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Borel Functors and Infinitary Interpretations
[article]

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*...

##
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Coding in graphs and linear orderings
[article]

2020
*
arXiv
*
pre-print

Any graph can be

arXiv:1903.06948v2
fatcat:vtqi7kju5ra7zajvdqcfhpemiu
*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]

2020
*
arXiv
*
pre-print

We introduce the category of "analytic ∞-

arXiv:2011.00437v1
fatcat:jncsemu5kvcuhcoayd7tcyima4
*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 ≡ := ≤ ∩ ≥. ...##
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A universal characterization of standard Borel spaces
[article]

2019
*
arXiv
*
pre-print

We prove that the category SBor of standard

arXiv:1908.10510v1
fatcat:zcbecb2juzectlze5zf767xywu
*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

2018
*
Journal of Logic and Computation
*

We extend Ł ukasiewicz logic obtaining the

doi:10.1093/logcom/exy011
fatcat:5ynrkxkxdbb2nknctgu2lem23i
*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]

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

doi:10.1109/lics.2013.38
dblp:conf/lics/KozenLMP13
fatcat:dix26nhntvfy7p56uv65gpa5ci
*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

doi:10.1007/978-3-642-40313-2_58
fatcat:s4tjn3zftzghdpdvyft5pofgke
*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]

2019
*
arXiv
*
pre-print

Other ingredients in our proof include the Lopez-Escobar theorem for continuous logic, a uniformization result for full

arXiv:1908.03268v1
fatcat:fmm6eu6khfemjfttq4ojvbwjgq
*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

2007
*
Theoretical Computer Science
*

In contrast to previous approaches, we do not assume that the endofunctors are locally contractive,

doi:10.1016/j.tcs.2007.02.059
fatcat:6vv5kwjt2nf2baaldejzbnqfi4
*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

2018
*
Bulletin of Symbolic Logic
*

Given f, g in C, < 1 a limit ordinal

doi:10.1017/bsl.2018.83
fatcat:ueqsznkxrjhudavtuwz7r24try
*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]

2007
*
Handbook of Spatial Logics
*

Heine-

doi:10.1007/978-1-4020-5587-4_8
fatcat:kqle33t67vh3rf5eeuvbt3dg4e
*Borel*Theorem) become false. ...*infinitary*disjunctions. ...##
###
SOME CONSTRUCTIVE ROADS TO TYCHONOFF
[chapter]

2005
*
From Sets and Types to Topology and Analysis
*

In categorical logic, points of the corresponding locale can be understood as certain cartesian

doi:10.1093/acprof:oso/9780198566519.003.0014
fatcat:cw3sbxe3ajdgxjgtj5zmmjsfqu
*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

2008
*
Journal of Logic and Computation
*

A notable feature of the deductive machinery is an

doi:10.1093/logcom/exn092
fatcat:uaonyza5urgsthfxssp7vi4ua4
*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 ...##
###
Quantitative Algebraic Reasoning

2016
*
Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science - LICS '16
*

In each case we have finitary

doi:10.1145/2933575.2934518
dblp:conf/lics/MardarePP16
fatcat:n7s6kwc65jhntfzs2q55jzemyy
*and*continuous versions. ... The four cases are: Hausdorff metrics from quantitive semilattices; p-Wasserstein metrics (hence also the Kantorovich metric) from barycentric algebras*and*also from pointed barycentric algebras*and*the ... We present both finitary*and**infinitary*versions of these constructions. ...
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