The subjects of category theory and topology are very intricately related. Alexander Grothendieck had a deep insight into this relationship. He viewed Homotopy theory as a branch of (n-) category theory. Grothendieck formulated the Homotopy hypothesis which allows one to view a topological space as a (higher) category and vice-versa. The aim of the Erie Categories and Topology seminar is to explore common ideas in the two subjects. One such idea is that of coherence which has its manifestations in both subjects. Let X be a topological monoid and Y be a topological space which is homotopy equivalent to X. One may transfer the product and the unit from the topological monoid X to Y however this transferred product on Y satisfies the monoid axioms only upto homotopy. For example, there will be a continuous path (associator) in the topological space Y connecting the two different ways of multiplying three elements of Y using the transferred product. Jim Stasheff showed that these associators satisfied a pentagon identity up to homotopy which satisfied its own coherence laws, again up to homotopy, and so on. This led to the discovery of the associahederon which turned out to be very useful in the study of coherence in category theory. In other words, the study of coherence in category theory has its roots in topology. A second concrete example of interaction between the two manifestations of coherence is the so-called Thomason’s theorem which states that, up to equivalence, every connective spectrum is weakly equivalent to a spectrum obtained by putting a (strict) symmetric monoidal category into an infinite loop space machine.

Another idea which is common to both subjects is that of dualizability. The cobordism hypothesis (which is perhaps still a conjecture) in the subject of differential topology makes the Bordism (higher) category a very fascinating object for category theorists because according to this hypothesis the smooth oriented Bordim category is the free symmetric monoidal higher category with duals. In other words, the category of smooth manifolds and bordisms between them is a fundamental object in the study of the still mysterious notion of (full)-dualizability in (higher) category theory.

Another objective of the Erie Categories and Topology seminar is to bring together category theorists and topologists in the areas surrounding Lake Erie and provide them a platform to share ideas and appretiate different perspectives.

January 30, 2020 Jonathan Scott (Cleveland State University)

Title: Interleavings and Gromov-Hausdorff Distance

Abstract: One of the central notions to emerge from the study of persistent homology is that of interleaving distance.
It has found recent applications in computational geometry, symplectic and contact geometry, sheaf theory, and phylogenetics.
Here we present a general study of this topic, considering interleavings of functors to be solutions to a certain extension problem.
By placing the problem in the context of (weighted) bicategories, we identify interleaving distance as a type of categorical generalization of Gromov--Hausdorff distance.
As an application we recover a definition of shift equivalences of discrete dynamical systems.
This is joint work with Vin de Silva (Pomona College) and Peter Bubenik (U Florida)

Venue: Yost 306, Case Western Reserve University.

February 6, 2020 Zhaoting Wei (Kent State University)

Title: The homotopy theory and homotopy limits of dg-categories

Abstract: In this talk we review the Dwyer-Kan model structure on the category of (small) differential graded categories (dg-categories).
We study some properties of the Dwyer-Kan model structure and then talk about the homotopy limit formula of cosimplial diagrams of dg-categories.

Venue: Mathematical Science Building #115, Kent State University.

February 27, 2020 Zhaoting Wei (Kent State University)

Title: Homotopy limits of dg-categories 2

Abstract: In this talk we first review homotopy limit in general and then discuss homotopy fiber products of dg-categories.
Then we talk about homotopy limits of cosimplicial diagrams of dg-categories, which arise naturally in geometry.

Venue: Yost 306, Case Western Reserve University.

March 5, 2020 Amit Sharma (Kent State University)

Title: Picard groupoids and Γ-categories

Abstract: In this talk we will describe a construction of a symmetric monoidal closed model
category of coherently commutative Picard groupoids. We will then construct another
model category structure on the category of (small) permutative categories
whose fibrant objects are (permutative) Picard groupoids. We will go on to show that
that the Segal′s nerve functor induces a Quillen equivalence between the two
aforementioned model categories. This result implies the classical result
that Picard groupoids model stable homotopy one-types.

Venue: Mathematical Science Building #115, Kent State University.

We suggest that the attendees who are driving to seminar meetings to be held at Kent State University park at the student center visitor parking lot which is only a short walk away from the Mathematical Sciences building. We suggest parking lot S-53 for attendees driving to Case Western Reserve seminar meetings. Parking lot maps for both universities are available on the following web pages:

Parking information for Kent State UniversityParking information for Case Western Reserve University

Nick Gurski

Amit Sharma