## Ahmad Al Yasry: Graph Homologies and Functoriality

We follow the same techniques we used before in arXiv:1308.2271 of extending knot Floer homology to embedded graphs in a 3-manifold, by using the Kauffman topological invariant of embedded graphs by associating a family of links and knots to a such graph by using some local replacements at each vertex in the graph. This new concept of graph Floer homology constructed to be the sum of the knot Floer homologies of all the links and knots associated to this graph and the Euler characteristic is the sum of all the Alexander polynomials of links in the family. We constructed three pre-additive categories, one for the graph under the cobordism, the second one is constructed in arXiv:0807.2924, the third one is a category of Floer homologies for graphs defined by Kauffman. Then we try to study the functoriality of graph categories and their graph homologies in two ways, under cobordism and under branched cover, we are trying to find the compatibility between them. [slides]

## Manuel Bärenz: Extending the Crane-Yetter TQFT

The Crane-Yetter TQFT is not quite as trivial as was believed. For modular categories, it is invertible, but for general ribbon fusion categories, it is not. It might still not give a strong 4-manifold invariant, but it is an interesting starting point when studying homotopy invariants of ribbon fusion categories, or when trying to understand 4d TQFTs in general. John Barrett and I have found a way to calculate the resulting 4-manifold invariant in an easy and beautiful way (arXiv:1601.03580), without resorting to triangulations. Alexander Kirillov Jr. recently has found a way to describe the state spaces assigned to 3-manifolds (possibly with boundary) in terms of string diagrams (still unpublished). So Crane-Yetter is understood in 4 dimensions, and it is separately understood in 2 and 3 dimensions. I will draw the connection between the two. I will draw nice pictures of handle decompositions and surgery diagrams, juggle with some fusion categories and bicategories and, at the end, make a wild guess about the Crane-Yetter TQFT and homotopy 2-types. [slides]

## Christian Blohmann: Higher Geometric Correspondences as Morita Equivalences of Higher Geometric Groupoids

For geometric groupoids such as Lie groupoids Morita equivalences can be given by principal groupoid bibundles. For higher groupoids in sets this generalizes readily to fibrant correspondences of simplicial sets. I will introduce higher geometric correspondences and show that they implement Morita equivalences of geometric $\infty$-groupoids. This requires new technical gadgets, in particular a finitary method of left fibrant replacement by universal simplicial subdivision. This is joint work with C. Zhu.

## Ronald Brown: Modelling and Computing Homotopy Types

The basic philosophy for this talk and its motivation is given in a paper with this title to appear in Indagationes Math, in 2017 in a volume in honor of J.E.L. Brouwer. The example developed there is that with Higgins on filtered spaces, crossed complexes and cubical $\omega$-groupoids.

Here I want to say something on how this scheme describes my work with J.-L. Loday and its development by Graham Ellis and others. The original impetus with Loday was a conjecture of his which could be translated to a triadic Hurewicz Theorem. In the work with Higgins the Relative Hurewicz Theorem, including the nonabelian case $n=2$, was deduced from an Excision Theorem itself deduced from a Higher Van Kampen Theorem. Could we do the analogous argument? Jean-Louis suggested a more general theorem should be easier to prove!

Such an Excision Theorem can be seen in the framework of a functor from dimension $n$ models to dimension $(n-1)$ models, which is both a fibration and cofibration of categories, giving a backdrop for the notion of constructions «induced» at level $n$ from morphisms at level $(n-1)$. [slides]

## David Carchedi: Higher Orbifolds as Structured Infinity Topoi

We will explain how to model higher smooth orbifolds as infinity-topoi with a structure sheaf. We will then explain a theorem giving a completely categorical characterization of these geometric objects.

## Nils Carqueville (for Catherine Meusburger): 3-Dimensional Defect TQFTs and Their Tricategories

I will discuss 2- and 3-dimensional «defect TQFTs», which are symmetric monoidal functors on stratified and decorated bordism categories. Such functors give rise to pivotal 2-categories and Gray categories with duals, respectively. After a short review of the relevant higher-categorical notions and their graphical calculus, I shall explain how they capture the algebraic essence of defect TQFTs. Reshetikhin-Turaev and Turaev-Viro theories embed into our framework, and they can be nontrivially extended to defect TQFTs with embedded surfaces. The talk is based on joint work with Catherine Meusburger and Gregor Schaumann (arXiv:1603.01171). [slides, external]

## Nils Carqueville: Orbifolds of $n$-Dimensional Defect TQFTs

Defect TQFTs are symmetric monoidal functors on decorated stratified bordisms of a fixed dimension $n$. In the talk (based on joint work with I. Runkel and G. Schaumann) this will be made precise by induction in $n$, controlling the allowed ways for strata to meet in terms of cylinders and cones over basic configurations. Once this is set, the main construction is that of «generalised orbifolds» for any $n$-dimensional defect TQFT, generalising both state sum models and gauging of finite symmetry groups, for any $n$: Given a defect TQFT $Z$, one obtains a new TQFT by decorating the Poincaré duals of triangulated bordisms with certain algebraic data $A$ and then evaluating with $Z$. The orbifold datum $A$ is constrained by demanding invariance under $n$-dimensional Pachner moves. [slides]

## Carles Casacuberta: Homotopy Algebras versus Algebras up to Homotopy

If a monad $T$ acts on a model category in such a way that a transferred model structure exists on the category of $T$-algebras, then the homotopy category of $T$-algebras is not equivalent, in general, to the Eilenberg-Moore category of the derived monad. In joint work with O. Raventós and A. Tonks, we find conditions under which homotopical localizations or cellularizations lift to $T$-algebras in both cases and discuss a number of examples, including algebras over (possibly coloured) operads. [slides]

## Hongyi Chu: Enriched Infinity Operads

A good theory of $\infty$-operads, where composition of multimorphisms is only associative up to a specified coherent choice of higher homotopies, is not only indispensable in many areas of mathematics, such as algebraic topology and derived algebraic geometry, but has also becomes interesting for mathematical physics. Nowadays many different approaches to $\infty$-operads such as Lurie's $\infty$-operads, complete dendroidal Segal spaces of Cisinski and Moerdijk, and Barwick's complete Segal operads are available. The goal of this talk is to introduce a theory of homotopy-coherently enriched $\infty$-operads, which can, for example, be used to set up an $\infty$-categorical version of Koszul duality. More precisely, I will first present enriched versions of complete dendroidal Segal spaces and Barwick's complete Segal operads. After discussing an extension of Rezk's completion theorem to this setting, I will show the equivalence of these two models. By using comparison results known from the literature, this result then implies that all the different established models of $\infty$-operads are equivalent. This is part of a collaborative effort together with Rune Haugseng and Gijs Heuts. [slides]

## Domenico Fiorenza: T-Duality in Rational Homotopy Theory

Forming the total space of a principal $U(1)$-bundle can be seen as a functor from topological spaces over the classifying space $BU(1)$ to topological spaces. This functor has an adjoint given by the cyclification; namely taking the (homotopy) quotient of the free loop space by the loop rotation action. If two spaces have isomorphic cyclifications, then there is an isomorphism between cocycles over the total spaces of the corresponding principal $U(1)$-bundles. Moreover, if the isomorphism between the cyclifications happens to cover the topological T-duality automorphism of the classifying space of the so-called topological T-duality 2-group, then the isomorphism between cocycles is given by a push-tensor-pull formula and so it is a topological T-duality isomorphism. While at the global level this picture is somehow heuristic as the rules for topological T-duality are guessed by physical considerations, at the infinitesimal/rational homotopy theory level, this can be neatly derived within the framework of the homotopy theory of (super-)$L_\infty$ algebras. As a consequence one rediscovers in a systematic way topological T-duality between $K^0$-cocycles in type IIA string theory and $K^1$-cocycles in type IIB string theory as given by the Buscher rules for Ramond-Ramond fields and Hori's formula. Joint work with Hisham Sati and Urs Schreiber (arXiv:1611.06536).

## Imma Gálvez-Carrillo: Restriction Species

We examine the close relation between restriction species, due to Schmitt, and the decomposition spaces (or unital 2-Segal spaces) of Gálvez-Kock-Tonks and Dyckerhoff-Kapranov. Any restriction species has an associated incidence coalgebra that we can interpret as an instance of our general construction of coalgebras from decomposition spaces. We also introduce a new notion of directed restriction species, also related to decomposition spaces, as presheaves on the category of finite posets and convex inclusions. Examples of this notion include rooted trees, directed graphs and posets; the associated incidence algebras include the Butcher—Connes—Kreimer Hopf algebra of rooted trees. [slides]

## Philip Hackney: The Homotopy Theory of Segal Cyclic Operads

Cyclic operads were introduced by Getzler and Kapranov as a suitable general setting for defining cyclic cohomology. Roughly, a cyclic operad is an operad-like structure where we relax the distinction between «inputs» and «outputs». Many familiar operads admit a cyclic structure, for instance the associative, Lie and commutative operads, the $A_\infty$-operad, and the framed little disks operads. In support of a project of Boavida, Horel, and Robertson on profinite completions of the framed little disks operad, we lay the foundations for homotopy-coherent versions of cyclic operads.

In pursuit of this goal, we take as inspiration the theory of dendroidal objects, which is used to model homotopy-coherent operads. There is a category of unrooted trees which is closely related to the Moerdijk-Weiss category of rooted trees used in the dendroidal picture. Cyclic operads can be regarded as those presheaves on the category of unrooted trees which satisfy a strict Segal condition. Segal cyclic operads are precisely those (reduced) presheaves satisfying a weak Segal condition. We show that there is a Quillen model category structure on this category of presheaves whose fibrant objects are precisely the Segal cyclic operads. This work is joint with Marcy Robertson and Donald Yau. [slides]

## Benjamin Hennion: Kac-Moody Algebras and Derived Algebraic Geometry

Kac-Moody algebras are used in mathematical physics for their link to 2D conformal field theories. We will introduce a higher dimensional version of Kac-Moody algebras (to be linked with higher dimensional field theories). Those algebras will then be Lie algebras up to homotopy. We will also link them to moduli problems of bundles on higher dimensional varieties, using derived algebraic geometry.

## Johannes Huebschmann: Multi Derivation Maurer-Cartan Algebras and sh-Lie-Rinehart Algebras

Given a commutative algebra $A$ over a ground ring $R$ and an $A$-module $L$, a Maurer-Cartan algebra relative to $A$ and $L$ is the graded $A$-algebra $\mathrm{Alt}_A(L, A)$ of $A$-valued $A$-multilinear alternating froms on $L$ together with an $R$-derivation $d$ that turns $(\mathrm{Alt}_A(L, A), d)$ into a differential graded $R$-algebra. An example of a Maurer-Cartan is the de Rham algebra of a smooth manifold; another example is the familiar differential graded algebra of alternating forms on a Lie algebra $\mathfrak{g}$ with values in the ground field, endowed with the standard Lie algebra cohomology operator.

We extend the classical characterization of a finite-dimensional Lie algebra $\mathfrak{g}$ in terms of its Maurer-Cartan algebra to sh Lie-Rinehart algebras. To this end, we first develop a characterization of sh Lie-Rinehart algebras in terms of differential graded cocommutative coalgebras and Lie algebra twisting cochains that extends the nowadays standard characterization of an ordinary sh Lie algebra (equivalently: $L_\infty$-algebra) in terms of its associated generalized Cartan-Chevalley-Eilenberg coalgebra. Our approach avoids any higher brackets but reproduces these brackets in a conceptual manner. The new technical tool we develop is a notion of filtered multi derivation chain algebra, somewhat more general than the standard concept of a multicomplex endowed with a compatible algebra structure. The crucial observation, just as for ordinary Lie-Rinehart algebras, is this: For a general sh Lie-Rinehart algebra, the generalized Cartan-Chevalley-Eilenberg operator on the corresponding graded algebra involves two operators, one coming from the sh Lie algebra structure and the other one from the generalized action on the corresponding algebra; the sum of the two operators is defined on the algebra while the operators are individually defined only on a larger ambient algebra. We illustrate the structure with quasi Lie-Rinehart algebras. Quasi Lie-Rinehart algebras arise from foliations.

References

[Hue05] Johannes Huebschmann. Higher homotopies and Maurer-Cartan algebras: quasi-Lie- Rinehart, Gerstenhaber, and Batalin-Vilkovisky algebras. In The breadth of symplec- tic and Poisson geometry, volume 232 of Progr. Math., pages 237–302. Birkhäuser Boston, Boston, MA, 2005. arXiv:math/0311294.

[Hue17] J. Huebschmann. Multi derivation Maurer–Cartan algebras and sh Lie–Rinehart algebras. J. Algebra, 472:437–479, 2017. arXiv:1303.4665.

## Theo Johnson-Freyd: The Moonshine Anomaly

Whenever a finite group $G$ acts on a holomorphic conformal field theory, there is a corresponding «anomaly» in $H^3(G,U(1))$ — a sort of «characteristic class» of the action — measuring the obstruction to gauging the action. After a brief review of the general story, I will describe a construction that I call «finite group T-duality», which allows for information about anomalies to be compared between different field theories. The most famous example of a finite group acting on a conformal field theory is surely the Monster group acting on its natural «moonshine» representation. I will explain how «T-duality» can be used to calculate the anomaly. Along the way I will also discuss the Conway group and the anomaly for its natural action, the fermionic version of anomalies, the relation to String structures, and how I hope to construct physically the 576-fold periodicity of $\mathrm{TMF}$.

## Branislav Jurčo: Homological Perturbation, Minimal Models and Effective Actions

Minimal models for quantum homotopy algebras will be constructed using homological perturbation theory. Their relation to effective actions in the BV formalism will be discussed.

## Vanessa Miemietz: Simple Transitive 2-Representations of Finitary 2-Categories

I will explain what I mean by a finitary 2-category and simple transitive 2-representations thereof. I will then survey results on the classification of the latter. [slides]

## Jeffrey Morton: Transformation Structures for 2-Group Actions in Higher Gauge Theory

2-groups, also known as categorical groups, are useful for describing the symmetries of categories. I will describe 2-groups, their actions, and a construction analogous to the transformation groupoids associated to group actions. A motivating example for this work comes from the geometry of moduli spaces of gerbes, and provides a connection to a natural construction for a double category of strict functors and pseudonatural transformations, in which the same example can be expressed in terms of transport functors. I will outline how these two constructions shed light on the symmetry of these higher-categorical moduli spaces. Based on joint work with Roger Picken (IST, Lisbon). [slides]

## Simona Paoli: Segal-Type Models of Higher Categories

Higher categorical structures find applications to diverse areas, such as homotopy theory, mathematical physics, logic and computer science, algebraic geometry. I will start this talk with an introduction to higher categories and to some of their connections with homotopy theory. I will then discuss a class of higher categories, called Segal-type models of weak $n$-categories, which are based on the combinatorics of multi-simplicial sets. This comprise the Tamsamani-Simpson model, as well as two new models which I have introduced, based on a new paradigm to weaken higher categorical structures. [slides]

## Dmitri Pavlov: Extended QFTs are Local

An extended QFT (not necessarily topological) is a functor from the $n$-category whose objects are $0$-dimensional manifolds and $k$-morphisms are $k$-dimensional bordisms with corners to some target $n$-category, e.g., $E_{n−1}$-algebras. «Extended» here refers to the fact that one starts at $0$-dimensional manifolds, in contrast to the more traditional definition of Atiyah, which can start at $d$-dimensional manifolds for some $d>0$. In typical applications bordisms are also equipped with a map to some manifold (more generally, higher stack) $X$.

The category of extended QFTs over $X$ is denoted by $\mathrm{QFT}(X)$. We prove that $\mathrm{QFT}(X)$ is a (higher) sheaf with respect to $X$. In the QFT world this property as known as locality and thus our result can be reformulated by saying that extended QFTs are local. We then combine this result with a result from another paper of ours and show that concordance classes of QFTs over $X$ are in bijection with the homotopy classes of maps from $X$ to a certain classifying space of QFTs, for which we give an explicit formula. This result is an important step toward the Stolz-Teichner conjecture, which claims that concordance classes of $2|1$-dimensional Euclidean QFTs over $X$ are in bijection with $\mathrm{TMF}^0(X)$, where $\mathrm{TMF}$ is the cohomology theory induced by the spectrum of topological modular forms of Hopkins and Miller.

## Timothy Porter: HQFTs and Beyond

The talk will recall the ideas of Turaev's notion of HQFTs and will address certain questions as to how to go beyond the known cases in which these have been studied. [slides]

## Ana Ros Camacho: Towards a Higher Categorical Statement for the Landau-Ginzburg/Conformal Field Theory Correspondence

The Landau-Ginzburg/conformal field theory (LG/CFT) correspondence is a result stated in the physics literature relating in particular categories of matrix factorizations and categories of representations of vertex operator algebras. Supported by recent examples developed in this setting (which were developed jointly with N. Carqueville and I. Runkel), we will describe some joint ongoing work with T. Wasserman to obtain a mathematical statement for the LG/CFT correspondence in the shape of 2- and 3-categories.

## Christian Saemann: The Self-Dual String and the $(2,0)$-Theory from Higher Structures

The six-dimensional superconformal field theory known as the «$(2,0)$-theory» plays an important role in string theory and beyond. It has been known to exist for over 20 years, but the available descriptions are still very incomplete. It is commonly believed that the $(2,0)$-theory is only meaningful as a quantum field theory. In this talk I argue that higher algebraic and differential geometric structures suggest otherwise. Using a 2-group model of the string group, I construct classical BPS equations for this $(2,0)$-theory and present explicit solutions. These can be regarded as a higher analogue of a non-abelian monopole. Moreover, I show that the resulting gauge structure fits very natural a previous incomplete guess of a classical description of the full theory. In particular, it passes many important consistency checks, such as the reduction to four-dimensional supersymmetric Yang-Mills theory. [slides]

## Alexander Schenkel: Towards Homotopical Algebraic Quantum Field Theory

An algebraic quantum field theory is an assignment of algebras to spacetimes. These algebras should be interpreted as quantizations of the algebras of functions on the moduli spaces of a classical field theory. In many cases of interest, especially in gauge theories, these moduli spaces are not conventional spaces but «higher spaces» such as stacks. Consequently, functions on such spaces do not form an algebra but a «higher algebra» which one may describe by homotopical algebra. This motivates us to study assignments of «higher algebras» to spacetimes, which is what I call homotopical algebraic quantum field theory. In this talk I will clarify the above picture and explain its advantages compared to traditional algebraic quantum field theory. For this I will also present simple toy-models related to Abelian gauge theory and homotopy Kan extensions. [slides]

## Daniel Scherl: Orbifolds and the Tricategory of Bimodule Categories

Defect TQFTs in $n$ dimensions are expected to give rise to $n$-categories with duality structures. «Generalized orbifolds» give a way to produce new TQFTs from old ones by taking state sum models internal to a given theory. (See also the talks of N. Carqueville and C. Meusburger.) On the side of the (higher) categories associated to defect TQFTs this is realized by first identifying «orbifolding data» in them and then forming the «orbifold completion» of the category with respect to this data. This has been made precise in two dimensions by N. Carqueville and I. Runkel but the three dimensional case is still work in progress. We will see how the two dimensional version of orbifold completion reappears in the three dimensional case and helps to clarify the situation there. In dimension two the orbifold completion of a pivotal bicategory is obtained by passing to its separable symmetric Frobenius algebras and their representation theory. In three dimensions one needs a categorified notion of such algebras to build a general theory of orbifold completion on. A class of sensible candidates is given by spherical fusion categories (ultimately internal to any Gray-category with duals). We show in what sense they constitute orbifolding data in the tricategory of spherical fusion categories and their bimodule categories (together with appropriate duality structure). As a special case of this we will see how $G$-extensions of spherical fusion categories give rise to orbifolding data.

## Stanisław Szawiel: Categories of Physical Processes

There is a morphism of monoidal stacks $\mathrm{GNS}∶ \mathrm{States} \longrightarrow ∗\mathrm{-Mod}$, assigning to every state on a ∗-algebra its Gelfand-Naimark-Segal representation. It is defined over appropriate algebro-geometric sites and certain topoi. From this single structure one can derive:

- The probabilistic interpretation of Quantum Mechanics
- The theory of composite systems
- The equivalence of the Heisenberg and Schrödinger pictures
- The relation between symmetries and unitary representations
- A theory of quantum Markov processes, including scattering and measurement
- A correctly-typed classical limit Generalizations of this setting seem suitable for discussing path integrals, renormalization flows, and duality conjectures.

These, as well as higher-categorical analogues of the GNS map, remain works in progress. [slides]

## Andrew Tonks: Tilings, Trees, DG2As and $B_\infty$-Algebras

Recall that a $B_\infty$-structure may be expressed as a DG bialgebra structure on a cofree coassociative algebra, or as a multibrace algebra with a compatible $A_\infty$-structure. Loday and Ronco showed that the primitive part of a cofree Hopf algebra is a multibrace algebra, constructed its universal enveloping 2-associative algebra, and gave descriptions of the free multibrace and 2-associative algebras in terms of trees. In joint work with Gálvez-Carrillo and Ronco we extend these results to $B_\infty$ and differential graded 2-associative algebras, and describe the free objects in terms of guillotine partitions in $d=3$ dimensions.

## Martin Wolf: Super Yang-Mills Theory from Higher Chern-Simons Theory

I describe a way of obtaining maximally supersymmetric Yang-Mills theories from higher Chern-Simons theories that correspond to topological sigma-models on supertwistorspaces. [slides]

## Funding Information

Benjamin Hennion, Ronald Brown and Daniel Scherl were funded by CMA/FCT/UNL under the project UID/MAT/00297/2013.