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Description: |
The category Met of metric spaces and nonexpanding maps is locally countably presentable in the enriched sense. It follows from the work of Kelly and Power that for countably accessible monads T on Met the category of algebras can be described as an equational category of S-algebras for a generalized signature S with countable arities.
This has been recently sharpened by Rosický [R] who used quantitative equations between terms s,t. These equations are indexed by a rational number e, and an algebra satisfies this iff every computation of s and t results in elements of distance at most e.
Mardare et al. work with classical finitary signatures S and with omega-basic equations [M]. This means quantitative equations with specified conditions on the distances of pairs of variables. These authors have exhibited in a series of articles several important examples of omega-varieties, i.e. classes of S-algebras presented by omega-equations. Can omega-varieties be characterized as monadic categories over Met? We do not have an answer. However, by moving from Met to the category of UMet ultrametric spaces, we present such a characterization of omega-varieties: they are precisely the monadic categories for enriched monads T on Umet preserving surjective morphisms and directed colimits of split subobjects.
References:
[M] R. Mardare, P. Panangaden, and G. Plotkin. Quantitative algebraic reasoning. In Proceedings of LICS 1916, pages 700-709
[R] J. Rosický, Metric monads, arXiv:2012.14641v5
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Date: |
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| Start Time: |
15:00 |
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Speaker: |
Jirí Adámek (Technische Universität Braunschweig, Germany)
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Institution: |
Technische Universität Braunschweig
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Place: |
Zoom
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| Research Groups: |
-Algebra, Logic and Topology
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See more:
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