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Category Theory in Physics,
Mathematics and Philosophy
Warsaw
Poland
16�17 November 2017
Category Theory in Physics, Mathematics and Philosophy
The Conference is organized by:
International Center for Formal Ontology, Faculty of Administration and SocialSciences, Warsaw University of Technology
Copernicus Center for Interdisciplinary Studies
Center for Theoretical Physics, Polish Academy of Sciences
Institute of Mathematics, Academy of Sciences of the Czech Republic
Edited by:
Bartªomiej Skowron
©Authors 2017
First published 2017
Contents
Krzysztof Bielas From Quantum-Mechanical Lattice of Projections to
Smooth Structure of R4 . . . . . . . . . . . . . . . . . . . . . . . 2Mikoªaj Boja«czyk Monads in the Theory of Formal Languages . . . . 2Tomasz Brengos Automata, Categories and Monads . . . . . . . . . . 3Benjamin Feintzeig Deduction and De�nability in In�nite Statistical
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Jan Gutt Topoi for Quantum Mechanics . . . . . . . . . . . . . . . . . 4Michael Heller & Jerzy Król Beyond the Space-Time Boundary . . . . 4Bartosz Klin Computation Theory in the Topos of Sets With Atoms . 5Zbigniew Król Category Theory and Philosophy: Inspirations, Prob-
lems and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 6Jerzy Król Remarks on Perturbative Categorical Quantum Gravity . . 6Wiesªaw Kubi± Generic Objects . . . . . . . . . . . . . . . . . . . . . . 7Józef Lubacz Possible Application of Category Theory in Epistemic
and Poietic Processes . . . . . . . . . . . . . . . . . . . . . . . . 7Michaª R. Przybyªek Beyond Sets With Atoms: De�nability in First-
Order Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Ji°í Rosický Accessible Categories . . . . . . . . . . . . . . . . . . . . . 8Zbigniew Semadeni Asymmetry of the Cantorian Mathematics From a
Categorical Standpoint: Is It Related to Irreversibilities in Nature? 9Bartªomiej Skowron A Defense of the Theory of Ideas . . . . . . . . . 10Piotr Suªkowski Categori�cation in Knot Theory and Physics . . . . . 10Krzysztof Wójtowicz Explanatory Virtues of Category Theory . . . . . 11Sebastian Zaj¡c Chord Diagrams Category and Its Limit . . . . . . . . 11
Abstracts
From Quantum-Mechanical Lattice of Projectionsto Smooth Structure of R4
Krzysztof BielasUniversity of Silesia, Katowice
Division of Astrophysics and Cosmology, Institute of Physics
Poland
Recently it was shown that mathematical formalism of quantum mechanicsprovides a unique way of thinking not only about real numbers in general,but also about the issue of exotic smoothness. The main point of departureis a set of complete Boolean algebras contained in the quantum-mechanicallattice of projections, used to build various Boolean-valued models of ZFC, thus�universes for mathematics� in the usual sense. As each of them provides itsown notion of real numbers, it gives a rather unique opportunity for consideringexotic smoothness questions. It is not a surprise that an important step inthe construction is a mapping between Boolean algebras and structures builtupon reals, such as manifold covers. The present work aims at a categoricalperspective on this subject.
Monads in the Theory of Formal Languages
Mikoªaj Boja«czykUniversity of Warsaw
Faculty of Mathematics, Informatics and Mechanics
Poland
In theoretical computer science, a �formal language� is simply any set of words,typically recognised by some computational device like a �nite automaton oran algorithm. Formal languages can be extended to objects beyond words,�e.g. trees or graphs�with associated computational models. In my talk, Iwill describe how category theory, speci�cally monads, can be used to �nda common language for describing formal languages over di�erent kinds ofobjects.
Automata, Categories and Monads
Tomasz BrengosWarsaw University of Technology
Faculty of Mathematics and Information Science
Poland
Automata theory is a branch of theoretical computer science studying abstractmodels of machines. Decades of research in this area introduced plethora ofautomata types. Their seemingly independent theories asked for uni�cation.
The purpose of this talk is to describe category theory as a common de-nominator language for many variants of automata. Our primary interest is inautomata with silent moves, i.e. machines with a special computation branchthat is allowed to take several steps and in some sense remain neutral to thestructure of the process. We show how di�erent automata with invisible movescan be modelled as morphisms of the type X → TX for a monad (T,m, e)on a given category (or equivalently, as endomorphisms in the Kleisli categoryfor T ). Moreover, we describe how the abstract language of category theorysimpli�es formulation of many de�nitions and facts in this area of research.
Deduction and De�nability in In�nite StatisticalSystems
Benjamin FeintzeigUniversity of Washington
Department of Philosophy
USA
Classical accounts of intertheoretic reduction involve two pieces: �rst, the newterms of the higher-level theory must be de�nable from the terms of the lower-level theory, and second, the claims of the higher-level theory must be deduciblefrom the lower-level theory along with these de�nitions. The status of eachof these pieces becomes controversial when the alleged reduction involves anin�nite limit, as in statistical mechanics. Can one de�ne features of or deducethe behavior of an in�nite idealized system from a theory describing only �nitesystems?
In this talk, I bring the tools of category theory to bear on this question ofreduction. I explicate analogs of the de�nability and deducibility requirementsin terms of category-theoretic questions about structure. I use these tools toanswer the questions: Are the properties of in�nite limiting systems forced
upon us by the properties of �nite systems? Do in�nite systems contain morestructure than �nite systems?
Topoi for Quantum Mechanics
Jan GuttCenter for Theoretical Physics, Warsaw
Polish Academy of Sciences
Poland
The phase space of a system is a fundamental object in classical mechanics.A suitable Boolean lattice of its subsets represents the system's logic�thephysically valid propositions regarding the state of the system�while a con-sistent assignment of probabilities to these propositions describes the statestatistically. Following J. von Neumann, quantum systems have their logics aswell, along with a suitable probability theory. Alas, the lattices representingthese quantum logics are non-distributive, re�ecting the non-existence of anhonest phase-space in the background. In the 90's however, C. Isham andhis collaborators have put forth the idea that a near-perfect analogue of theclassical picture may nevertheless be established in the quantum setting, aslong as one interprets its components in a suitable topos. This approach hasbeen further reworked by C. Heunen et al., leading to a deep re-examinationof the structural foundations of physics.
Beyond the Space-Time Boundary
Michael HellerCopernicus Center for Interdisciplinary Studies, Kraków
Poland
Jerzy KrólUniversity of Silesia, Katowice
Institute of Physics
Poland
In General Relativity a space-timeM is regarded singular if there is an obstaclethat prevents an incomplete curve in M to be continued. Usually, such aspace-time is completed to form M̄ = M∪∂M where ∂M is a singular boundaryofM . The standard geometric tools onM do not allow �to cross the boundary�.
However, the so-called Synthetic Di�erential Geometry (SDG), a categoricalversion of standard di�erential geometry based on intuitionistic logic, has at itsdisposal tools permitting doing so. Owing to the existence of in�nitesimals oneis able to penetrate �germs of manifolds� that are not visible from the standardperspective. We present a simple model showing what happens �beyond theboundary� and when the singularity is �nally attained. The model is purelymathematical and is mathematically rigorous but it does not pretend to referto the physical universe.
Computation Theory in the Topos of Sets WithAtoms
Bartosz KlinUniversity of Warsaw
Faculty of Mathematics, Informatics and Mechanics
Poland
Sets with atoms have been known in mathematics for almost a century, un-der several names: permutation models, nominal sets, or the Schanuel topos.Among other features, they o�er a relaxed notion of �niteness, encompassingstructures that are in principle in�nite, but exhibit enough symmetries to bede�nable by �nite means.
A considerable amount of computation theory applies to such �nitely de-�nable structures, including notions such as automata or Turing machines,but also certain algorithms, programming languages and parts of complexitytheory. Some classical techniques fail though, including results that rely onthe fact that a �nite set has only �nitely many subsets (e.g. determinization of�nite automata), or on even very limited principles of choice (e.g. the Gaussianelimination algorithm).
I will give a brief sketch of the computational landscape of sets with atoms,and try to motivate the more general quest of formulating computation theoryin a topos.
Category Theory and Philosophy: Inspirations,Problems and Methods
Zbigniew KrólWarsaw University of Technology
International Center for Formal Ontology, Faculty of Administration and Social Sciences
Institute of Philosophy and Sociology, Polish Academy of Sciences
Poland
The role of category theory for philosophy is considered. Category theory is asource of problems, methods and inspiration for considering some new as well asclassic philosophical problems. Category theory decides also some hypothesesconcerning the laws of the development of mathematics. The in�uence ofcategory theory on the foundations and ontology of mathematics is brie�yinquired.
Remarks on Perturbative Categorical QuantumGravity
Jerzy KrólUniversity of Silesia, Katowice
Institute of Physics
Poland
In categories allowing for synthetic and intuitionistic reasoning (e.g. in smoothtoposes, E), points of spacetime manifold acquire in�nitesimal structure. Thereare monads that smear every point x ∈M4. Monads are naturally parametrizedby in�nitesimal subobjects Dk ⊂ R, k = 1, 2, ... of the real line. Suppose thatspacetime M4 in SET allows for monads at the microscale (when looked uponfrom the perspective of E). It is natural to assume that (quantum) gravitationalinteractions hold on monads. This means that vertices of Feynman diagrams ofthe canonical quantum gravity live on monads. We choose the parametrizationby D2; then we show that
1. In categorical spacetimes with monads, perturbative quantum gravitygains cuto� into at most 1-loop diagrams. We thus expect that such a purequantum gravity (without sources) is a renormalizable theory.
2. The reference to monads in E from a spacetime in SET can be madesmooth. However, the smoothness agreeing with the process has to be exotic.In particular smooth R4 becomes exotic smooth R4. This follows from aninterplay of intuitionistic reasoning, allowing for monads, with classical onepresupposed in SET.
Generic Objects
Wiesªaw Kubi±Institute of Mathematics, Czech Academy of Sciences, Praha, Czechia
Institute of Mathematics, Cardinal Stefan Wyszy«ski University in Warsaw, Poland
We present a category-theoretic framework for investigating special objects,called generic. We shall work in a �xed category S contained in a larger cate-gory L whose objects are limits of certain sequences from S. Some additionalaxioms are imposed. Roughly speaking, an L-object is generic with respect toS if it can be described in terms of a winning strategy in a natural in�nitetwo-player game producing sequences in S.
Generic objects can be regarded as the most complicated limits of certainin�nite evolution processes, where a single evolution step is just an arrow fromone object to another. That is why the language of category theory seems tobe very well-suited here.
Known examples of generic objects occur in model theory, although recentresearch shows that they actually show up in several areas of mathematics, forexample, in topology and functional analysis.
The aim of the talk will be explaining the details and axioms of our frame-work and then presenting selected examples of generic objects.
Possible Application of Category Theory inEpistemic and Poietic Processes
Józef LubaczWarsaw University of Technology
International Center for Formal Ontology & Faculty of Electronics and Information
Technology
Poland
The presentation considers generic models of epistemic and poietic processeswhich distinguish the following types of conceptual presentations: D � de-scription of what is hypothesized about something in the case of epistemicprocesses, P � prescription of what is to be created in the case of poieticprocesses, E � description of how empirical data concerning what is described(D) or prescribed (P ) is conceptualized. The relation of D and E, and of Pand E, have to meet some prede�ned criteria if epistemic and poietic processesare to be considered as meeting their intended goals. It is argued that in somecircumstances and contexts category theory may be helpful in de�ning and/orinterpreting such criteria.
Beyond Sets With Atoms: De�nability inFirst-Order Logic
Michaª R. PrzybyªekPolish-Japanese Academy of Information Technology, Warsaw
Poland
In recent years more and more results in automata theory has been carriedover from the classical set theory to various theories of sets with atoms. Mostof these results come from a single group of researchers led by prof. MikoªajBoja«czyk. The results concerning sets with atoms are not, however, in thevacuum. Formalized in a weird language and obfuscated by misleading termi-nology, they are, in fact, crippled versions of well-known theorems and conceptsfrom topos theory. The aim of this talk is three-fold: to present the resultsof prof. Mikoªaj Boja«czyk's team in a broader context of topos theory, tospeculate what the team can possibly discover in the future, and to give answersto questions that are beyond of the tools they use. During the talk, I shall focuson answering the following question:
�What are the necessary and su�cient conditions on a �rst-ordertheory to allow for e�ective computations in de�nable sets of thetheory?�
Accessible Categories
Ji°í RosickýMasaryk University, Brno
Department of Mathematics and Statistics
Czechia
We explain the concept of an accessible category and outline its usefullness invarious areas of mathematics.
Asymmetry of the Cantorian Mathematics Froma Categorical Standpoint: Is It Related to
Irreversibilities in Nature?
Zbigniew SemadeniUniversity of Warsaw, Warsaw
Institute of Mathematics
Poland
By the Cantorian mathematics we mean here the basic mathematical theories:algebraic, topological, functional analysis etc. built up in terms of set theory.The following problem will be analyzed. While category theory is fully �sym-metric� in the sense that each notion and each theorem has its uniquely de�neddual, the Cantorian mathematics is speci�cally asymmetric. E.g., if we considertypical categories as Gr, the category of groups and their homomorphisms,category Ab of abelian groups, the category Rinc1 of commutative rings withunits and unit-preserving homomorphisms, the category Comp of compactspaces and continuous maps, the category Ban1 of Banach spaces and linearoperators of norm ||T || ≤ 1, then the product in each such category is basicallythe Cartesian product (the product in the category Set of sets) provided with asuitable structure (in the case of Ban1 it is the product of unit balls) whereasthe coproducts (the free product in Gr, the direct sum in Ab, the Stone-�echβ of the disjoint union in Comp, the l1 sum in Ban1) are markedly distinct.Put di�erently, the forgetful functors from each of these categories to Set
commute with products and do not commute with coproducts. Analogousasymmetry concerns equalizers and coequalizers and�more generally�limitsand colimits in those categories, and also in the category Aut of Mealy au-tomata (A,S, Y, δ : S ×X → S, λ : S ×X → Y ) with suitable morphisms.
The problem is to explain deep hidden reasons of this asymmetry of theCantorian mathematics. It may be a consequence of the asymmetry of themembership relation ∈ (relation element-set is akin to the Platonic many-one)and/or of the asymmetry of domain-codomain of morphisms f : X → Y inthose theories, which in turn is related to the one-way direction of time and,generally, to irreversibilities in Nature.
A Defense of the Theory of Ideas
Bartªomiej SkowronWarsaw University of Technology
International Center for Formal Ontology, Faculty of Administration and Social Sciences
Poland
I propose the new incarnation of the theory of Ideas and I try to defendthe theory against traditional counterarguments. The starting point are thetheories of Ideas of Plato and Ingarden and an ontology of Ideas proposedby Kaczmarek; these theories are paraphrased�using a modi�ed method ofsemantic paraphrases of Ajdukiewicz�and presented in terms of the basicconcepts of category theory.
To paraphrase Ideas as categories I propose recognized category theory as apattern for the theory of Ideas. This recognition is based on an analogy betweenmathematical structures and philosophical structures. It could also be called amathematical philosophy or mathematical modeling in metaphysics. I invokean arrows-like, i.e. no-object-oriented, formulation of a category and I base theproposed theory of Ideas on that formulation. The components of an Idea arearrows and their compositions (equivalents of changes and transformations);objects in this approach are special arrows namely the identity arrows. Usingthe category of higher dimensions I introduce the concept of the dimension ofan Idea (and other concepts) which allows me to refute the argument of the�third man�.
Categori�cation in Knot Theory and Physics
Piotr SuªkowskiUniversity of Warsaw
Faculty of Physics
Poland
Knot theory is a branch of mathematics that characterizes and classi�es knots(such as those, which we can tie on a piece of a rope). To this aim mathemati-cians introduce so called (classical) knot invariants�objects such as numbersof polynomials, which characterize knots. These invariants can be categori�ed,which means that they can be interpreted as dimensions of certain spacesassociated to knots. Surprisingly, it turns out that classical knot invariantscan be also interpreted and computed as certain amplitudes in quantum �eldtheory. Moreover, quantum �eld theory and string theory also enable tointerpret categori�cation in physical terms, by assigning quantum mechanicalHilbert spaces to knots. In this talk I will discuss these ideas and some of theirconsequences, in particular an unexpected predictions in number theory.
Explanatory Virtues of Category Theory
Krzysztof WójtowiczUniversity of Warsaw
Faculty of Philosophy and Sociology
Poland
For any mathematical theory T , we might ask, whether T has explanatoryvalue�and how this explanatory value manifests itself. The notion of explana-tion is vague, but nevertheless it is one of the philosophical notions of a greatimportance (and the discussion concerning explanation in mathematics is livelyin the last years).
The question might be formulated with respect to category theory: areclaims concerning the (potential) explanatory power of category theory justi-�ed? I propose to examine three possible �elds of �explanatory application� ofCT: (i) in mathematics; (ii) in physics; (iii) in metaphysics.
The problem of explanation within mathematics is often analyzed in thecontext of the explanatory/non-explanatory character of proof, or in the con-text of conceptual recasting of a discipline. The problem of the potentialexplanatory role of mathematics in physics is often stated within the broadercontext of explanatory virtues of a scienti�c theory, and the potential explana-tory role of mathematics.
My working thesis is that category theory has rather limited explanatorypower within mathematics and physics in the senses indicated above (even ifmany mathematical facts can be formulated with the use of category-theoreticalconcepts)�but it reveals its power rather when applied to fundamental ques-tions, which might be called �metaphysical�. So, in conclusion, the answer is(moderately) negative for cases (i) and (ii), but it is positive for (iii).
Chord Diagrams Category and Its Limit
Sebastian Zaj¡cCardinal Stefan Wyszynski University in Warsaw, Warsaw
Faculty of Mathematics and Natural Studies
Poland
Chord diagrams may be used to represent the inter-reletionships between someobjects. They occur in many branches of matematics and physics like: geom-etry, topology, random matrix models, moduli space of Riemann surfaces. Inphysics they may be used to represent interactions between particles (Feynman
diagrams). Chord diagrams are also very important in a complicated problemof RNA and protein structure prediction in molecular biology. They representso called secondary structure interactions and can be used to describe prop-erties of this structures by a topological characteristic called genus. Categorytheory is a natural framework for a description of chord diagrams. Because ofamalgamation properties of this category it is possible to �nd so called Fraisselimit and apply Fraisse theory to them. I will also present some questions aboutcategory theory and they role in mathematical physics, logic and philosophy.
