscillations of the channel, the system exhibits a directed transport in…
The filter quotient construction is a particular instance of a filtered colimit of categories. It has primarily been considered in the context of categorical logic, where it has been used effectively to construct non-trivial models, for…
Doctrines are categorical structures very apt to study logics of different nature within a unified environment: the 2-category Dtn of doctrines. Modal interior operators are characterised as particular adjoints in the 2-category Dtn. We…
We introduce constructible directed complexes, a combinatorial presentation of higher categories inspired by constructible complexes in poset topology. Constructible directed complexes with a greatest element, called atoms, encompass common…
Usually, mathematical objects have highly parallel interpretations. In this paper, we consider them as sequential constructors of other objects. In particular, we prove that every reflexive directed graph can be interpreted as a program…
We present a categorical model for intuitionistic linear logic where objects are polynomial diagrams and morphisms are simulation diagrams. The multiplicative structure (tensor product and its adjoint) can be defined in any locally…
Skew-monoidal categories arise when the associator and the left and right units of a monoidal category are, in a specific way, not invertible. We prove that the closed skew-monoidal structures on the category of right R-modules are…
Mott noted a one-to-one correspondence between saturated multiplicatively closed subsets of a domain D and directed convex subgroups of the group of divisibility D. With this, we construct a functor between inclusions into saturated…
A matrix (and any associated linear system) will be referred to as structured if it has a small displacement rank. It is known that the inverse of a structured matrix is structured, which allows fast inversion (or solution), and reduced…
This is a book on higher-categorical diagrams, including pasting diagrams. It aims to provide a thorough and modern reference on the subject, collecting, revisiting and expanding results scattered across the literature, informed by recent…
Topological spaces - such as classifying spaces, configuration spaces and spacetimes - often admit extra temporal structure. Qualitative invariants on such directed spaces often are more informative yet more difficult to calculate than…
Categories, n-categories, double categories, and multicategories (among others) all have similar definitions as collections of cells with composition operations. We give an explicit description of the information required to define any…
The field of directed type theory seeks to design type theories capable of reasoning synthetically about (higher) categories, by generalizing the symmetric identity types of Martin-L\"of Type Theory to asymmetric hom-types. We articulate…
We show that the categories of directed and undirected reflexive graphs carry exactly two (up to isomorphism) biclosed monoidal structures.
We study polynomial comonads and polynomial bicomodules. Polynomial comonads amount to categories. Polynomial bicomodules between categories amount to parametric right adjoint functors between corresponding copresheaf categories. These may…