The New York City
Category Theory Seminar
Department of Computer Science
Department of Mathematics
The Graduate Center of The City University of New York
365 Fifth Avenue (at 34th Street) map
(Diagonally across from the Empire State Building)
New York, NY 10016-4309
Wednesdays 7:00 - 8:30 PM.
Room 6417 .
Contact N. Yanofsky to
schedule a speaker
or to add a name to the
seminar mailing list.
Fall 2015 - Spring 2016
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 book is about the field of Topos Theory, which is a vast generalization of a number of fields, giving a categorical intersection between type theory, logic, and also geometry. No prior knowledge of category theory will be required, although the introductory sections on the basics are a bit whirlwind, so at least mild familiarity with the definition of a category, a functor, and a natural transformation will certainly help. If you have been interested in learning more about categories, this will be a good place to start to see them in action towards interesting ends.
The list for announcements and scheduling regarding this group is also at the hott-nyc google group:
While prior, people may have understood that acronym to mean "homotopy type theory" (which was our prior group read project, now completed), with a slight change in perspective the acronym can be read to refer to "Homotopy or Topos Theory" :-). As with that group read, we plan to meet the first and third wednesday of every month, working as slowly as necessary through bite-sized pieces.
Also, as with the prior group read, this is meant to be a place where people with mild background in any number of different fields can interact, and which is welcoming to people with only modest knowledge to begin with. The aim is to read and learn together, each bringing our own intuitions, backgrounds, and experiences to the table.
We meet the first and third Wednesdays of each month from 7:00 to 8:30 PM in Room 6417. The meetings are run by Gershom Bazerman and Dustin Mulcahey.
Speaker: Gershom Bazerman.
Date and Time:   Wednesday September 16, 2015, 7:00 - 8:30 PM., Room 6417.
Title: A General Introduction to Topos Theory and Orginizational Meeting.
Speaker: Jeff Rosenbluth.
Date and Time:   Wednesday October 7, 2015, 7:00 - 8:30 PM., Room 6417.
Title: Topos Theory Reading Group --- Categorical Preliminaries.
Speaker: Louis Thrall, York College.
Date and Time:   Wednesday October 21, 2015, 7:00 - 8:30 PM., Room 6417.
Title: Topos Theory Reading Group --- The category of G-sets and M-sets.
Speaker: Ben Doyle.
Date and Time:   Wednesday November 4, 2015, 7:00 - 8:30 PM., Room 6417.
Title: Topos Theory Reading Group --- The Yonada Lemma.
Speaker: Larry Moss, Indiana University.
Date and Time:   Wednesday November 11, 2015, 7:00 - 8:30 PM., Room 6417.
Title: Thirty Years of Coalgebra: What Have we Learned?
Peter Aczel's book _Non-Well-Founded Sets_ dates from 1984-85. The most lasting contribution of that book was not to set theory itself but rather to other areas. These other areas include theoretical computer science, especially those parts of semantics where one needs circular processes and discrete dynamical systems of various sorts. The point is that Aczel phrased his results in the more general language of coalgebras and also worked with some associated concepts from category theory. When people saw that the more general concepts had applications to things they were interested, the subject took off. Coalgebra has been extensively developed and now offers a lot of ideas and connections which should be interesting to logicians. To name some examples: generalizations of modal logic in various ways, corecursion, algebraic characterizations of the real unit interval and of some fractal sets, and constructions of reflexive objects in domain theory and of types spaces in economics.
This talk will survey some of what has been done, centered on constructions of final coalgebras for various functors.
For background, I suggest http://plato.stanford.edu/entries/nonwellfounded-set-theory/
especially Section 4.
Speaker: Gershom Bazerman.
Date and Time:   Wednesday November 18, 2015, 7:00 - 8:30 PM., Room 6417.
Title: Topos Theory Reading Group --- Limits and Colimits.
Speaker: Raymond Puzio.
Date and Time:   Wednesday December 2, 2015, 7:00 - 8:30 PM., Room 6417.
Title: Topos Theory Reading Group --- The Completeness Theorem and Subobject Classifiers.
Speaker: David Ellerman, University of California Riverside.
Date and Time:   Wednesday January 13, 2016, 7:00 - 8:30 PM., Room 6417.
Title: The Joy of Hets:
Heteromorphisms and Category Theory
Abstract: Click Here.
Speaker: Jonathan Funk, Queensborough Community College .
Date and Time:   Wednesday January 27, 2016, 7:00 - 8:30 PM., Room 6417.
Title: Isotropy theory
Abstract: The isotropy of a topos is about its internal symmetry. By definition, an element of symmetry, or isotropy, of an object X of a topos EE is an automorphism of the geometric morphism EE/X--->EE. Any (Grothendieck) topos has a group, denoted Z, that classifies this symmetry, which we call the isotropy group of the topos. For example, if G is a group, then the isotropy group of the topos B(SET;G) of G-sets is the group G itself under conjugation. We present some of the basic aspects of isotropy theory such as the universal action that the isotropy group has in every object of its topos. Then our specific aim is to explain the following structure theorem for toposes, and to discuss some of its applications and consequences. Terminology: we say that an object is isotropically free if no (non-trivial) element of isotropy fixes any element of the object; by contrast, we say that an object is isotropically trivial if every element of isotropy fixes every element of the object. We say that a topos is anisotropic if it has no isotropy whatsoever, so that its isotropy group is trivial.
THEOREM: A topos EE has a globally supported isotropically free object iff it has a globally supported isotropically trivial object O such that EE/O is equivalent to B(FF;G), where G is group internal to an anisotropic topos FF.
The interpretation of the theorem for inverse semigroups reveals some basic, but nevertheless illuminating facts about inverse semigroups and their relationship with toposes.
Joint work with Pieter Hofstra (University of Ottawa) and Ben Steinberg (City College, CUNY)
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