**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

Thursdays 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 - Fall 2016**

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?
**

Abstract: **
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)

** Speaker: **** Martin Allen.** **
**

Date and Time: ** Wednesday March 2, 2016, 7:00 - 8:30 PM., Room 6417.**

Title:** Topos Theory Reading Group --- Sheaves and Bundles. **

Speaker: ** Gershom Bazerman.**

Date and Time: ** Wednesday March 16, 2016, 7:00 - 8:30 PM., Room 6417.**

Title:** Topos Theory Reading Group --- Sheaves and Cross Sections. **

Speaker: ** Noson S. Yanofsky, Brooklyn College.**

Date and Time: ** Wednesday April 6, 2016, 7:00 - 8:30 PM., Room 6417.**

Title:** Topos Theory Reading Group --- Sheaves and Etale Spaces. **

Speaker: ** Noson S. Yanofsky, Brooklyn College.**

Date and Time: ** Wednesday April 20, 2016, 7:00 - 8:30 PM., Room 6417.**

Title:** Topos Theory Reading Group --- Sheaves and Grothendieck Toposes. **

Speaker: ** Martin Allen.**

Date and Time: ** Wednesday May 4, 2016, 7:00 - 8:30 PM., Room 6417.**

Title:** Topos Theory Reading Group --- The Zariski Site. **

Speaker: ** Raymond Puzio.**

Date and Time: ** Wednesday October 5, 2016, 7:00 - 8:30 PM., Room 6417.**

Title:** Topos Theory Reading Group --- Monads and Beck's Monadicity Theorem. **

Speaker: ** Raymond Puzio.**

Date and Time: ** Wednesday October 19, 2016, 7:00 - 8:30 PM., Room 6417.**

Title:** Topos Theory Reading Group --- Beck's Monadicity Theorem. **

Speaker: ** Lou Thrall.**

Date and Time: ** Wednesday November 2, 2016, 7:00 - 8:30 PM., Room 6417.**

Title:** Topos Theory Reading Group --- Colimits in Elementary Topoi. **

Speaker: ** Gershom Bazerman.**

Date and Time: ** Wednesday November 16, 2016, 7:00 - 8:30 PM., Room 6417.**

Title:** Topos Theory Reading Group --- Factorization and Images in Elementary Toposes. **

Speaker: ** Pieter Hofstra, University of Ottawa.**

Date and Time: ** Wednesday December 7, 2016, 7:00 - 8:30 PM., Room 6417.**

Title:** Topos-theoretic invariants of algebraic theories
**** Abstract: **** To every algebraic theory T (monoids, groups, rings, etc.) we can associate a classifying topos Set[T] with the property that geometric morphisms E->Set[T] correspond to T-models in E. This means that we can apply topos-theoretic concepts to the study of algebraic theories. In this talk, I will focus on the study of a group-theoretic invariant of toposes called the isotropy group, and interpret this invariant for various algebraic theories. We will consider various examples, and also touch upon the various methods for combining algebraic theories (sum, tensor).
Based on joint work with my PhD student Jason Parker.
**