# On Finitary Functors and Their Presentation

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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) (J. Adámek & V. Trnková 1990) Our results. 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)

Locally finitely presentable (lfp) categories
„Definition.“ Examples. fp objects = finite sets, posets, graphs, groups presented by finitely many generators and relations etc.

Locally finitely presentable (lfp) categories
Definition. every finite subcategory of the diagram scheme has a cocone in it fp objects = finite sets, posets, graphs, groups presented by finitely many generators and relations etc. Examples.

Example: presentation of the finite power-set functor
Before we talk about presentations of functors on lfp cats, let‘s talk about example in Set. How does this present P_f?  H_\Sigma, parallel pair, coequalizer

From Set to lfp categories
Following Kelly & Power (1993) Construction

Example in posets

Finitary functors and presentations
Theorem. Proof. The same bijective correspondence as before  H_\Sigma left Kan-extension of \Sigma 2nd Theorem: See paper! Theorem.

The Hausdorff functor Non-determinism for systems with complete metric state space. 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 distances

Accessability of the Hausdorff functor
Theorem. van Breugel, Hermida, Makkai, Worrell (2007) Makkai & Pare (1989) The Theorem definitely says that H has a finitary flavor to it. Makkai & Pare argument: functors with a left- or right-adjoint are accessible. associative commutative idempotent

Yes, we can! Proposition. Bad news. But: preserves colimits
Namely, using algebraic theories

Finitaryness of the Hausdorff functor
Theorem. Proof. 2. from: x commutes with filtered colimits in PMS

Presentation of the Hausdorff functor
separable spaces = countably presentable locally countably presentable Proposition. First line: just a slogan H_\Sigma involves Cauchy completion of the spaces of tuples with max-metric Proof.

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 sifted colimit preserving  finitary

Agenda Kapiteltrennung
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Hier steht eine einzeilige Headline
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