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 .

Some of the talks are videoed and available here.

Contact N. Yanofsky to schedule a speaker

or to add a name to the seminar mailing list.

The NYC Category Theory Seminar and NY Haskell Meetup are organizing a

We will be reading:

We meet on at least two Wednesdays (exact schedule to be determined) of each month from 7:00 to 8:30 PM in Room 6417, CUNY Grad Center, 365 Fifth Avenue (at 34th Street) The first meeting will be on Wednesday, September 18, 2019. This book introduces many fundamental concepts in category theory, beginning with the character of cartesian-closedness.

From there, it develops the connection between category theory and logic via the lambda calculus and higher order type theories.

As such, this material should be of interest to students of computer science and programming language semantics, as well as categorists and logicians.

These reading groups proceed slowly and methodically, and we make sure to cover the basic material as we go, so even those with little-to-no background in category theory should be able to follow along and acquire the necessary tools as we go.

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

In the present talk, I will review the definitions and will present yet another application of stabilization. First, we shall redefine and extend the classical torsion over commutative domains. This algebraic concept goes back to Poincaré, who described (and named) it in a topological setting around 1900. In 1959, Bass observed that the kernel of the canonical bidualization map, which we call the Bass torsion, from a finitely generated module over a commutative domain coincides with the classical torsion of the module. Using the injective stabilization of the tensor product, Jeremy Russell and I defined, for arbitrary modules over arbitrary rings, a new torsion radical, which agrees with both the classical torsion over commutative domains and the Bass torsion for finitely presented modules over arbitrary rings. The functorial nature of the new torsion makes it amenable to dualization, yielding, for the first time, a notion of cotorsion, also applicable to arbitrary modules over arbitrary rings. The informal, metamathematical dualization process used to define the cotorsion can be effected by a purely mathematical tool, known as the Auslander-Gruson-Jensen functor.

Category
theory gives us the duality between subsets and quotient sets (= partitions =
equivalence relations). Certain ‘classical’ theories are based on the subset or
subobject side of the duality: Boolean logic of *subsets* (usually
presented as “propositional logic”) and the Birkhoff-von-Neumann quantum logic
of *subspaces* (of a separable Hilbert space). Hence there are two dual
theories: 1) Since partitions are dual to subsets, there is a dual *logic of
partitions*, and 2) for vector spaces, direct-sum decompositions are the
partitional dual to subspaces, so there is a *quantum logic of direct-sum
decompositions*.

The
quantitative version of Boolean subset logic is finite discrete probability
theory, and the quantitative version of partition logic is 3) the *logical
theory of information*—where the Shannon notions of simple, compound,
conditional, and mutual entropies are derived by a uniform requantifying
transformation from the corresponding natural logical notions of entropy. And
the standard quantum information theory notion of von Neumann entropy can be
arrived at from Shannon entropy by substituting density matrices for
probability distributions and traces for sums. Similarly, the new 4) *quantum
logical entropy* is arrived at from the notion of logical entropy by the
same substitutions. Those are the four new partition-related theories that will
be sketched in the talk.

References (downloadable from www.ellerman.org) :

1)
Ellerman, David. 2010. “The Logic of Partitions:
Introduction to the Dual of the Logic of Subsets.” *Review of Symbolic Logic*
3 (2 June): 287–350., or Ellerman, David. 2014. “An Introduction to Partition
Logic.” *Logic Journal of the IGPL* 22 (1): 94–125. https://doi.org/10.1093/jigpal/jzt036.
See also Brendan Fong’s MIT Category Theory Seminar talk on partition logic at:
https://www.youtube.com/watch?v=5I7v9mvOC2E

2)
Ellerman, David. 2018. “The Quantum Logic of
Direct-Sum Decompositions: The Dual to the Quantum Logic of Subspaces.” *Logic
Journal of the IGPL* 26 (1 (January)): 1–13. https://doi.org/10.1093/jigpal/jzx026.

3)
Ellerman, David. 2017. “Logical Information Theory:
New Foundations for Information Theory.” *Logic Journal of the IGPL* 25 (5
Oct.): 806–35. https://doi.org/10.1093/jigpal/jzx022.

4)
Ellerman, David. 2018. “Logical Entropy:
Introduction to Classical and Quantum Logical Information Theory.” *Entropy*
20 (9): Article ID 679. https://doi.org/10.3390/e20090679.

Previous Semesters:

##### Fall 2018 - Spring 2019 Topos Theory Reading Group

##### Spring 2017 - Spring 2018 Topos Theory Reading Group

##### Fall 2015 - Fall 2016 Topos Theory Reading Group

##### Fall 2013 - Spring 2015 Homotopy Type Theory Reading Group

##### Spring 2012

##### Fall 2011

##### Spring 2011

##### Fall 2010

##### Spring 2010

##### Fall 2009

##### Spring 2009

Other Category Theory Seminars: