Pullback-power axiom
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University of Regina Topology Seminar

Winter 2022

Topic: Enriched model categories

Time: Thursdays 1:00 - 2:20 PM, unless otherwise noted.
Location: on Zoom.

The Zoom link will be emailed to members of the Department of Mathematics and Statistics. Please contact Martin Frankland if you would like to receive the announcements but are not a member of the department.

In this learning seminar, we will study enriched model categories and their applications in topology.

The lecture series will be interspersed with guest research talks, which are posted on our Youtube channel.

Date Speaker Title and abstract References
January 20 Martin Frankland Enriched categories

Linear maps between vector spaces themselves form a vector space. Similarly, continuous maps between topological spaces themselves form a topological space. In many contexts, the morphisms between objects form not just a set, but carry some extra structure. The purpose of enriched categories is to encode that extra structure.

In this talk, I will introduce the basics of enriched categories. We will focus on examples of enrichments: in abelian groups, vector spaces, chain complexes, simplicial sets, and topological spaces.

Notes from the talk
A pleasantry
January 27 Martin Frankland Model categories

In classical homotopy theory, one can look at continuous maps up to homotopy and consider two topological spaces as essentially the same if they are homotopy equivalent. Model categories, introduced by Quillen, allow us to make similar homotopical constructions in a much broader context. I will present the definition of model category and some basic properties. We will look at the examples of topological spaces, chain complexes, and simplicial sets.

W.G. Dwyer and J. Spaliński, Homotopy theories and model categories (1995).
Notes from the talk
February 3 Carlos Gabriel Valenzuela Simplicial model categories

In this talk I’ll talk about simplicial sets and introduce the basics of simplicial model categories. We’ll be focusing on the enriched structure and go over some general results.

P.G. Goerss and J.F. Jardine, Simplicial homotopy theory (1999), Chapter II.
Notes from the talk
A pleasantry
February 10 Hana Jia Kong
(Institute for Advanced Study)
Motivic image-of-J spectrum via the effective slice spectral sequence

The "image-of-J" spectrum in the classical stable homotopy category has been well-studied; Bachmann–Hopkins defined its motivic analogue. In this talk, I will start with the classical story, and then move to the motivic version. I will talk about the effective slice computation of the motivic "image-of-J" spectrum over the real numbers. Unlike the classical case, the map from the motivic sphere is not surjective on homotopy groups. Still, it captures a regular pattern that appears in the R-motivic stable stems. This is joint work with Eva Belmont and Dan Isaksen.

February 17 Arnaud Ngopnang Ngompe Monoidal model categories

In this talk we are going to describe the structures of monoidal model category and of enriched model category. Also we will provide some examples and we will see that any monoidal model category is an enriched model category over itself, via the enrichment of its underlying closed monoidal category.

Notes from the talk
February 24   Reading Week. No seminar.  
March 3 Larry So Monoidal model categories of spectra

In this talk I will introduce diagram spaces and diagram spectra, which provide a framework to study and compare different constructions of spectra. I will also describe the levelwise model structures and the stable model structures, and model categories of ring spectra and module spectra over a ring spectrum.

M. Mandell, J.P. May, S. Schwede, and B. Shipley, Model categories of diagram spectra (2001).
Notes from the talk
March 10 William Balderrama
(University of Virginia)
The motivic lambda algebra and Hopf invariant one problem

Current best approaches to understanding the stable homotopy groups of spheres at the prime 2 make use of the Adams spectral sequence, which computes stable stems starting with information about the cohomology of the Steenrod algebra. The first major success of the Adams spectral sequence was in Adams' resolution of the Hopf invariant one problem, which proceeded via an analysis of secondary cohomology operations. Later, J.S.P. Wang used a certain algebraic device, the lambda algebra, to give a more thorough computation of the cohomology of the Steenrod algebra, and used this to give a slick almost entirely algebraic derivation of the Hopf invariant one theorem.

In this talk, I will go over some of the above history, and then describe work (joint with Dominic Culver and J.D. Quigley) on analogues in motivic stable homotopy theory. In particular, I will describe a mod 2 motivic lambda algebra, defined over any base field of characteristic not equal to 2, as well as some of what can be said about the 1-line of the motivic Adams spectral sequence for various base fields.

March 17 Don Stanley Enrichments on additive categories

We will look at various ways of getting enrichments on additive categories.

D. Dugger and B. Shipley, Enriched model categories and an application to additive endomorphism spectra (2007).
March 24 Larry So Spectral enrichment of stable model categories

It is known that any model category can be enriched over simplicial sets. Dugger shows an analogous result concerning stable model categories. In this talk I will talk about his work on the spectral enrichment of stable model categories over symmetric spectra and its applications.

D. Dugger, Spectral enrichments of stable model categories (2006).
Notes from the talk
March 31 Don Stanley Replacing model categories with simplicial ones

We follow the proof of Dugger that a left proper combinatorial model category can be replaced by a Quillen equivalent simplicial model category. We review some of the terms and then go on to define the Reedy model category on simplicial objects. A localization of it will be the simplicial model category we seek.

D. Dugger, Replacing model categories with simplicial ones (2001).
April 7 Matthew Alexander Model categories of functors

In this talk we will examine the projective, injective, and Reedy model structures on functor categories, and their relationship to one another. Examples and motivation for the study of such model categories will be given, focusing on the simplicial picture.

April 14
Room RI 208, livestreamed on Zoom
Sacha Ikonicoff
(University of Calgary)
Divided power algebras over an operad

Divided power algebras were defined by H. Cartan in 1954 to study the homology of Eilenberg-MacLane spaces. They are commutative algebras endowed, for each integer n, with an additional monomial operation. Over a field of characteristic 0, this operation corresponds to taking each element to its n-th power divided by factorial of n. This definition does not make sense if the base field is of prime characteristic, yet Cartan's definition of divided power algebra applies in this situation as well. The notion of divided power algebra over a field of prime characteristic allows us to describe algebraic structures that appear in homology and homotopical​ algebra and has found applications in a wide array of mathematical domains, for instance in crystalline cohomology, and deformation theory.

In this talk, we will introduce the generalised definition of a divided power algebra over an operad given by B. Fresse in 2000. We will give a complete characterisation for generalised divided power algebras in terms of monomial operations and relations. We will show how to improve this characterisation to particular cases, including the case of a product of operads with distributive laws.


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