Christina Vasilakopoulou | University of Patras

Christina Vasilakopoulou

University of Patras, Mathematics, Post-Doc

University of Patras, Mathematics, Post-Doc |

Category Theory

127 Followers | 105 Following | 5 Co-authors | —Total Views

Ph.D. Thesis

Generalization of Algebraic Operations Via Enrichment

Ph.D Thesis, University of Cabridge, 2014

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Papers

A Larson-Sweedler Theorem for Hopf V-categories

The aim of this paper is to extend the classical Larson-Sweedler theorem, namely that a k-bialgeb... more The aim of this paper is to extend the classical Larson-Sweedler theorem, namely that a k-bialgebra has a non-singular integral (and in particular is Frobenius) if and only if it is a finite dimensional Hopf algebra, to the 'many-object' setting of Hopf categories. To this end, we provide new characterizations of Frobenius V-categories and we develop the integral theory for Hopf V-categories. Our results apply to Hopf algebras in any braided monoidal category as a special case, and also relate to Turaev's Hopf group algebras and particular cases of weak and multiplier Hopf algebras.

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Monoidal Grothendieck Construction

We lift the standard equivalence between fibrations and indexed categories to an equivalence betw... more We lift the standard equivalence between fibrations and indexed categories to an equivalence between monoidal fibrations and monoidal indexed categories, namely weak monoidal pseudofunctors to the 2-category of categories. In doing so, we investigate the relation between this `global' monoidal structure where the total category is monoidal and the fibration strictly preserves the structure, and a `fibrewise' one where the fibres are monoidal and the reindexing functors strongly preserve the structure, first hinted by Shulman. In particular, when the domain is cocartesian monoidal, lax monoidal structures on the functor to the 2-category of categories correspond to lifts of the functor to the 2-category of monoidal categories. Finally, we give examples where this correspondence appears, spanning from the fundamental and family fibrations to network models and systems.

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On Enriched Fibrations

Cahiers de Topologie et Géométrie Différentielle Catégoriques, 2018

We introduce the notion of an enriched fibration, i.e. a fibration whose total category and base... more We introduce the notion of an enriched fibration, i.e. a fibration
whose total category and base category are enriched in those of a monoidal fibration in an appropriate way. Furthermore, we provide a way to obtain such a structure, starting from actions of monoidal categories with parameterized adjoints. The motivating goal is to capture certain example cases, like the fibration of modules over algebras enriched in the opfibration of comodules over coalgebras.

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Oplax Hopf Algebras

by Mitchell A Buckley and Christina Vasilakopoulou

We introduce the notion of (op)lax Hopf monoids in a monoidal 2-category and show that Hopf V-cat... more We introduce the notion of (op)lax Hopf monoids in a monoidal 2-category and show that Hopf V-categories introduced in [BCV16] are a particular type of oplax Hopf monoids in the 2-category Span|V described in [Böh16]. We also introduce Frobe-nius V-categories as the Frobenius objects in this category.

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Enriched Duality in Double Categories: V-categories and V-cocategories

Journal of Pure and Applied Algebra, 2019

In this work, we explore a double categorical framework for categories of enriched graphs, catego... more In this work, we explore a double categorical framework for categories of enriched graphs, categories and the newly introduced notion of cocategories. A fundamental goal is the establishment of an enrichment of V-categories in V-cocategories, which generalizes the so-called Sweedler theory relatively to an enrichment of algebras in coalgebras. The language employed is that of V-matrices, and an interplay between the double categorical and bicategorical perspective provides a high-level flexibility for demonstrating essential features of these dual structures. Furthermore, we investigate enriched fibration aspects of categories of monads and comon-ads in fibrant double categories a.k.a. proarrow equipments, which lead to natural applications for their materialization as categories and cocategories in the enriched matrices setting.

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Measuring Comodules and Enrichment

by Christina Vasilakopoulou and Ignacio Franco

We study the existence of universal measuring comodules Q(M, N) for a pair of modules M , N in a ... more We study the existence of universal measuring comodules Q(M, N) for a pair of modules M , N in a braided monoidal closed category, and the associated enrichment of the global category of modules over the monoidal global category of comodules. In the process, we use results for general fibred adjunctions encompassing the fibred structure of modules over monoids and the opfibred structure of comodules over comonoids. We also explore applications to the theory of Hopf modules.

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Digree: A Middleware for a Graph Databases Polystore

by vasilis spiropoulos and Christina Vasilakopoulou

2016 IEEE International Conference on Big Data (Big Data)

— In this paper we present Digree, an experimental middleware system that can execute graph patte... more — In this paper we present Digree, an experimental middleware system that can execute graph pattern matching queries over databases hosting voluminous graph datasets. First, we formally present the employed data model and the processes of rewriting a query into an equivalent set of subqueries and subsequently combining the partial results into the final result set. Our framework guarantees the correctness and completeness of the produced answers. Then, we present a prototype implementation of Digree, which is agnostic to the underlying data processing engines used at the endpoints. As the experimental results show, in many cases Digree outper-forms a single node graph database deployment in execution speed, up to 20 times depending on the query type.

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Dynamical Systems and Sheaves

Applied Categorical Structures, 2019

A categorical framework for modeling and analyzing systems in a broad sense is proposed. These s... more A categorical framework for modeling and analyzing systems in a broad sense
is proposed. These systems should be thought of as ‘machines’ with inputs and outputs,
carrying some sort of signal that occurs through some notion of time. Special cases include continuous and discrete dynamical systems (e.g. Moore machines). Additionally,
morphisms between the different types of systems allow their translation in a common
framework. A central goal is to understand the systems that result from arbitrary interconnection of component subsystems, possibly ofdifferent types,aswell as establish conditions that ensure totality and determinism compositionally. The fundamental categorical tools used here include lax monoidal functors, which provide a language of compositionality, as well as sheaf theory, which flexibly captures the crucial notion of time.

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Algebraic Databases

by Christina Vasilakopoulou and Ryan Wisnesky

Theory and Applications of Categories, 2017

We also show how all of the components of our model---including schemas, instances, change-of-sch... more

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Generalization of Algebraic Operations Via Enrichment

Ph.D Thesis, University of Cabridge, 2014

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A Larson-Sweedler Theorem for Hopf V-categories

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Monoidal Grothendieck Construction

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On Enriched Fibrations

Cahiers de Topologie et Géométrie Différentielle Catégoriques, 2018

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Oplax Hopf Algebras

by Mitchell A Buckley and Christina Vasilakopoulou

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Enriched Duality in Double Categories: V-categories and V-cocategories

Journal of Pure and Applied Algebra, 2019

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Measuring Comodules and Enrichment

by Christina Vasilakopoulou and Ignacio Franco

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Digree: A Middleware for a Graph Databases Polystore

by vasilis spiropoulos and Christina Vasilakopoulou

2016 IEEE International Conference on Big Data (Big Data)

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Dynamical Systems and Sheaves

Applied Categorical Structures, 2019

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Algebraic Databases

by Christina Vasilakopoulou and Ryan Wisnesky

Theory and Applications of Categories, 2017

We also show how all of the components of our model---including schemas, instances, change-of-sch... more

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Hopf measuring comonoids and enrichment

by Christina Vasilakopoulou and Martin Hyland

Proc. London Math. Soc., 2017

We study the existence of universal measuring comonoids $P(A,B)$ for a pair of monoids $A$, $B$ i... more We study the existence of universal measuring comonoids $P(A,B)$ for a pair of monoids $A$, $B$ in a braided monoidal closed category, and the associated enrichment of a category of monoids over the monoidal category of comonoids. In symmetric categories, we show that if $A$ is a bimonoid and $B$ is a commutative monoid, then $P(A,B)$ is a bimonoid; in addition, if $A$ is a cocommutative Hopf monoid then $P(A,B)$ always is Hopf. If $A$ is a Hopf monoid, not necessarily cocommutative, then $P(A,B)$ is Hopf if the fundamental theorem of comodules holds; to prove this we give an alternative description of the dualizable $P(A,B)$-comodules and use the theory of Hopf (co)monads. We explore the examples of universal measuring comonoids in vector spaces, graded spaces and differential graded spaces.

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