Related papers: Methods of constructive category theory
Starting from a generalized Reedy category $R$ satisfying a simple condition, we construct an absolutely dense functor $\mathbf{D}_R \to R$ with domain a strict Reedy category. In the case of a generalized inverse category $R$, and given…
In these lecture notes, we give a brief introduction to some elements of category theory. The choice of topics is guided by applications to functional programming. Firstly, we study initial algebras, which provide a mathematical…
Representation theorems relate seemingly complex objects to concrete, more tractable ones. In this paper, we take advantage of the abstraction power of category theory and provide a general representation theorem for a wide class of…
Category theory has foundational importance because it provides conceptual lenses to characterize what is important in mathematics. Originally the main lenses were universal mapping properties and natural transformations. In recent decades,…
Let $\mathcal{X}$ be a resolving and contravariantly finite subcategory of $\rm{mod}\mbox{-}\Lambda$, the category of finitely generated right $\Lambda$-modules. We associate to $\mathcal{X}$ the subcategory…
We introduce judgemental theories and their calculi as a general framework to present and study deductive systems. As an exemplification of their expressivity, we approach dependent type theory and natural deduction as special kinds of…
There are many ways to present model categories, each with a different point of view. Here we'd like to treat model categories as a way to build and control resolutions. This an historical approach, as in his original and spectacular…
Recently, the Johnson-McCarthy discrete calculus for homotopy functors was extended to include functors from an unbased simplicial model category to spectra. This paper completes the constructions needed to ensure that there exists a…
Toy models have been used to separate important features of quantum computation from the rich background of the standard Hilbert space model. Category theory, on the other hand, is a general tool to separate components of mathematical…
Bayes' rule tells us how to invert a causal process in order to update our beliefs in light of new evidence. If the process is believed to have a complex compositional structure, we may observe that the inversion of the whole can be…
It is proved that MacLane's coherence results for monoidal and symmetric monoidal categories can be extended to some other categories with multiplication; namely, to relevant, affine and cartesian categories. All results are formulated in…
This paper is the fourth in a series whose goal is to develop a fundamentally new way of building theories of physics. The motivation comes from a desire to address certain deep issues that arise in the quantum theory of gravity. Our basic…
Let $R$ be a commutative unital ring. We construct a category $\mathcal{C}_R$ of fractions $X/G$, where $G$ is a finite group and $X$ is a finite $G$-set, and with morphisms given by $R$-linear combinations of spans of bisets. This category…
We introduce the notion of a diagram category and discuss its application to the invariant theory of classical groups and super groups, with some indications concerning extensions to quantum groups and quantum super groups. Tensor functors…
We give a criterion for a functor \(F:C\rightarrow B\) between small categories to generate a small presentation of the universal model category \(U(B)\) in the sense of Dugger.
Given an initial family of sets, we may take unions, intersections and complements of the sets contained in this family in order to form a new collection of sets; our construction process is done recursively until we obtain the last family.…
We compute certain Ext and Tor groups in the category of all functors from an Z/p-linear additive category A to vector spaces in terms of Ext and Tor computed in the full subcategory of additive functors from A to vector spaces. We thus…
We define a differential Tannakian category and show that under a natural assumption it has a fibre functor. If in addition this category is neutral, that is, the target category for the fibre functor are finite dimensional vector spaces…
This article gives an elementary and formal 2-categorical construction of a bicategory of right fractions analogous to anafunctors, starting from a 2-category equipped with a family of covering maps that are fully faithful and co-fully…
We generalize the notion of ends and coends in category theory to the realm of module categories over finite tensor categories. We call this new concept "module (co)end". This tool allows us to give different proofs to several known results…