Presentation is loading. Please wait.

Presentation is loading. Please wait.

On Finitary Functors and Their Presentation

Similar presentations


Presentation on theme: "On Finitary Functors and Their Presentation"— Presentation transcript:

1 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

2 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)

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

4 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.

5 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

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

7 Example in posets

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

9 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

10 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

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

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

13 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.

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 sifted colimit preserving  finitary

15 Agenda Kapiteltrennung
Kisuaeli antux in weimi kameran Populario falst Quol damnarin Tropi zu klenne perdi Utilira regau socht mol sunt Her mitant dur Wolche to illemit

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.


Download ppt "On Finitary Functors and Their Presentation"

Similar presentations


Ads by Google