28 citations · 159 across the 26 of their papers we have counts for
12 papers · 1 filter
Categorical algebra of conditional probability
Mika Bohinen, Paolo Perrone
In the field of categorical probability, one uses concepts and techniques from category theory, such as monads and monoidal categories, to study the structures of probability and s…
How to Represent Non-Representable Functors
Paolo Perrone
The arrows of a category are elements of particular sets, the hom-sets. These sets are functorial, and their functoriality specifies how to compose the arrows with other arrows of…
Convergence of martingales via enriched dagger categories
Paolo Perrone, Ruben Van Belle
We provide a categorical proof of convergence for martingales and backward martingales in mean, using enriched category theory. The enrichment we use is in topological spaces, with…
Weakly Markov categories and weakly affine monads
Tobias Fritz, Fabio Gadducci, Paolo Perrone +1
Introduced in the 1990s in the context of the algebraic approach to graph rewriting, gs-monoidal categories are symmetric monoidal categories where each object is equipped with the…
Dilations and information flow axioms in categorical probability
Tobias Fritz, Tomáš Gonda, Nicholas Gauguin Houghton-Larsen +3
We study the positivity and causality axioms for Markov categories as properties of dilations and information flow in Markov categories, and in variations thereof for arbitrary sem…
Lifting couplings in Wasserstein spaces
Paolo Perrone
This paper makes mathematically precise the idea that conditional probabilities are analogous to path liftings in geometry. The idea of lifting is modelled in terms of the category…
Weak cartesian properties of simplicial sets
Carmen Constantin, Tobias Fritz, Paolo Perrone +1
Many special classes of simplicial sets, such as the nerves of categories or groupoids, the 2-Segal sets of Dyckerhoff and Kapranov, and the (discrete) decomposition spaces of Gálv…
Kan extensions are partial colimits
Paolo Perrone, Walter Tholen
One way of interpreting a left Kan extension is as taking a kind of "partial colimit", whereby one replaces parts of a diagram by their colimits. We make this intuition precise by…
Partial Evaluations and the Compositional Structure of the Bar Construction
Carmen Constantin, Paolo Perrone, Tobias Fritz +1
The algebraic expression can be evaluated to , but it can also be partially evaluated to . In categorical algebra, such partial evaluations can be defined in…
Notes on Category Theory with examples from basic mathematics
Paolo Perrone
These notes were originally developed as lecture notes for a category theory course. They should be well-suited to anyone that wants to learn category theory from scratch and has a…
Monads, partial evaluations, and rewriting
Tobias Fritz, Paolo Perrone
Monads can be interpreted as encoding formal expressions, or formal operations in the sense of universal algebra. We give a construction which formalizes the idea of "evaluating an…
A Criterion for Kan Extensions of Lax Monoidal Functors
Tobias Fritz, Paolo Perrone
In this mainly expository note, we state a criterion for when a left Kan extension of a lax monoidal functor along a strong monoidal functor can itself be equipped with a lax monoi…