Schedule for: 18w5147 - Infinity-Categories, Infinity-Operads, and their Applications

The alpha form of factorization homology is based on the topology of Ran spaces, where $Ran(X)$ is the space of finite subsets of $X$, topologized so that points can collide. This alpha factorization homology takes as input a manifold or variety $X$ together with a suitable algebraic coefficient system $A$, and it outputs the cosheaf homology of $Ran(X)$ with coefficients defined by $A$. Factorization homology simultaneously generalizes singular homology, Hochschild homology, and conformal blocks or observables in conformal field theory. I'll discuss applications of this alpha form of factorization homology in the study of mapping spaces in algebraic topology, bundles on algebraic curves, and perturbative quantum field theory. I'll then describe a beta form of factorization homology, where one replaces $Ran(X)$ with a moduli space of stratifications of $X$, designed to overcome certain strict limitations of the alpha form. A main result, joint with Ayala and Rozenblyum, is that an $(\infty,n)$-category defines a cosheaf on the moduli space of vari-framed stratifications. A theorem-in-progress is that an $(\infty,n)$-category with adjoints defines a cosheaf on the moduli space of solidly $n$-framed stratifications. An immediate consequence is a proof of the cobordism hypothesis (after Baez–Dolan, Costello, Hopkins–Lurie, and Lurie) exactly in the manner of Pontryagin–Thom theory. This is joint work with David Ayala.

I'll describe the multiplicative structure of Thom spectra, and the theory of multiplicative orientations of stable fibrations from the $\infty$-categorical perspective pioneered by Ando, Blumberg, Gepner, Hopkins and Rezk. This is joint work with Tobias Barthel.

Modular operads were originally constructed by Getzler and Kapronov to model operations similar to the gluing of boundaries of a genus $g$ Riemann surface with $n$ boundaries. Variations on this original definition have found importance in various geometric problems. In this talk we give two models for up to homotopy, or $\infty$, versions of modular operads. This is joint work with Philip Hackney and Donald Yau.

Normed motivic spectra are motivic spectra equipped with a coherent system of multiplicative norms along finite etale maps. Many motivic spectra of interest admit canonical normed structures, e.g. the motivic cohomology spectrum, the algebraic K-theory spectrum, and the algebraic cobordism spectrum. For example, the normed structure on HZ underlies Fulton and MacPherson’s norm maps on Chow groups as well as Voevodsky’s power operations in motivic cohomology. Among other things, the formalism of normed spectra allows us to extend the Fulton-MacPherson norms to Chow groups in mixed characteristic and to Chow-Witt groups. This is joint work with Tom Bachmann.

Algebraic $K$-theory -- the analog of topological $K$-theory for varieties and schemes -- is a deep and far-reaching invariant, but it is notoriously difficult to compute. To date, the primary means of understanding $K$-theory is through its "cyclotomic trace" map $K \to TC$ to topological cyclic homology. This map is usually advertised as an analog of the Chern character, but this is something of a misnomer: $TC$ is a further refinement of any flavor of de Rham cohomology (even "topological", i.e. built from $THH$), though this discrepancy disappears rationally. However, despite the enormous success of so-called "trace methods" in $K$-theory computations, the algebro-geometric nature of $TC$ has remained mysterious. In this talk, I will describe a new construction of $TC$ that affords a precise interpretation of the cyclotomic trace at the level of derived algebraic geometry, which is based on nothing but universal properties (coming from Goodwillie calculus) and the geometry of 1-manifolds (via factorization homology). This is joint work with David Ayala and Nick Rozenblyum.

Let $G$ be a finite group and $X$ a topos with homotopy coherent $G$-action. From this, we construct a stable homotopy theory $Sp^G(X)$ which recovers and extends the theory of genuine $G$-spectra. We explain what our construction yields when: (i) $X$ is the topos of sheaves on a topological space with $G$-action (ii) $X$ is the etale $C_2$-topos of a scheme $S$ adjoined a square root of -1. This is a preliminary report on joint work with Elden Elmanto.

Étale homotopy theory, as originally introduced by Artin and Mazur in the late 60s, is a way of associating to a suitably nice scheme a pro-object in spaces. We will explain how, when working over a base field $k$, a modern reformulation in terms of the theory of infinity-topoi leads to a more refined invariant, which takes into account the action of the absolute Galois group. We will then explain joint work of ours with Elden Elmanto that extends the étale realization functor of Isaksen, which provides a bridge between unstable motivic homotopy theory and étale homotopy theory, to a functor taking into account the Galois group action. Time permitting, we will explain work in progress extending this to the stable setting.

This talk is themed around the idea that stratifications are a powerful source of correspondences between categorical and geometric structure. I'll discuss connections between stratifications and the categorical property of orbitality, which is central to parametrized higher algebra. I'll also introduce stratified higher topos theory, giving a new computation of an étale homotopy type as an application. There will be a couple of open questions. Most of what I'll say belongs to joint work with Clark Barwick and Peter Haine.

In this talk I will articulate and contextualize the following sequence of results.

• The Bruhat decomposition of the general linear group defines a stratification of the orthogonal group.
• Matrix multiplication defines an algebra structure on its exit-path category in a certain Morita category of categories.
• In this Morita category, this algebra acts on the category of $n$-categories -- this action is given by adjoining adjoints to $n$-categories.

This result is extracted from a larger program -- entirely joint with John Francis, some parts joint with Nick Rozenblyum -- which proves the cobordism hypothesis.

The Dunn--Lurie additivity theorem states that an algebra with $n$ compatible multiplications is the same as an algebra over the operad of little $n$-disks, i.e. an $E_n$-algebra. I will describe a similar statement for shifted Poisson algebras. Namely, an $E_n$-algebra in the category of $m$-shifted Poisson algebras is the same as an $(n+m)$-shifted Poisson algebra. I will explain motivitations for such a result coming from mathematical physics and derived Poisson geometry.

I will report on joint work with Thomas Nikolaus. We construct a t-structure on the stable infinity-category of cyclotomic spectra and show that the homotopy objects with respect to this t-structure on naturally Dieudonné modules, i.e., abelian groups equipped with operations $V$ and $F$ such that $FV=p$. For smooth schemes over perfect fields of characteristic $p>0$, these cyclotomic homotopy objects are the terms in the de Rham—Witt complex. As an application, I will explain how certain formal moduli problems associated to Calabi-Yau varieties admit cyclotomic interpretations, which can be used to strengthen past results about the behavior of these invariants under Fourier—Mukai equivalence.

In this talk I will explain how one can use geometric arguments to obtain results on dualizablity in a "factorization version" of the Morita category. We will relate our results to previous dualizability results (by Douglas-Schommer-Pries-Snyder and Brochier-Jordan-Snyder on Turaev-Viro and Reshetikin-Turaev theories). We will discuss applications of these dualiability results: one is to construct examples of low-dimensional field theories "relative" to their observables. An example will be given by Azumaya algebras, for example polynomial differential operators (Weyl algebra) in positive characteristic and its center. (This is joint work with Owen Gwilliam.)

We introduce enriched $\infty$-operads as certain presheaves which generalize Barwick's complete Segal operads. The simple structure of the indexing categories allows us to define algebras between enriched $\infty$-operads. By comparing this presheaf model with an enriched version of Moerdijk-Cisinski's dendroidal Segal spaces, we then prove a rectification theorem which states that the homotopy theory of $\infty$-operads enriched in a nice symmetric monoidal model category is equivalent to that of strictly enriched operads. From this we can easily infer that all established models for $\infty$-operads are equivalent to simplicially enriched operads. The talk is based on joint work with Rune Haugseng.

I will discuss an infinity-category obtained from that of pointed spaces by inverting the maps inducing isomorphisms in $v_n$-periodic homotopy groups. The case $n = 0$ corresponds to rational homotopy theory. In analogy with Quillen’s results in the rational case, I will outline how this $v_n$-periodic homotopy theory is equivalent to the homotopy theory of Lie algebras in $T(n)$-local spectra (or a variant for $K(n)$-local spectra). One can also compare it to the homotopy theory of cocommutative coalgebras in $T(n)$-local spectra, where there is only an equivalence up to a certain "Goodwillie convergence" issue. I will describe the relevant operadic and cooperadic structures and a form of Koszul duality relevant to this setting.

In classical algebra, the Frobenius provides a natural endomorphism of every ring in characteristic $p$. In higher algebra, one can do away with the characteristic $p$ assumption and define a Frobenius for every $E_{\infty}$ ring spectrum; it has the defect that it takes values in the Tate cohomology of the ring, rather than the ring itself. I will discuss a dual construction in the case of coalgebras which has better formal properties and answer a question of Nikolaus about the homotopy coherence of the coalgebra Frobenius. I will then describe applications of this work to $p$-adic homotopy theory and time permitting, discuss the case of algebras.

In joint work with Dominic Verity we prove that four models of (∞,1)-categories — quasi-categories, complete Segal spaces, Segal categories, and 1-complicial sets — are equivalent for the purpose of developing ∞-category theory. To prove this we first introduce the notion of an ∞-cosmos, a category in which ∞-categories live as objects, an example of which is given by each of the four models mentioned above. We then explain how the category theory of ∞-categories can be developed inside any ∞-cosmos; eg, we define right adjoints and limits and prove that the former preserve the latter. We conclude by arguing that the four above mentioned ∞-cosmoi all biequivalent, the upshot being that ∞-categorical structures are preserved, reflected, and created by a number of “change-of-model” functors. More precisely, we show that each of these ∞-cosmoi have a biequivalent “calculus of modules,” modules between ∞-categories being a vehicle to express ∞-categorical universal properties.