The New York City
Category Theory Seminar
The New York Category Theory Seminar and the New York Haskell Meetup is sponsoring a
Topos Theory Reading Group
We will be reading
Sheaves in Geometry and Logic: A First Introduction to Topos Theory
by Saunders MacLane, Ieke Moerdijk
The list for announcements and scheduling regarding this group is also at the hott-nyc google group:
https://groups.google.com/forum/#!forum/hott-nyc
Spring 2017- Spring 2018
Speaker: Sam van Gool, University of Amsterdam and ILLC
Date and Time: Wednesday April 18, 2018, 7:00 - 8:30 PM., Room 6417.
Title: Duality, sheaves, and semigroups.
Abstract: Distributive lattices are the order-algebraic structures which are at the basis of any logical system containing disjunction (or) and conjunction (and). Stone-Priestley duality shows that the category of distributive lattices is dually equivalent to a certain category of ordered topological spaces. The fact that this equivalence is dual means that the direction of morphisms is reversed: subobjects in the category of algebras correspond to quotient objects in the category of spaces, and vice versa. This duality was previously known to be useful for obtaining semantics in logic and theoretical computer science.
In this talk, I will first give a slightly more detailed description of the Stone-Priestley duality theorem. I will then broadly describe two recent mathematical contributions, which we obtained by building on this duality: (1) an analysis of sheaf representations of lattice-ordered algebraic structures (joint work with Gehrke, CNRS); (2) a model-theoretic perspective on profinite semigroups (joint work with Steinberg, CCNY).
Speaker: Stéphane Dugowson, Supméca Paris. Website.
Date and Time: Monday (NOTE SPECIAL DAY!), March 12, 2018, 7:00 - 8:30 PM., Room 6417.
Title: From connectivity spaces to a general interactivity theory : a categorical exploration.
Abstract: This lecture will start by explaining how we got interested in connectivity spaces, and it will give some definitions and properties about them : representation of connectivity spaces, connectivity structures of quantum entanglement, connectivity structures of multiple relations, topos of a connectivity space... Then we will show how the theme of connectivity spaces leads very naturally to that of dynamical systems, and we will explain the principles of the general categorical theory of interactions that we have developed. Finally, we will sketch some perspectives on the future development of this theory. During this presentation, we will sometimes make some philosophical remarks about the topics covered.
References:
→"On Connectivity Spaces", Cahiers de topologie et géométrie différentielle catégoriques, LI, 4 (2010) 282-315. arXiv : 1001.2378.
→"Espaces connectifs : représentations, feuilletages, ordres, difféologies", Cahiers de topologie et géométrie différentielle catégoriques, LIV (2013). arXiV : 1610.07366.
→"Structures connectives de l'intrication quantique", juillet 2014, arXiv : 1407.5920.
→"Définition du topos d'un espace connectif", octobre 2016, arXiv : 1610.04422.
→"Dynamiques en interaction : une introduction à la théorie des dynamiques sous-fonctorielles ouvertes", août 2016, arXiv : 1608.07938.
Speaker: David Ellerman
University of California at Riverside
Date and Time: Wednesday January 17, 2018, 7:00 - 8:30 PM., Room 6417.
Title: New Foundations for Information Theory: The logical theory of information.
Abstract: There is a new foundational theory of information based on logic. That logic is the logic of partitions category-theoretically dual to the usual Boolean logic of subsets. It is the theory of information-as-distinctions, where a distinction of a partition is defined as an ordered pair of elements in different blocks of the partition. Logical probability theory is the quantitative version of Boolean subset logic, and logical information theory is the quantitative version of the logic of partitions. Logical probability of subset (event) is the normalized size of the subset, and the logical information of a partition is the normalized size of the partition where the partition is represented by its set of distinctions.
What is the relation to the usual Shannon theory of information? The definition of Shannon entropy as well as the notions on joint, conditional, and mutual entropy as defined by Shannon can all be derived by a uniform requantifying transformation from the corresponding formulas of logical information theory. The transformation replaces the counting of distinctions with the counting of the number of binary partitions (bits) it takes, on average, to make the same distinctions by uniquely encoding the distinct elements--which is why the Shannon theory perfectly dovetails into coding and communications theory.
Moreover, logical entropy is a measure (in the sense of measure theory) while Shannon entropy is not--even though Shannon carefully defined the compound notions of his entropy so that they would satisfy the usual Venn diagram formulas (e.g., inclusion-exclusion) as if it were a measure on a set. Since logical entropy automatically satisfies those formulas as a (probability) measure, and the uniform requantifying transform preserves Venn diagrams, that explains how the Shannon notions can have those relationships.
In short, the logical theory of information-as-distinctions is grounded on logic and it displaces the Shannon theory from being the foundational theory of information to being a higher-order specialized theory that requantifies logical information for the purposes of coding and communications theory, where it has been so successful.
Speaker: David Ellerman
University of California at Riverside
Date and Time: Wednesday January 18, 2017, 7:00 - 8:30 PM., Room 6417.
Title: From Abstract Objects in Mathematics to
Indefinite ''Blobs'' in Quantum Mechanics
Abstract: Given an equivalence relation ~ on a set U, an abstraction operator @ takes two equivalent entities to the same abstract thing: u~u' iff @(u)=@(u'). In mathematics there two different notions of 'abstraction' at work. The #1 version is just the equivalence class, e.g., a homotopy type is just the equivalence class of homotopic spaces. The #2 version is a more abstract object that is, for example, definite on what is common to all the spaces homotopic to each other but is indefinite on where they differ.
We show how to mathematically model the two versions starting just with a subset S of U and then focus on the less familiar #2 interpretation where S is viewed not as a set of distinct elements but as a more abstract entity 'S-ness' that is definite on what is common between the elements of S and indefinite on how they differ.
The point is that the #2 case dovetails precisely with the quantum mechanics notion of a superposition of distinct eigenstates of some observable which is a state definite only on what is common to the superposed eigenstates and is objectively or onticly indefinite between them. The process in general of classifying by some attribute to make the #2 indefinite state more definite then emerges as the process of measurement in QM.
Speaker: Raymond Puzio.
Date and Time: Wednesday February 1, 2017, 7:00 - 8:30 PM., Room 6417.
Title: Topos Theory Reading Group --- Slice Categories.
Speaker: Gershom Bazerman.
Date and Time: Wednesday March 8, 2017, 7:00 - 8:30 PM., Room 6417.
Title: Topos Theory Reading Group --- Lattice and Heyting algebra objects in a topos.
Speaker: Mikael Vejdemo-Johansson, College of Staten Island.
Date and Time: Wednesday March 29, 2017, 7:00 - 8:30 PM., Room 6417.
Title: Persistent homology and the persistence topos.
Abstract: Persistent homology is a fundamental technique in topological data analysis, summarizing the variations in homology groups in a parametrized topological space as the parameter sweeps through all valid values. Since the first definition of persistent homology in 2000, several algebraic frameworks have been proposed to capture the underlying ideas of persistent homology. For each new framework, the scope of data analyses possible has increased.
Joint with Primoz Skraba and Joao Pita Costa, we have developed a topos-based foundation for persistent homology, where the underlying Heyting algebra P captures the *shape* of the persistent homology theory, and persistent homology emerges as the internal theory of (semi)simplicial homology within the set theory determined by the topos of sheaves over P.
In this talk, I will give an introduction to persistent homology, and demonstrate how the theory fits with a topos-based foundation.
Speaker: Raymond Puzio.
Date and Time: Wednesday April 5, 2017, 7:00 - 8:30 PM., Room 6417.
Title: Topos Theory Reading Group --- Lawvere-Tierney topologies.
Speaker: Noson Yanofsky, Brooklyn College and The GC.
Date and Time: Wednesday May 10, 2017, 7:00 - 8:30 PM., Room 6417.
Title: Theoretical Computer Science for the Working Category Theorist.
Abstract: We will describe certain
categories of computable functions and categories of models of computations such as Turing machines,
circuits, and register machines. There will be functors between these various categories.
Surprisingly, many of the results of computability theory, complexity theory, and Kolmogorov complexity theory are simple consequences of
functoriality and composition. Most of this talk will be understandable if you only know the definition of a category and a functor.
Speaker: Emily Riehl, Johns Hopkins University.
Date and Time: Wednesday May 17, 2017, 5:00 - 6:30 PM. NOTE SPECIAL TIME, Room TBA.
Title: Towards a synthetic theory of (∞,1)-categories.
Abstract:
This talk will discuss joint work in progress with Michael Shulman that aims to develop a synthetic theory of (∞,1)-categories in
homotopy type theory. This work is motivated by a particular model of homotopy type theory in the category of simplicial spaces,
which is an example of an ∞-cosmos, a "universe" in which one can develop a the basic category theory of infinite-dimensional
categories "synthetically" (i.e. in ignorance that the objects being discussed are simplicial spaces) as opposed to "analytically"
(as is the strategy of similar work by Joyal and Lurie).
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