Abstracts

Fernando Abellán García (Universität Hamburg)
Marked colimits and higher cofinality

Thursday, February 4th, 17:30-18:45

In this talk I will introduce the notion of marked colimits in ∞-bicategories, providing a natural framework to interpret Theorem A^† from the previous talk as a higher cofinality statement. Relevant examples will be discussed, as a well the relation of marked colimits with weighted colimits. The main result that I will discuss is the following:

Let C ^†, D^† be ∞-categories equipped with a collection of marked morphisms and let f be a functor from C ^† to D^† that preserves the marking. Then f is a marked cofinal functor (i.e., restriction along f preserves marked colimits) if and only if the conditions of Theorem A^† are satisfied.

Daniel Berwick-Evans (University of Illinois at Urbana-Champaign)
How do field theories detect the torsion in topological modular forms?

Monday, February 1st, 18:30-19:45

Since the mid 1980s, there have been hints of a connection between 2-dimensional field theories and elliptic cohomology. This led to Stolz and Teichner's conjectured geometric model for the universal elliptic cohomology theory of topological modular forms (TMF), for which cocycles are 2-dimensional (supersymmetric) field theories. Properties of these field theories lead naturally to the expected integrality and modularity properties of classes in TMF. However, the abundant torsion in TMF has always been mysterious from the field theory point of view. In this talk, we will describe a map from 2-dimensional field theories to a cohomology theory that approximates TMF. This map affords a cocycle description of certain torsion classes. In particular, we will explain how a choice of anomaly cancelation for the supersymmetric sigma model with target S³ determines a cocycle representative of the generator of π₃(TMF)=Z/24.

Guillermo Carrión Santiago (Universitat Autònoma de Barcelona)​

Approaching higher limits from homotopy theory

Friday, February 5th, 16:45-18:00

Higher limits are the higher derived functors of the inverse limit construction for functors taking values in abelian groups. Classically, they are computed using tools from homological algebra.

In algebraic topology, the cohomology of a homotopy colimit of a diagram spaces can be approach via a spectral sequence whose E₂ page consists precisely of the higher limits of the functor obtained from applying cohomology to the diagram of spaces. In particular, if the higher limits vanish, then the cohomology of the homotopy colimit is just the inverse limit of the cohomologies. There are many vanishing results, for example the Mittag-Leffler property [2, Section 3.5] or the pseudo-projectivity property [1].

We study the case where the category P is a poset with an order preserving map d: P → N. If we consider the injective model category on the functor category Fun(P, Ch(Ab)), a functor is pseudo-projective if it is cofibrant.

In this talk we will show how we can use the techniques from model categories, inspired by homotopy theory, to describe higher limits in this situation when the indexing category is a poset. We will give explicit bounds for the vanishing of higher limits in terms of properties of the functor improving previous results.

References:

[1] Diaz Ramos, A., A family of acyclic functors, Journal of Pure and Applied Algebra, 213.5 (2009), 783–794.

[2] Weibel, C., An Introduction to Homological Algebra, Cambridge University Press, 1994.

Emanuele Dotto (University of Warwick)
Hermitian K-theory of stable ∞-categories

Friday, February 5th, 15:15-16:30

The talk will give an overview of Grothendieck–Witt theory in the higher categorical formalism of stable ∞-categories equipped with a Poincaré structure. As an example of the flexibility of this framework, we will see how to relate the Grothendieck–Witt groups to Ranicki's L-groups and how to prove a strong version of Karoubi's periodicity theorem without assuming that 2 is invertible in the base ring.

This is joint work with Calmès, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and Steimle.

Daniel Fuentes-Keuthan (Johns Hopkins University)
Goodwillie towers of ∞-categories and desuspension

Tuesday, February 9th, 16:15-17:30

We reconceptualize the process of forming n-excisive approximations to ∞-categories, in the sense of Heuts [1], as inverting the suspension functor lifted to A_n-cogroup objects. We characterize n-excisive ∞-categories as those ∞-categories in which A_n-cogroup objects admit desuspensions. Applying this result to pointed spaces we reprove a theorem of Klein–Schwänzl–Vogt [2]: every 2-connected cogroup-like A_∞-space admits a desuspension.

This is joint work with Klaus O. Johnson and Michael River.

References:

[1] G. Heuts, Goodwillie approximations to higher categories, arXiv:1510.03304, 2018.

[2] J. Klein, R. Schwänzl, and R. Vogt, Comultiplication and suspension, Topology and its Applications, 1997.

Drew Heard (NTNU Trondheim)
Support theory for triangulated categories in algebra and topology

Wednesday, February 3rd, 12:30-13:45

We will survey the support theory of triangulated categories through the machinery of tensor-triangulated geometry. We will discuss the stratification theory of Benson–Iyengar–Krause for triangulated categories, the construction by Balmer of the spectrum of a tensor-triangulated category, and the relation between the two. Time permitting, we will discuss a recent application to the category of derived Mackey functors.

Ralph Kaufmann (University of Purdue​​)

Progress in operad-like theories with a focus on Feynman categories

Monday, February 8th, 16:15-17:30

In the past years, several genereralization of operads have been considered. We will briefly give a quick overview and then turn to a particularly effective formulation —Feynman categories.

The basic setup is categorical, as the name suggests, and this allows to consider many natural constructions. One important aspect are representations, which are functors in this setting. These include algebras, operads and other more intricate or less sophisticated gadgets. In this respect the theory is analogous to representations of groups with restriction, induction and Frobenius reciprocity.

We will give a gentle introduction and as time allows, we may highlight various other constructions and applications achieved ranging from categorical considerations like comprehension schemes and Hopf-algebraic aspects to moduli spaces of curves.

These results are partly in collaboration with B. Ward, J. Lucas, I. Gálvez-Carrillo, A. Tonks, C. Berger, M. Monaco, M. Markl and M. Batanin (in chronological order).

Martin Markl (Akademie věd České republiky)
Operads, properads and more

Monday, February 1st, 11:00-12:15

We will provide a guided tour through the menagerie of various operad- and PROP-like structures. Our approach will be based on pasting schemes, although other approaches will also be mentioned.

Ieke Moerdijk (Universiteit Utrecht)

Varieties of trees

Monday, February 1st, 9:30-10:45

In this lecture we will present various categories of trees, together with Quillen model structures on the associated categories of simplicial presheaves. The central example is formed by the category Omega and the complete Segal model structure on its simplicial presheaves, which provides a model for the homotopy theory of infinity-operads analogous to (and in fact, in a strict sense, containing) Rezk's complete Segal model for infinity categories. But unlike this simplicial case, the case of trees allows for many variations of the underlying category of trees as well as of the model structure. To mention a few variations, one obtains in this way model categories for the homotopy category of unital operads, that of algebras over a given infinity-operad, and that of infinite loop spaces, for example.

The lecture will survey these topics in a hopefully generally accessible way. Later in the programme, we will go into some more technical aspects of the theory.

Reference: G. Heuts, I. Moerdijk, Trees in Algebra and Geometry – An introduction to dendroidal homotopy theory (draft of book available on our websites).

Thomas Nikolaus (Westfälische Wilhelms-Universität Münster)

Polynomial functors and K-theory

Wednesday, February 3rd, 11:00-12:15

We will report on (long overdue) joint work with Clark Barwick, Saul Glasman and Akhil Mathew. Algebraic K-theory of a ring or more generally an additive category is, by its definition as a group completion, functorial in additive functors. We prove that it is in fact functorial in more functors: the so-called polynomial functors (in the sense of Eilenberg–Mac Lane) and still satisfies a universal property. This generalizes previous results by Passi, Dold and others. We will in fact show this for a stable ∞-category and polynomial (= n-excisive) functors in the sense of Goodwillie. If time permits, we will explain applications of this result for lambda-ring structures on algebraic K-theory and give the definition of a spectral lambda ring (i.e., a higher algebra version of a lambda ring).

Angélica Osorno (Reed College)
2-categorical opfibrations, Quillen's Theorem B, and S ^–1S

Thursday, February 11th, 19:15-20:30

Quillen recognized the higher algebraic K-groups of a commutative ring R as the homotopy groups of the topological group completion of the classifying space of the category of finitely generated projective R-modules. He moreover proved that the topological group completion could be obtained categorically via his S ^–1S construction. In this talk we will present a 2-categorical version of this result. As part of the proof, we will give a comparison between strict and lax pullbacks for 2-categorical opfibrations, which gives a version of Quillen's Theorem B amenable to applications. This is joint work with Nick Gurski and Niles Johnson.

Irakli Patchkoria (University of Aberdeen)

Classification of module spectra and Franke's algebraicity conjecture

Friday, February 5th, 9:30-10:45

This is all joint work with Piotr Pstrągowski. Given an E₁-ring R such that the graded homotopy ring π_* R is q-sparse and the global projective dimension d of π_* R is less than q, we show that the homotopy (q – d)-category of Mod(R) is equivalent to the homotopy (q – d)-category of differential graded modules over π_* R. Thus for such E₁-rings the homotopy theory of their modules is algebraic up to the level (q – d). Examples include appropriate Morava K-theories, Johnson–Wilson theories, truncated Brown–Peterson theories and some variations of topological K-theory spectra. We also show that the result is optimal in the sense that (q – d) is the best possible level in general where algebraicity happens. At the end of the talk we will outline how the results for modules can be generalised to the settings where we do not have compact projective generators. This proves Franke's algebraicity conjecture, which provides a general result when certain nice homology theories provide algebraic models for homotopy theories.

Manuel Rivera (University of Purdue)
The fundamental group of a simplicial cocommutative coalgebra

Monday, February 1st, 17:00-18:15

In this talk I will describe a functor F from the category of connected simplicial cocommutative coalgebras to differential graded bialgebras satisfying the following properties:

1) if C is the simplicial cocomutative coalgebra of chains on a reduced simplicial set X, then the dg bialgebra F (C ) is naturally quasi-isomorphic to the chains on the based loop space of |X|;

2) F is homotopical in the sense that it sends "Koszul weak equivalences" (also called "Ω -quasi-isomorphisms") of simplicial cocommutative coalgebras to quasi-isomorphisms of dg bialgebras.

The composition Π₁ = G ◦ H₀ ◦ F, where H₀ denotes zero-th homology and G denotes group-like elements, gives rise to a functor from connected simplicial cocomutative coalgebras to the category of groups, which recovers the fundamental group when applied to chains on a simplicial set. We use this construction to extend theorems of Quillen, Sullivan, Mandell, and Goerss to the setting of non-simply connected spaces. The end goal of the program is to provide a complete algebraic (homological) characterization of homotopy types. Some of the results discussed are a joint work with Mahmoud Zeinalian and Felix Wierstra [1].

References:

[1] Manuel Rivera, Felix Wierstra, Mahmoud Zeinalian, The simplicial coalgebra of chains determines homotopy types rationally and one prime at a time, arXiv:2006.05362.

Martina Rovelli (University of Massachusetts)

Exploring (∞, n)-categories through n-complicial sets – Part 1

Friday, February 12th, 11:00-12:15

With the rising significance of (∞, n)-categories, it is important to have easy-to-handle models for those and understand them as much as possible. In these talks we will discuss the model of n-complicial sets, and study how one can realize convenient representatives of strict n-categories, which encode universal indexing shapes for diagrams valued in (∞, n)-categories. We will focus on n = 2, for which more results are available, but keep an eye towards the general case.

Pedro Tamaroff (Trinity College Dublin)

Tangent complexes and the Diamond Lemma: homotopical methods for term rewriting

Monday, February 8th, 17:45-19:00

Term rewriting has been an indispensable tool to approach various computational problems involving associative algebras and algebraic operads, their homology theories and their deformation theory [1, 2, 3, 5, 6, 8, 9]. One of the cornerstones of the theory, the celebrated Diamond Lemma [4], gives an effectively verifiable criterion of uniqueness of normal forms for term rewriting in associative algebras.

In joint work with V. Dotsenko [7], we presented a new way to interpret and prove this result from the viewpoint of homotopical algebra. Our main result states that every multiplicative free resolution of an algebra with monomial relations gives rise to its own Diamond Lemma, so that Bergman's condition of "resolvable ambiguities" becomes the first non-trivial component of the Maurer–Cartan equation in the corresponding tangent complex. Our approach works for many other algebraic structures, such as algebraic operads, highlighting the importance of computing multiplicative resolutions of algebras presented by monomial relations, as it was done in [10].

For those whose intuition comes from homotopical algebra, our work presents a conceptual explanation of useful (but seemingly technical) criteria of "resolvable ambiguities" for uniqueness of normal forms. For those with a background in Gröbner bases or term rewriting, our work offers intuition behind both the Diamond Lemma and its optimisations, as well as precise guidance on how to generalise those for other algebraic structures. Specifically, our work means that computing models of algebras with monomial relations explicitly helps both to state the relevant Diamond Lemmas and to optimise them.

With the aim of bringing the effective methods of term rewriting closer to the powerful methods of homotopical algebra and higher structures, prior knowledge of the techniques involved in our work is not assumed: they will be explained along the way, with an emphasis in providing a working dictionary to go from term rewriting to deformation theory and homotopical algebra, and back.

References:

[1] D. J. Anick. On the homology of associative algebras. Trans. Amer. Math. Soc., 296(2):641–659, 1986.

[2] J. Backelin. La série de Poincaré–Betti d'une algèbre graduée de type fini à une relation est rationnelle. C. R. Acad. Sci. Paris Ser. A-B, 287(13): A843–A846, 1978.

[3] M. J. Bardzell. Noncommutative Gröbner bases and Hochschild cohomology. In: Symbolic Computation: Solving Equations in Algebra, Geometry, and Engineering (South Hadley, MA, 2000), vol. 286 of Contemp. Math., pp. 227–240. Amer. Math. Soc., Providence, RI, 2001.

[4] G. M. Bergman. The diamond lemma for ring theory. Adv. Math., 29(2):178–218, 1978.

[5] V. Dotsenko, V. Gelinas, P. Tamaroff. Finite generation for Hochschild cohomology of Gorenstein monomial algebras. arXiv:1909.00487.

[6] V. Dotsenko and A. Khoroshkin. Quillen homology for operads via Gröbner bases. Doc. Math., 18:707–747, 2013.

[7] V. Dotsenko and P. Tamaroff. Tangent complexes and the Diamond Lemma. arXiv:2010.14792.

[8] E. S. Golod. Standard bases and homology. In: Algebra – Some Current Trends (Varna, 1986), vol. 1352 of Lecture Notes in Math., pp. 88–95. Springer, Berlin, 1988.

[9] E. L. Green, D. Happel, and D. Zacharia. Projective resolutions over Artin algebras with zero relations. Illinois J. Math., 29(1):180–190, 1985.

[10] P. Tamaroff. Minimal models for monomial algebras. Homology Homotopy Appl., 23(1): 341–366, 2021.

Tashi Walde (Technische Universität München)

Higher Segal spaces via higher excision

Tuesday, February 2nd, 15:30-16:45

Starting from the classical Segal spaces, Dyckerhoff and Kapranov introduced a hierarchy of what they call higher Segal structures. While the first new level (2-Segal spaces) has been well studied in recent years, not much is known about the higher levels and the hierarchy as a whole.

In this talk I explain how this hierarchy can be understood conceptually in close analogy to the manifold calculus of Goodwillie and Weiss. I describe a natural “discrete manifold calculus" on the simplex category and on the cyclic category, for which the polynomial functors are precisely the higher Segal objects. Furthermore, this perspective yields intrinsic categorical characterizations of higher Segal objects in the spirit of higher excision.

Matt Young (Utah State University)
Real categorical representation theory in topology and physics

Tuesday, February 9th, 19:15-20:30

I will give an overview of the theory of Real categorical representations of a finite group, as developed in [1, 2, 3]. In this theory, a Z₂-graded finite group acts on a category by autoequivalences or anti-autoequivalences, according to the Z₂-grading. There is a natural geometric character theory of such representations which is most naturally formulated in terms of unoriented mapping spaces. In this way, one obtains an unoriented generalization of the 2-character theory of Ganter–Kapranov and a candidate for a Hopkins–Kuhn–Ravenel-type character theory for a conjectural Real equivariant elliptic cohomology theory. Time permitting, I will explain how Real categorical representation theory is related to unoriented Dijkgraaf–Witten theory, a three dimensional topological quantum field theory [4].

References:

[1] D. Rumynin and M. Young. Burnside rings for Real 2-representations: The linear theory. Commun. Contemp. Math., to appear. arXiv:1906.11006.

[2] B. Noohi and M. Young. Twisted loop transgression and higher Jandl gerbes over finite groupoids. arXiv:1910.01422, 2019.

[3] M. Young. Real representation theory of finite categorical groups. Higher Struct., to appear. arXiv:1804.09053.

[4] M. Young. Orientation twisted homotopy field theories and twisted unoriented Dijkgraaf–Witten theory. Commun. Math. Phys., 374(3):1645–1691, 2020.