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Added by
James Hefford
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Modelling concept representation is a foundational problem in the study of cognition and linguistics. This work builds on the confluence of conceptual tools from Gardenfors semantic spaces, categorical compositional linguistics, and applied category theory to present a domain-independent and categorical formalism of concept.
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Markov categories are a recent categorical approach to the mathematical foundations of probability and statistics. Here, this approach is advanced by stating and proving equivalent conditions for second-order stochastic dominance, a widely used way of comparing probability distributions by their spread. Furthermore, we lay foundation for the theory of comparing statistical experiments within Markov categories by stating and proving the classical Blackwell-Sherman-Stein Theorem. Our version not only offers new insight into the proof, but its abstract nature also makes the result more general, automatically specializing to the standard Blackwell-Sherman-Stein Theorem in measure-theoretic probability as well as a Bayesian version that involves prior-dependent garbling. Along the way, we define and characterize representable Markov categories, within which one can talk about Markov kernels to or from spaces of distributions. We do so by exploring the relation between Markov categories and Kleisli categories of probability monads.
Restriction categories were introduced to provide an axiomatic setting for the study of partially defined mappings; they are categories equipped with an operation called restriction which assigns to every morphism an endomorphism of its domain, to be thought of as the partial identity that is defined to just the same degree as the original map. In this paper, we show that restriction categories can be identified with emph{enriched categories} in the sense of Kelly for a suitable enrichment base. By varying that base appropriately, we are also able to capture the notions of join and range restriction category in terms of enriched category theory.
We develop the mathematical theory of separable and unitary $n$-categories based on Gaiotto and Johnson-Freyds theory of condensation completion. We use it to study the categories of topological orders by including gapless quantum phases and defects. In particular, we show that all the topological features of a potentially gapless quantum phase can be captured by its topological skeleton, and that the category of the topological skeletons of higher dimensional gapped/gapless quantum phases can be explicitly computed categorically from a simple coslice 1-category.
We show that a well behaved Noetherian, finite dimensional, stable, monoidal model category is equivalent to a model built from categories of modules over completed rings in an adelic fashion. For abelian groups this is based on the Hasse square, for chromatic homotopy theory this is based on the chromatic fracture square, and for rational torus-equivariant homotopy theory this is the model of Greenlees-Shipley arXiv:1101.2511.
We prove that topological Hochschild homology (THH) arises from a presheaf of circles on a certain combinatorial category, which gives a universal construction of THH for any enriched infinity category. Our results rely crucially on an elementary, model-independent framework for enriched higher category theory, which may be of independent interest.
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