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On Finitary Functors and Their Presentation. Jiří Adámek , Stefan Milius and Larry Moss. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A. Why finitary functors are interesting. (J. Adámek 1974). (J. Worrell 1999). - PowerPoint PPT Presentation

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Folie 1

Ji Admek, Stefan Milius and Larry MossOn Finitary Functors and Their Presentation

TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: APlatzhalter fr Bild, Bild auf Titelfolie hinter das Logo einsetzenCMCS 2012 | Stefan Milius | April 1, 2012 | S. Nr.Why finitary functors are interesting

Our results.

(J. Worrell 1999)(J. Admek 1974)(J. Admek & V. Trnkov 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. Nr.Locally finitely presentable (lfp) categoriesDefinition.

Examples.

CMCS 2012 | Stefan Milius | April 1, 2012 | S. Nr.fp objects = finite sets, posets, graphs, groups presented by finitely many generators and relations etc. 3Locally finitely presentable (lfp) categoriesDefinition.

every finite subcategory of the diagram scheme has a cocone in it

Examples.

CMCS 2012 | Stefan Milius | April 1, 2012 | S. Nr.fp objects = finite sets, posets, graphs, groups presented by finitely many generators and relations etc. 4Example: presentation of the finite power-set functor

CMCS 2012 | Stefan Milius | April 1, 2012 | S. Nr.Before we talk about presentations of functors on lfp cats, lets talk about example in Set.

How does this present P_f? H_\Sigma, parallel pair, coequalizer5From Set to lfp categoriesFollowing Kelly & Power (1993)

Construction

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

CMCS 2012 | Stefan Milius | April 1, 2012 | S. Nr.Finitary functors and presentationsTheorem.

Proof.

Theorem.

CMCS 2012 | Stefan Milius | April 1, 2012 | S. Nr.The same bijective correspondence as before H_\Sigma left Kan-extension of \Sigma

2nd Theorem: See paper!8

The Hausdorff functor

Non-determinism for systems with complete metric state space.

CMCS 2012 | Stefan Milius | April 1, 2012 | S. Nr.H is like a finite power set for CMS models non-determinism in the metric approach to semantics or for systems CMS state space

Hausdorff metric:For any point on M take the nearest point in N and then the sup over those distances9Accessability of the Hausdorff functorTheorem. van Breugel, Hermida, Makkai, Worrell (2007)

idempotentassociativecommutative

Makkai & Pare (1989)

CMCS 2012 | Stefan Milius | April 1, 2012 | S. Nr.The Theorem definitely says that H has a finitary flavor to it.

Makkai & Pare argument: functors with a left- or right-adjoint are accessible.10

Yes, we can!Proposition.

preserves colimits

Bad news.

But:

CMCS 2012 | Stefan Milius | April 1, 2012 | S. Nr.Namely, using algebraic theories11Finitaryness of the Hausdorff functorTheorem.

Proof.

CMCS 2012 | Stefan Milius | April 1, 2012 | S. Nr.2. from: x commutes with filtered colimits in PMS12Presentation of the Hausdorff functor

locally countably presentableseparable spaces= countably presentable

Proposition.

Proof.

CMCS 2012 | Stefan Milius | April 1, 2012 | S. Nr.First line: just a slogan

H_\Sigma involves Cauchy completion of the spaces of tuples with max-metric13Conclusions and future workFinitary 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

CMCS 2012 | Stefan Milius | April 1, 2012 | S. Nr.sifted colimit preserving finitary14Kisuaeli antux in weimi kameran Populario falstQuol damnarin Tropi zu klenne perdi Utilira regau socht mol suntHer mitant dur Wolche to illemitAgenda Kapiteltrennung CMCS 2012 | Stefan Milius | April 1, 2012 | S. Nr.Hier steht eine einzeilige HeadlineNullam 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.

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CMCS 2012 | Stefan Milius | April 1, 2012 | S. Nr.

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