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Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Jiří Adámek, Stefan Milius and Larry Moss On Finitary Functors and Their Presentation.

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1 Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Jiří Adámek, Stefan Milius and Larry Moss On Finitary Functors and Their Presentation TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A

2 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)

3 CMCS 2012 | Stefan Milius | April 1, 2012 | S. 3 Locally finitely presentable (lfp) categories Definition. Examples.

4 CMCS 2012 | Stefan Milius | April 1, 2012 | S. 4 Locally finitely presentable (lfp) categories Definition. every finite subcategory of the diagram scheme has a cocone in it Examples.

5 CMCS 2012 | Stefan Milius | April 1, 2012 | S. 5 Example: presentation of the finite power-set functor

6 CMCS 2012 | Stefan Milius | April 1, 2012 | S. 6 From Set to lfp categories Following Kelly & Power (1993) Construction

7 CMCS 2012 | Stefan Milius | April 1, 2012 | S. 7 Example in posets

8 CMCS 2012 | Stefan Milius | April 1, 2012 | S. 8 Finitary functors and presentations Theorem. Proof. Theorem.

9 CMCS 2012 | Stefan Milius | April 1, 2012 | S. 9 The Hausdorff functor Non-determinism for systems with complete metric state space.

10 CMCS 2012 | Stefan Milius | April 1, 2012 | S. 10 Accessability of the Hausdorff functor Theorem. van Breugel, Hermida, Makkai, Worrell (2007) idempotent associative commutative Makkai & Pare (1989)

11 CMCS 2012 | Stefan Milius | April 1, 2012 | S. 11 Yes, we can! Proposition. preserves colimits Bad news. But:

12 CMCS 2012 | Stefan Milius | April 1, 2012 | S. 12 Finitaryness of the Hausdorff functor Theorem. Proof.

13 CMCS 2012 | Stefan Milius | April 1, 2012 | S. 13 Presentation of the Hausdorff functor locally countably presentable separable spaces = countably presentable Proposition. Proof.

14 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

15 CMCS 2012 | Stefan Milius | April 1, 2012 | S. 15 Kisuaeli antux in weimi kameran Populario falst Quol damnarin Tropi zu klenne perdi Utilira regau socht mol sunt Her mitant dur Wolche to illemit Agenda Kapiteltrennung

16 CMCS 2012 | Stefan Milius | April 1, 2012 | S. 16 Hier steht eine einzeilige Headline Nullam pulvinar lorem sed enim placerat vel malesuada purus laoreet. Nulla ultrices urna sapien, venenatis ultrices risus. Pellentesque congue, magna laoreet congue semper, eros dui commodo metus, sed fringilla nisi massa a justo. Vivamus gravida accumsan nibh, at semper enim egestas in. Proin eget blandit quam. Duis semper scelerisque semper Lorem ipsum dolor sit amet, consectetur adipiscing elit. Proin eget blandit quam. Duis semper scelerisque semper. Mauris cursus orci iaculis libero Imperdiet ac egestas odio fermentum. Orci iaculis libero imperdiet ac egestas odio fermentum.


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