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View article: Di- is for Directed: First-Order Directed Type Theory via Dinaturality
Di- is for Directed: First-Order Directed Type Theory via Dinaturality Open
We show how dinaturality plays a central role in the interpretation of directed type theory where types are given by (1-)categories and directed equality by hom-functors. We introduce a first-order directed type theory where types are sema…
View article: Directed First-Order Logic
Directed First-Order Logic Open
We present a first-order logic equipped with an "asymmetric" directed notion of equality, which can be thought of as transitions/rewrites between terms, allowing for types to be interpreted as preorders. We then provide a universal propert…
View article: Monads and limits in bicategories of circuits
Monads and limits in bicategories of circuits Open
We study monads in the (pseudo-)double category $\mathbf{KSW}(\mathcal{K})$ where loose arrows are Mealy automata valued in an ambient monoidal category $\mathcal{K}$, and the category of tight arrows is $\mathcal{K}$. Such monads turn out…
View article: Fibrations of algebras
Fibrations of algebras Open
We study fibrations arising from indexed categories of the following form: fix two categories $\mathcal{A},\mathcal{X}$ and a functor $F : \mathcal{A} \times \mathcal{X} \longrightarrow\mathcal{X} $, so that to each $F_A=F(A,-)$ one can as…
View article: Profunctor Optics, a Categorical Update
Profunctor Optics, a Categorical Update Open
Optics are bidirectional data accessors that capture data transformation patterns such as accessing subfields or iterating over containers. Profunctor optics are a particular choice of representation supporting modularity, meaning that we …
View article: Automata and coalgebras in categories of species
Automata and coalgebras in categories of species Open
We study generalized automata (in the sense of Adámek-Trnková) in Joyal's category of (set-valued) combinatorial species, and as an important preliminary step, we study coalgebras for its derivative endofunctor $\partial$ and for the "Eule…
View article: Bicategories of Automata, Automata in Bicategories
Bicategories of Automata, Automata in Bicategories Open
We study bicategories of (deterministic) automata, drawing from prior work of\nKatis-Sabadini-Walters, and Di Lavore-Gianola-Rom\\'an-Sabadini-Soboci\\'nski,\nand linking their bicategories of `processes' to a bicategory of Mealy machines\…
View article: Differential 2-rigs
Differential 2-rigs Open
We study the notion of a "differential 2-rig", a category R with coproducts and a monoidal structure distributing over them, also equipped with an endofunctor D : R -> R that satisfies a categorified analogue of the Leibniz rule. This is i…
View article: Fibrational Linguistics (FibLang): Language Acquisition
Fibrational Linguistics (FibLang): Language Acquisition Open
In this work we show how FibLang, a category-theoretic framework concerned with the interplay between language and meaning, can be used to describe vocabulary acquisition, that is the process with which a speaker acquires new vocabulary (t…
View article: Adjoint functor theorems for lax-idempotent pseudomonads
Adjoint functor theorems for lax-idempotent pseudomonads Open
For each pair of lax-idempotent pseudomonads $R$ and $I$, for which $I$ is locally fully faithful and $R$ distributes over $I$, we establish an adjoint functor theorem, relating $R$-cocontinuity to adjointness relative to $I$. This provide…
View article: The semibicategory of Moore automata
The semibicategory of Moore automata Open
We study the semibicategory $\textsf{Mre}$ of "Moore automata": an arrangement of objects, 1- and 2-cells which is inherently and irredeemably nonunital in dimension one. Between the semibicategory of Moore automata and the better behaved …
View article: Completeness for categories of generalized automata
Completeness for categories of generalized automata Open
We present a slick proof of completeness and cocompleteness for categories of $F$-automata, where the span of maps $E\leftarrow E\otimes I \to O$ that usually defines a deterministic automaton of input $I$ and output $O$ in a monoidal cate…
View article: A Categorical Semantics for Bounded Petri Nets
A Categorical Semantics for Bounded Petri Nets Open
We provide a categorical semantics for bounded Petri nets, both in the\ncollective- and individual-token philosophy. In both cases, we describe the\nprocess of bounding a net internally, by just constructing new categories of\nexecutions o…
View article: Fibrational Linguistics (FibLang): Language Acquisition
Fibrational Linguistics (FibLang): Language Acquisition Open
In this work we show how FibLang, a category-theoretic framework concerned with the interplay between language and meaning, can be used to describe vocabulary acquisition, that is the process with which a speaker acquires new vocabulary (t…
View article: Fibrational linguistics: First concepts
Fibrational linguistics: First concepts Open
We define a general mathematical framework for linguistics based on the theory of fibrations, called FibLang. We start by modelling the interaction between linguistics and cognition in the most general way possible, with a heavy focus on c…
View article: (Co)end Calculus
(Co)end Calculus Open
The language of ends and (co)ends provides a natural and general way of expressing many phenomena in category theory, in the abstract and in applications. Yet although category-theoretic methods are now widely used by mathematicians, since…
View article: Escrows are optics
Escrows are optics Open
We provide a categorical interpretation for escrows, i.e. trading protocols in trustless environment, where the exchange between two agents is mediated by a third party where the buyer locks the money until they receive the goods they want…
View article: Escrows are optics
Escrows are optics Open
We provide a categorical interpretation for _escrows_, i.e. trading protocols in trustless environment, where the exchange between two agents is mediated by a third party where the buyer locks the money until they receive the goods they wa…
View article: Differential 2-rigs
Differential 2-rigs Open
We explore the notion of a category with coproducts $\cup$ and a monoidal structure $\otimes$ distributing over it, endowed with an endo-functor $\partial$ which is linear and Leibniz. Such $\partial$ can be legitimately called a *derivati…
View article: A Categorical Semantics for Bounded Petri Nets
A Categorical Semantics for Bounded Petri Nets Open
We provide a categorical semantics for bounded Petri nets, both in the collective- and individual-token philosophy. In both cases, we describe the process of bounding a net internally, by just constructing new categories of executions of a…
View article: Rosen's no-go theorem for regular categories
Rosen's no-go theorem for regular categories Open
The famous biologist Robert Rosen argued for an intrinsic difference between biological and artificial life, supporting the claim that `living systems are not mechanisms'. This result, understood as the claim that life-like mechanisms are …
View article: Categorical Ontology I - Existence
Categorical Ontology I - Existence Open
The present paper is the first piece of a series whose aim is to develop an approach to ontology and metaontology through category theory. We exploit the theory of elementary toposes to claim that a satisfying ``theory of existence'', and …
View article: t-structures on stable infinity-categories
t-structures on stable infinity-categories Open
The present work re-enacts the classical theory of t-structures reducing the classical definition given in *Faisceaux Pervers* to a rather primitive categorical gadget: suitable reflective factorization systems. This translation is only po…
View article: Profunctor Optics, a Categorical Update
Profunctor Optics, a Categorical Update Open
Optics are bidirectional data accessors that capture data transformation patterns such as accessing subfields or iterating over containers. Profunctor optics are a particular choice of representation supporting modularity, meaning that we …
View article: A Fubini rule for $\infty$-coends
A Fubini rule for $\infty$-coends Open
We prove a Fubini rule for $\infty$-co/ends of $\infty$-functors $F : \mathcal C^\text{op}\times\mathcal C\to \mathcal D$. This allows to lay down "integration rules", similar to those in classical co/end calculus, also in the setting of $…