Erie Categories and Topology Seminar

The regular meeting time of the seminar is Thursday 2:30 - 3:30 PM on dates specified below

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.

Fall 2019 Schedule

September 12, 2019 Nick Gurski (Case Western Reserve University)
Title: Signs and spheres
Abstract: There are two symmetries for the tensor product of graded abelian groups: you can introduce minus signs (how many depends on the degrees of the elements) or not when you swap the order of two elements. An observation of Kapranov's is that you can extract a model for the 1-truncated sphere spectrum from the category of graded abelian groups by restricting to the invertible objects and using the former sign convention. This talk will be about employing the same philosophy for building the 2-truncated sphere spectrum, but now using invertible graded Picard categories. This is joint work with Niles Johnson and Angélica Osorno.
Venue: Mathematical Science Building #376, Kent State University.

September 26, 2019 Amit Sharma (Kent State University)
Title: The homotopy theory of coherently commutative monoidal quasi-categories
Abstract: A quasi-category is a generalization of the notion of a category by allowing the composition law to be only coherently associative and unital. We propose a similar generalization of the notion of a symmetric monoidal category to be a quasi-category equipped with a coherently commutative multiplicative structure. We construct a symmetric monoidal closed model category structure on the category of $\Gamma$-spaces whose fibrant objects are called coherently commutative monoidal quasi-categories and which provide a model for the desired quasi-categories equipped a coherently commutative multiplicative structure. The Joyal model category structure on simplicial sets is an extension of the natural model category structure on Cat. In particular, the nerve functor defines a homotopy reflection between the two model category structures. The guiding principle of our construction is a similar extension of the natural model category structure on the category of permutative (or strict symmetric monoidal) categories Perm to the category of $\Gamma$-spaces such that the Segal's Nerve functor becomes a homotopy reflection.
Venue: Yost 306, Case Western Reserve University.

October 10, 2019 Rhiannon Griffiths (Case Western Reserve University)
Title: Simplicial functor calculus
Abstract: Functor calculus is a categorification of differential calculus. In Goodwillie's original series of papers smooth functions are replaced by homotopy functors between categories of spaces and spectra. His results have since been generalised to the broader context of simplicial homotopy theory. In the recent Johnson-McCarthy discrete calculus, a homotopy functor is approximated by a universal degree n-functor, analogous to the degree n-approximation of a smooth function. We attempt to define degree n-approximations for simplicial functors, together with corresponding degree n-model structures on categories of simplicial functors. This is joint work with Lauren Bandklayder, Julie Bergner, Brenda Johnson and Rekha Santhaman.
Venue: Mathematical Science Building #376, Kent State University.

October 24, 2019 Mark Meckes
Title: From the Euler characteristic of a category to the magnitude of a metric space (and beyond)
Abstract: Around 2006, Tom Leinster defined a notion of Euler characteristic of a finite category which generalizes both the classical Euler characteristic (applied to the classifying space of a category) and Rota's notion of the Euler characteristic of a partially ordered set. Leinster's definition can be further generalized to enriched categories, where it is renamed magnitude. A particularly interesting case of enriched categories is metric spaces; in this setting magnitude turns out to be related to various classical geometric invariants, such as dimension, volume, and surface area. In another direction, it was recently discovered that there is a homology theory for metric spaces (and more generally, for enriched categories) for which magnitude functions as a kind of Euler characteristic. Recent work has shown that this magnitude homology is closely related to persistent homology, which plays a key role in topological data analysis. I will attempt to give, in one hour or less, a broad overview of what is and is not known about all these topics.
Venue: Yost 306, Case Western Reserve University.

November 7, 2019 Michael Horst (Ohio State University)
Title: 2-Homological Algebra and Picard Cohomology
Abstract: In this talk I will outline the basics of a 2-categorical homological algebra, including the homology of 2-chain complexes and constructions of derived functors. This theory will be applied to defining and computing examples of a categorification of group cohomology.
Venue: Mathematical Science Building #376, Kent State University.

November 21, 2019 Niles Johnson (Ohio State University)
Title: Homotopy theory of Picard categories
Abstract: In this talk we discuss some parallels between low-dimensional stable homotopy theory and low-dimensional symmetric monoidal algebra. On the homotopical side we will review a computation of stable homotopy groups via cohomology operations in Postnikov towers. On the categorical side, we will describe the corresponding structure in dimensions 0, 1, and 2. We will then outline a categorical analogue of the Toda bracket <2,eta,2>. This work is joint with N. Gurski and A. Osorno.
Venue: Yost 306, Case Western Reserve University.

December 5, 2019 David White (Denison University)
Title: Baez-Dolan Stabilization and Bousfield localization without left properness
Abstract: We will begin with an overview of an old problem, due to Baez and Dolan, that applied higher category theory to the study of mathematical physics (via topological quantum field theories). I will show how to reduce the statement of the Baez-Dolan Stabilization Hypothesis from a statement in higher category theory (we choose the setting of Rezk’s Theta_n spaces) to one in abstract homotopy theory. I’ll then sketch how to prove the Stabilization Hypothesis using semi-model categories, n-operads, and Cisinski’s locally constant presheaves. A key new ingredient is a result that does left Bousfield localization for non-left proper model categories. This is joint work with Michael Batanin.
Venue: Mathematical Science Building #376, Kent State University.