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### Borel Functors and Infinitary Interpretations [article]

Matthew Harrison-Trainor, Russell Miller, Antonio Montalbán
2016 pre-print
interpretation, and every Baire-measurable functor between the set of copies of two countable structures is induced by an infinitary interpretation.  ...  In general, an interpretation between structures induces a continuous homomorphism between their automorphism groups, and furthermore, it induces a functor between the categories of copies of each structure  ...  For every Borel adjoint equivalence of categories (F, G) between Iso(B) and Iso(A) there is an infinitary bi-interpretation (I, J ) between A and B, such that F and G are naturally isomorphic to the functors  ...

### Coding in graphs and linear orderings [article]

Julia Knight, Alexandra Soskova, Stefan Vatev
2020 arXiv   pre-print
Any graph can be interpreted in a linear ordering using computable Σ_3 formulas. Friedman and Stanley gave a Turing computable embedding L of directed graphs in linear orderings.  ...  Harrison-Trainor and Montalbán have also shown this, by a quite different proof.  ...  A structure A is interpreted in B using L ω 1 ω -formulas iff there is a Borel functor (Φ, Ψ) from B to A.  ...

### Borel and analytic sets in locales [article]

Ruiyuan Chen
2020 arXiv   pre-print
We introduce the category of "analytic ∞-Borel locales" as the regular completion under images of the unary site of locales and ∞-Borel maps, and prove analogs of several classical results about analytic  ...  We give a detailed analysis of various notions of image, and prove that a continuous map need not have an ∞-Borel image.  ...  Corollary 2.10.11 (infinitary transitivity/cut). In the infinitary calculus, if A∪B, and A∪{¬b} for every b ∈ B, then A. a ≤ b :⇐⇒ {a} {b}, and let ≡ := ≤ ∩ ≥.  ...

### A universal characterization of standard Borel spaces [article]

Ruiyuan Chen
2019 arXiv   pre-print
We prove that the category SBor of standard Borel spaces is the (bi-)initial object in the 2-category of countably complete Boolean (countably) extensive categories.  ...  Letting Bor denote the category of Borel spaces and Borel maps, we thus have a functor B : Bor op → σBool, where B(f ) := f −1 : B(Y ) → B(X) for a Borel map f : X → Y ∈ Bor.  ...  Standard Borel spaces and Borel maps (i.e., preimages of Borel sets are Borel) are ubiquituous in descriptive set theory as a basic model of "definable sets" and "definable functions" between them; see  ...

### Infinitary logic and basically disconnected compact Hausdorff spaces

Antonio Di Nola, Serafina Lapenta, Ioana LeuŞtean
2018 Journal of Logic and Computation
We extend Ł ukasiewicz logic obtaining the infinitary logic IRŁ whose models are algebras C(X,[0,1]), where X is a basically disconnected compact Hausdorff space.  ...  The Lindenbaum-Tarski algebra of IRŁ is, up to isomorphism, an algebra of [0,1]-valued Borel functions. Finally, our system enjoys standard completeness with respect to the real interval [0,1].  ...  The functors that give the equivalence, denoted by Γ and Γ R , are defined similarly to each other: in the case of MV-algebras, for any ℓu-group (G, u), Γ(G, u) = {x ∈ G | 0 ≤ x ≤ u} = [0, u] G and for  ...

### Borel functors, interpretations, and strong conceptual completeness for L_ω_1ω [article]

Ruiyuan Chen
2019 arXiv   pre-print
Borel naturally isomorphic to the functor induced by some L'_ω_1ω-interpretation of T in T'.  ...  This implies that given two theories ( L, T) and ( L', T') (in possibly different languages L, L'), every Borel functor Mod( L', T') →Mod( L, T) between the respective groupoids of countable models is  ...  Every Borel functor Mod(L , T ) → Mod(L, T ) is Borel naturally isomorphic to F * for some interpretation F : (L, T ) → (L , T ).  ...

### Stone Duality for Markov Processes

2013 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science
We prove a Stone-type duality theorem between countable Aumann algebras and countably-generated continuous-space Markov processes.  ...  Acknowledgments We would like to thank Jean Goubault-Larrecq, Ernst-Erich Doberkat, Robert Goldblatt and Larry Moss for helpful discussions.  ...  This is called the Borel algebra of the space, and the measurable sets are called Borel sets.  ...

### Strong Completeness for Markovian Logics [chapter]

2013 Lecture Notes in Computer Science
We propose a new infinitary rule that replaces the so-called Countable Additivity Rule (CAR) currently used in the literature to address the problem of proving strong completeness for these and similar  ...  These logics are not compact so one needs infinitary rules in order to obtain strong completeness results.  ...  Mardare's research was supported by VKR Center of Excellence MT-LAB and by the Sino-Danish Basic Research Center IDEA4CPS.  ...

### Representing Polish groupoids via metric structures [article]

Ruiyuan Chen
2019 arXiv   pre-print
Other ingredients in our proof include the Lopez-Escobar theorem for continuous logic, a uniformization result for full Borel functors between open quasi-Polish groupoids, a uniform Borel version of Katětov's  ...  construction of U, groupoid versions of the Pettis and Birkhoff–Kakutani theorems, and a development of the theory of non-Hausdorff topometric spaces and their quotients.  ...  interpretations, and Proof.  ...

### Recursively defined metric spaces without contraction

Franck van Breugel, Claudio Hermida, Michael Makkai, James Worrell
2007 Theoretical Computer Science
In contrast to previous approaches, we do not assume that the endofunctors are locally contractive, and our results do not depend on Banach's fixed-point theorem.  ...  In this paper we use the theory of accessible categories to find fixed points of endofunctors on the category of 1-bounded complete metric spaces and nonexpansive functions.  ...  The first and third authors were supported by Natural Sciences and Engineering Research Council of Canada.  ...

### Functions of the first Baire class

Raphaël Carroy
2018 Bulletin of Symbolic Logic
Given f, g in C, < 1 a limit ordinal and n a natural, we have 1. If CB(f) = CB(g) = then f ≡ g, 2. If CB(f) = + n and CB(g) = + 2n + 1, then f ≤ g.  ...  Here X, Y, X , and Y are variables for Polish 0-dimensional (P0D for short) spaces, considered as closed subspaces of the Baire space of infinite sequences of natural numbers.  ...  We isolate two infinitary operations on P0D spaces, called the gluing and the pointed gluing.  ...

### Locales and Toposes as Spaces [chapter]

Steven Vickers
2007 Handbook of Spatial Logics
Heine-Borel Theorem) become false.  ...  infinitary disjunctions.  ...

### SOME CONSTRUCTIVE ROADS TO TYCHONOFF [chapter]

Steven Vickers
2005 From Sets and Types to Topology and Analysis
In categorical logic, points of the corresponding locale can be understood as certain cartesian functors from P to Set, and in the flat site these generalize to the flat functors from P to Set.  ...  In category theory there is a notion of flat functor from C to Set such if C is cartesian (has all finite limits) then flatness is equivalent to the functor being cartesian (preserves finite limits).  ...

### Deduction Systems for Coalgebras Over Measurable Spaces

R. Goldblatt
2008 Journal of Logic and Computation
A notable feature of the deductive machinery is an infinitary Countable Additivity Rule. A deductive construction of canonical spaces and coalgebras leads to completeness results.  ...  These functors have been shown by Moss and Viglizzo to have final coalgebras that represent certain universal type spaces in game-theoretic economics.  ...  Relevant functors include those that are polynomial : constructible from constant functors and the identity functor Id by forming products T 1 × T 2 , coproducts T 1 + T 2 , and exponential functors T  ...