Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Jiří Adámek, Stefan Milius...
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Transcript of Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Jiří Adámek, Stefan Milius...
Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen
Jiří Adámek, Stefan Milius and Larry Moss
On Finitary Functors and Their Presentation
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)
CMCS 2012 | Stefan Milius | April 1, 2012 | S. 3
Locally finitely presentable (lfp) categories
„Definition.“
Examples.
CMCS 2012 | Stefan Milius | April 1, 2012 | S. 5
Example: presentation of the finite power-set functor
CMCS 2012 | Stefan Milius | April 1, 2012 | S. 6
From Set to lfp categories Following Kelly & Power (1993)
Construction
CMCS 2012 | Stefan Milius | April 1, 2012 | S. 7
Example in posets
CMCS 2012 | Stefan Milius | April 1, 2012 | S. 8
Finitary functors and presentations
Theorem.
Proof.
Theorem.
CMCS 2012 | Stefan Milius | April 1, 2012 | S. 9
The Hausdorff functor
Non-determinism for systems with complete metric state space.
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)
CMCS 2012 | Stefan Milius | April 1, 2012 | S. 11
Yes, we can!
Proposition.
preserves colimits
Bad news.
But:
CMCS 2012 | Stefan Milius | April 1, 2012 | S. 12
Finitaryness of the Hausdorff functor
Theorem.
Proof.
CMCS 2012 | Stefan Milius | April 1, 2012 | S. 13
Presentation of the Hausdorff functor
locally countably presentableseparable spaces= countably presentable
Proposition.
Proof.
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