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Exploring Finitary Functors: Presentations and Applications

This article delves into the significance and applications of finitary functors, featuring results from notable researchers and extending to category theory and non-determinism. The focus on locally finitely presentable (lfp) categories is exemplified through the presentation of the finite power-set functor. The Hausdorff functor is explored in depth for systems with complete metric state space, with discussions on its finitary nature and accessibility. The paper also touches upon the preservation of colimits, separable spaces, and countably presentable properties. Research conclusions and future directions are outlined, including the exploration of the Kantorovich functor in modelling probabilistic non-determinism and the relevance of presentations in rank-1 methodologies.

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Exploring Finitary Functors: Presentations and Applications

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  1. On FinitaryFunctorsandTheirPresentation JiříAdámek, Stefan Miliusand Larry Moss TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A

  2. Whyfinitaryfunctorsareinteresting (J. Adámek 1974) (J. Worrell 1999) (J. Adámek & V. Trnková 1990) Ourresults. Applicationof G.M. Kelly & A.J. Power 1993 Relatedto: Bonsangue & Kurz (2006); Kurz & Rosicky (2006); Kurz & Velebil (2011) Strengtheningof: van Breugel, Hermida, Makkai, Worrell (2007)

  3. Locallyfinitelypresentable (lfp) categories „Definition.“ Examples.

  4. Example: presentationofthe finite power-set functor

  5. From Set tolfpcategories Following Kelly & Power (1993) Construction

  6. Example in posets

  7. Finitaryfunctorsandpresentations Theorem. Proof. Theorem.

  8. The Hausdorfffunctor Non-determinismforsystemswithcompletemetricstatespace.

  9. AccessabilityoftheHausdorfffunctor Theorem. van Breugel, Hermida, Makkai, Worrell (2007) Makkai & Pare (1989) commutative idempotent associative

  10. Yes, wecan! preservescolimits Proposition. Bad news. But:

  11. FinitarynessoftheHausdorfffunctor Theorem. Proof.

  12. PresentationoftheHausdorfffunctor separablespaces = countablypresentable locallycountablypresentable Proposition. Proof.

  13. Conclusionsandfuturework • Finitaryfunctors on lfpcategoriesarepreciselythosehaving a finitarypresentation • The Hausdorfffunctorisfinitaryandhas a presentationbyoperationswith finite arity • Future work • Kantorovich functor on CMS (formodellingprobabilistic non-determinism) • Relation ofourpresentationsto rank-1 presentationsas in Bonsangue & Kurz

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