Core Concepts
Studying generalized automata and coalgebras in the category of species reveals rich applications in modern combinatorics and category theory.
Abstract
The content delves into the theory of combinatorial species, exploring automata and coalgebras within this framework. It discusses the historical context, fundamental problems, and various applications in mathematics. The study focuses on categorical structures, endofunctors, adjunctions, and their implications for abstract state machines. Detailed analyses of monoidal structures, algebraic properties, co/algebras, and total categories of automata are provided.
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Introduction to Combinatorial Species
- Originating from Joyal's work as a categorification of generating functions.
- Significance in modern combinatorics explained.
- Applications across diverse mathematical domains highlighted.
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Abstract State Machines
- Interpretation inside general categories explored.
- Transition from determinism to non-determinism discussed.
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Categorical Automata Theory
- Evolution from Cartesian to monoidal machines detailed.
- Introduction to F-automatons for abstract machine dynamics.
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Automata Structures in Spc
- Definitions of species and V-species outlined.
- Examples like subsets, total orders, permutations elucidated.
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Algebraic Structures in Spc
- Monoids, comonoids, Hopf monoids characterized.
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Co/Algebras for Endofunctors
- Exploration of algebraic structures for interesting endofunctors over species.
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Abstract Automata in Spc
- Definition of MlyK(F,B) and MreK(F,B) categories provided.
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Fibrational Properties
- Total Mealy 2-category Mly examined with its functorial properties.
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Total Categories of Automata
- Construction of total Mealy and Moore categories explained based on fixed domain K.
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Reindexings and Left Regular Representation
- Effects on reindexing functors under natural transformations discussed.
Stats
Fosco Loregian was supported by the Estonian Research Council grant PRG1210.
Quotes
"Categories that naturally arise organizing computational machines share a universal property."
"The operation of plethystic substitution is recognized as a fundamental building block."