On Finitary Functors and Their Presentation
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Jiří Adámek, Stefan Milius and Larry Moss
On Finitary Functors and Their Presentation
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CMCS 2012 | Stefan Milius | April 1, 2012 | S. 2
Why finitary functors are interesting
Our results.
(J. Worrell 1999)
(J. Adámek 1974)
(J. Adámek & V. Trnková 1990)
Application of G.M. Kelly & A.J. Power 1993
Related to: Bonsangue & Kurz (2006); Kurz & Rosicky (2006); Kurz & Velebil (2011)
Strengthening of: van Breugel, Hermida, Makkai, Worrell (2007)
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CMCS 2012 | Stefan Milius | April 1, 2012 | S. 3
Locally finitely presentable (lfp) categories
„Definition.“
Examples.
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CMCS 2012 | Stefan Milius | April 1, 2012 | S. 5
Example: presentation of the finite power-set functor
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CMCS 2012 | Stefan Milius | April 1, 2012 | S. 6
From Set to lfp categories Following Kelly & Power (1993)
Construction
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CMCS 2012 | Stefan Milius | April 1, 2012 | S. 7
Example in posets
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CMCS 2012 | Stefan Milius | April 1, 2012 | S. 8
Finitary functors and presentations
Theorem.
Proof.
Theorem.
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CMCS 2012 | Stefan Milius | April 1, 2012 | S. 9
The Hausdorff functor
Non-determinism for systems with complete metric state space.
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CMCS 2012 | Stefan Milius | April 1, 2012 | S. 10
Accessability of the Hausdorff functor
Theorem. van Breugel, Hermida, Makkai, Worrell (2007)
idempotentassociative commutative
Makkai & Pare (1989)
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CMCS 2012 | Stefan Milius | April 1, 2012 | S. 11
Yes, we can!
Proposition.
preserves colimits
Bad news.
But:
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CMCS 2012 | Stefan Milius | April 1, 2012 | S. 12
Finitaryness of the Hausdorff functor
Theorem.
Proof.
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CMCS 2012 | Stefan Milius | April 1, 2012 | S. 13
Presentation of the Hausdorff functor
locally countably presentableseparable spaces= countably presentable
Proposition.
Proof.
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CMCS 2012 | Stefan Milius | April 1, 2012 | S. 14
Conclusions and future work
Finitary functors on lfp categories are precisely those having a finitary presentation
The Hausdorff functor is finitary and has a presentation by operations with finite arity
Future work
Kantorovich functor on CMS (for modelling probabilistic non-determinism)
Relation of our presentations to rank-1 presentations as in Bonsangue & Kurz