111 research outputs found
Enriched Fell Bundles and Spaceoids
We propose a definition of involutive categorical bundle (Fell bundle)
enriched in an involutive monoidal category and we argue that such a structure
is a possible suitable environment for the formalization of different
equivalent versions of spectral data for commutative C*-categories.Comment: 12 pages, AMS-LaTeX2e, to be published in "Proceedings of 2010 RIMS
Thematic Year on Perspectives in Deformation Quantization and Noncommutative
Geometry
The Formal Theory of Monads, Univalently
We develop the formal theory of monads, as established by Street, in univalent foundations. This allows us to formally reason about various kinds of monads on the right level of abstraction. In particular, we define the bicategory of monads internal to a bicategory, and prove that it is univalent. We also define Eilenberg-Moore objects, and we show that both Eilenberg-Moore categories and Kleisli categories give rise to Eilenberg-Moore objects. Finally, we relate monads and adjunctions in arbitrary bicategories. Our work is formalized in Coq using the https://github.com/UniMath/UniMath library
Univalent Double Categories
Category theory is a branch of mathematics that provides a formal framework
for understanding the relationship between mathematical structures. To this
end, a category not only incorporates the data of the desired objects, but also
"morphisms", which capture how different objects interact with each other.
Category theory has found many applications in mathematics and in computer
science, for example in functional programming. Double categories are a natural
generalization of categories which incorporate the data of two separate classes
of morphisms, allowing a more nuanced representation of relationships and
interactions between objects. Similar to category theory, double categories
have been successfully applied to various situations in mathematics and
computer science, in which objects naturally exhibit two types of morphisms.
Examples include categories themselves, but also lenses, petri nets, and spans.
While categories have already been formalized in a variety of proof assistants,
double categories have received far less attention. In this paper we remedy
this situation by presenting a formalization of double categories via the proof
assistant Coq, relying on the Coq UniMath library. As part of this work we
present two equivalent formalizations of the definition of a double category,
an unfolded explicit definition and a second definition which exhibits
excellent formal properties via 2-sided displayed categories. As an application
of the formal approach we establish a notion of univalent double category along
with a univalence principle: equivalences of univalent double categories
coincide with their identitie
Univalent Double Categories
Category theory is a branch of mathematics that provides a formal framework for understanding the relationship between mathematical structures. To this end, a category not only incorporates the data of the desired objects, but also "morphisms", which capture how different objects interact with each other. Category theory has found many applications in mathematics and in computer science, for example in functional programming.Double categories are a natural generalization of categories which incorporate the data of two separate classes of morphisms, allowing a more nuanced representation of relationships and interactions between objects. Similar to category theory, double categories have been successfully applied to various situations in mathematics and computer science, in which objects naturally exhibit two types of morphisms. Examples include categories themselves, but also lenses, petri nets, and spans.While categories have already been formalized in a variety of proof assistants, double categories have received far less attention. In this paper we remedy this situation by presenting a formalization of double categories via the proof assistant Coq, relying on the Coq UniMath library. As part of this work we present two equivalent formalizations of the definition of a double category, an unfolded explicit definition and a second definition which exhibits excellent formal properties via 2-sided displayed categories. As an application of the formal approach we establish a notion of univalent double category along with a univalence principle: equivalences of univalent double categories coincide with their identities
Constructing Higher Inductive Types as Groupoid Quotients
In this paper, we study finitary 1-truncated higher inductive types (HITs) in
homotopy type theory. We start by showing that all these types can be
constructed from the groupoid quotient. We define an internal notion of
signatures for HITs, and for each signature, we construct a bicategory of
algebras in 1-types and in groupoids. We continue by proving initial algebra
semantics for our signatures. After that, we show that the groupoid quotient
induces a biadjunction between the bicategories of algebras in 1-types and in
groupoids. Then we construct a biinitial object in the bicategory of algebras
in groupoids, which gives the desired algebra. From all this, we conclude that
all finitary 1-truncated HITs can be constructed from the groupoid quotient.
We present several examples of HITs which are definable using our notion of
signature. In particular, we show that each signature gives rise to a HIT
corresponding to the freely generated algebraic structure over it. We also
start the development of universal algebra in 1-types. We show that the
bicategory of algebras has PIE limits, i.e. products, inserters and equifiers,
and we prove a version of the first isomorphism theorem for 1-types. Finally,
we give an alternative characterization of the foundamental groups of some
HITs, exploiting our construction of HITs via the groupoid quotient. All the
results are formalized over the UniMath library of univalent mathematics in
Coq
Formalizing Monoidal Categories and Actions for Syntax with Binders
We discuss some aspects of our work on the mechanization of syntax and
semantics in the UniMath library, based on the proof assistant Coq. We focus on
experiences where Coq (as a type-theoretic proof assistant with decidable
typechecking) made us use more theory or helped us to see theory more clearly.Comment: Abstract for a talk at CoqPL 2023,
https://popl23.sigplan.org/details/CoqPL-2023-papers/7/Formalizing-Monoidal-Categories-and-Actions-for-Syntax-with-Binder
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