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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