Operads often arise from geometry. The standard $A_\infty$ operad can be derived from the cellular chains on the Stasheff associahedra, and an $A_\infty$ algebra is an algebra over this operad. The notion of an $\mathbf{fc}$-multicategory, also called a virtual double category, is a two-dimensional generalization of operads and multicategories. Here $\mathbf{fc}$ stands for the free category monad.
We establish a natural $\mathbf{fc}$-multicategory structure on the collection of moduli spaces of pseudo-holomorphic polygons with boundary on sequences of Lagrangian submanifolds in a symplectic manifold. These moduli spaces are known to underlie the construction of Fukaya categories. Based on this, we develop the theory of differential graded (dg) variants of $\mathbf{fc}$-multicategories and show that a broad range of $A_\infty$-type structures, such as $A_\infty$ algebras, $A_\infty$ (bi)modules, and $A_\infty$ categories (possibly curved), admit a uniform operadic formulation as algebras over dg $\mathbf{fc}$-multicategories.
We construct a new cylinder object for semifree differential graded (dg) categories in the category of dg categories. Using this, we give a practical formula computing homotopy colimits of semifree dg categories. Combining it with the result of Ganatra, Pardon, and Shende, we get a formula computing wrapped Fukaya categories of Weinstein manifolds using their sectorial coverings. This formula has lots of applications including a practical computation of the wrapped Fukaya category of any cotangent bundle or plumbing space. In this paper, we compute wrapped Fukaya categories of cotangent bundle…
Let $M$ be an exact symplectic manifold with contact type boundary such that $c_1(M)=0$. In this paper we show that the cyclic cohomology of the Fukaya category of $M$ has the structure of an involutive Lie bialgebra. Inspired by a work of Cieliebak-Latschev we show that there is a Lie bialgebra homomorphism from the linearized contact homology of $M$ to the cyclic cohomology of the Fukaya category. Our study is also motivated by string topology and 2-dimensional topological conformal field theory.
We endow categories of non-symmetric operads with natural model structures. We work with no restriction on our operads and only assume the usual hypotheses for model categories with a symmetric monoidal structure. We also study categories of algebras over these operads in enriched non-symmetric monoidal model categories.
We prove, under mild assumptions, that a Quillen equivalence between symmetric monoidal model categories gives rise to a Quillen equivalence between their model categories of (non-symmetric) operads, and also between model categories of algebras over operads. We also show left properness results on model categories of operads and algebras over operads. As an application, we prove homotopy invariance for (unital) associative operads.
We provide an explicit formula for localizing $A^1$-homotopy invariants of topological Fukaya categories of marked surfaces. Following a proposal of Kontsevich, this differential $\mathbb Z$-graded category is defined as global sections of a constructible cosheaf of dg categories on any spine of the surface. Our theorem utilizes this sheaf-theoretic description to reduce the calculation of invariants to the local case when the surface is a boundary-marked disk. At the heart of the proof lies a theory of localization for topological Fukaya categories which is a combinatorial analog of Thomason-…
We introduce twisted arrow categories of operads and of algebras over operads. Up to equivalence of categories, the simplex category $Δ$, Segal's category $Γ$, Connes cyclic category $Λ$, Moerdijk-Weiss dendroidal category $Ω$, and categories similar to graphical categories of Hackney-Robertson-Yau are twisted arrow categories of symmetric or cyclic operads. Twisted arrow categories of operads admit Segal presheaves and 2-Segal presheaves, or decomposition spaces. Twisted arrow category of an operad $P$ is the $(\infty, 1)$-localization of the corresponding category $Ω/P$ by the boundary prese…
In this paper we prove a local-to-global principle for the Fukaya category of a closed Riemann surface $Σ$ of genus $g \geq 2$. We show that $\mathrm{Fuk}(Σ)$ can be glued from the Fukaya categories of the pairs-of-pants making up a pants decomposition of $Σ$. This extends our earlier results for the case of punctured Riemann surfaces. Our result has several interesting consequences: we obtain simple proofs of old and new HMS statements for Riemann surfaces, and establish a geometrization theorem for the objects of $\mathrm{Fuk}(Σ)$.
I describe a generalization of the notion of operadic category due to Batanin and Markl. For each such operadic category I describe a skew monoidal category of collections, such that a monoid in this skew monoidal category is precisely an operad over the operadic category. In fact I describe two skew monoidal categories with this property. The first has the feature that the operadic category can be recovered from the skew monoidal category of collections; the second has the feature that the right unit constraint is invertible. In the case of the operadic category S of finite sets and functions…
We introduce a new way to encode semicyclic structures using a stack of broken cycles. (We also prove an analogue for paracyclic structures.) This was motivated not only by higher algebra but also by Fukaya-categorical considerations. We also openly speculate about some Fukaya-categorical implications. For example, this stack sees moduli of stopped Liouville disks, and hence yields another platform for gluing together Fukaya categories. We also see that Lagrangian cobordisms with multiple ends may not only serve to detect $K_0$ groups of Fukaya categories, but higher $K$-theory groups as well.…
Higher-dimensional category theory is the study of n-categories, operads, braided monoidal categories, and other such exotic structures. It draws its inspiration from areas as diverse as topology, quantum algebra, mathematical physics, logic, and theoretical computer science. This is the first book on the subject and lays its foundations. Many examples are given throughout. There is also an introductory chapter motivating the subject for topologists.
Operads were originally defined as V-operads, that is, enriched in a symmetric or braided monoidal category V. The symmetry or braiding in V is required in order to describe the associativity axiom the operads must obey, as well as the associativity that must be a property of the action of an operad on any of its algebras. After a review of the role of operads in loop space theory and higher categories we go over definitions of iterated monoidal categories and introduce a large family of simple examples. Then we generalize the definition of operad by defining n-fold operads and their algebras …
We decribe the correspondence between normalised $ω$-operads and certain lax monoidal structures on the category of globular sets. As with ordinary monoidal categories, one has a notion of category enriched in a lax monoidal category. Within the aforementioned correspondence, we provide also an equivalence between the algebras of a given normalised $ω$-operad, and categories enriched in globular sets for the induced lax monoidal structure.
The purpose of this paper is two-fold. In Part 1 we introduce a new theory of operadic categories and their operads. This theory is, in our opinion, of an independent value. In Part 2 we use this new theory together with our previous results to prove that multiplicative 1-operads in duoidal categories admit, under some mild conditions on the underlying monoidal category, natural actions of contractible 2-operads. The result of D. Tamarkin on the structure of dg-categories, as well as the classical Deligne conjecture for the Hochschild cohomology, is a particular case of this statement.
We define the notion of a 2-operad relative to an operad, and prove that the 2-associahedra form a 2-operad relative to the associahedra. Using this structure, we define the notions of an $(A_\infty,2)$-category and $(A_\infty,2)$-algebra in spaces and in chain complexes over a ring. Finally, we show that for any continuous map $A \to X$, we can associate an $(A_\infty,2)$-algebra $θ(A \to X)$ in $\textsf{Top}$, which specializes to $θ(\text{pt} \to X) = Ω^2 X$ and $θ(A \to \text{pt}) = ΩA \times ΩA$.
Fix a suitably convex, exact symplectic manifold M. We consider the stable oo-category Lag(M) of non-compact Lagrangians whose (higher) morphisms are (higher) Lagrangian cobordisms between them. We show that this oo-category pairs with the Fukaya category Fuk(M) of compact branes. In fact, we also show that there is a subcategory of Lag(M) which pairs with the wrapped Fukaya category of M. This is a first step in a project to enrich wrapped Fukaya categories over cobordism spectra. As a corollary, we show that cobordant compact branes are equivalent in the Fukaya category. We will also mention…
We explain how any cofibrantly generated weak factorisation system on a category may be equipped with a universally and canonically determined choice of cofibrant replacement. We then apply this to the theory of weak omega-categories, showing that the universal and canonical cofibrant replacement of the operad for strict omega-categories is precisely Leinster's operad for weak omega-categories.
We prove the existence of Morita model structures on the categories of small simplicial categories, simplicial sets, simplicial operads and dendroidal sets, modelling the Morita homotopy theory of $(\infty,1)$-categories and $\infty$-operads. We give a characterization of the weak equivalences in terms of simplicial presheaves, simplicial algebras and slice categories. In the case of the Morita model structure for simplicial categories and simplicial operads, we also show that each of these model structures can be obtained as an explicit left Bousfield localization of the Bergner model structu…
We use Lurie's symmetric monoidal envelope functor to give two new descriptions of $\infty$-operads: as certain symmetric monoidal $\infty$-categories whose underlying symmetric monoidal $\infty$-groupoids are free, and as certain symmetric monoidal $\infty$-categories equipped with a symmetric monoidal functor to finite sets (with disjoint union as tensor product). The latter leads to a third description of $\infty$-operads, as a localization of a presheaf $\infty$-category, and we use this to give a simple proof of the equivalence between Lurie's and Barwick's models for $\infty$-operads.
In this paper we develop the theory of operads, algebras and modules in cofibrantly generated symmetric monoidal model categories. We give J-semi model strucures, which are a slightly weaker version of model structures, for operads and algebras and model structures for modules. In a second part we develop the thoery of S-modules of [EKMM]., which allows a general homotopy theory for commutative algebras and pseudo unital symmetric monoidal categories of modules over them. Finally we prove a base change and projection formula.
We extend the theory of d-categories, by providing an explicit description of the right mapping spaces of the d-homotopy category of an $\infty$-category. Using this description, we deduce an invariant $\infty$-categorical characterization of the d-homotopy category. We then proceed to develop an analogous theory of d-operads, which model $\infty$-operads with (d -1)-truncated multi-mapping spaces, and prove analogous results for them.
In this paper we study duoidal structures on $\infty$-categories of operadic modules. Let $\mathcal{O}^{\otimes}$ be a small coherent $\infty$-operad and let $\mathcal{P}^{\otimes}$ be an $\infty$-operad. If a $\mathcal{P}\otimes\mathcal{O}$-monoidal $\infty$-category $\mathcal{C}^{\otimes}$ has a sufficient supply of colimits, then we show that the $\infty$-category ${\rm Mod}_A^{\mathcal{O}}(\mathcal{C})$ of $\mathcal{O}$-$A$-modules in $\mathcal{C}^{\otimes}$ has a structure of $(\mathcal{P},\mathcal{O})$-duoidal $\infty$-category for any $\mathcal{P}\otimes\mathcal{O}$-algebra object $A$.
We extend the W-construction of Boardman and Vogt to operads of an arbitrary monoidal model category with suitable interval, and show that it provides a cofibrant resolution for well-pointed sigma-cofibrant operads. The standard simplicial resolution of Godement as well as the cobar-bar chain resolution are shown to be particular instances of this generalised W-construction.
In this paper we introduce the notion of an operator category and two different models for homotopy theory of $\infty$-operads over an operator category -- one of which extends Lurie's theory of $\infty$-operads, the other of which is completely new, even in the commutative setting. We define perfect operator categories, and we describe a category $Λ(Φ)$ attached to a perfect operator category $Φ$ that provides Segal maps. We define a wreath product of operator categories and a form of the Boardman--Vogt tensor product that lies over it. We then give examples of operator categories that provid…
Over suitable monoidal model categories, we construct a Dwyer-Kan model category structure on the category of algebras over an augmented operadic collection. As examples we obtain Dwyer-Kan model category structure on the categories of enriched wheeled props, wheeled properads, and wheeled operads, among others. This result extends known model category structure on the categories of operads, properads, and props enriched in simplicial sets and other monoidal model categories. We also show that our Dwyer-Kan model category structure is well behaved with respect to simultaneous changes of the un…
We establish a novel relation between the cluster categories associated with marked surfaces and the topological Fukaya categories of the surfaces. We consider a generalization of the triangulated cluster category of the surface by a $2$-Calabi-Yau extriangulated/exact $\infty$-category, which arises via Amiot's construction from the relative Ginzburg algebra of the triangulated surface. This category is shown to be equivalent to the $1$-periodic version of the topological Fukaya category of the marked surface, as well as to Wu's Higgs category. We classify the cluster tilting objects in this …
We establish a highly flexible condition that guarantees that all colored symmetric operads in a symmetric monoidal model category are admissible, i.e., the category of algebras over any operad admits a model structure transferred from the original model category. We also give a necessary and sufficient criterion that ensures that a given weak equivalence of admissible operads admits rectification, i.e., the corresponding Quillen adjunction between the categories of algebras is a Quillen equivalence. In addition, we show that Quillen equivalences of underlying symmetric monoidal model categori…
We establish a Quillen equivalence relating the homotopy theory of Segal operads and the homotopy theory of simplicial operads, from which we deduce that the homotopy coherent nerve functor is a right Quillen equivalence from the model category of simplicial operads to the model category structure for infinity-operads on the category of dendroidal sets. By slicing over the monoidal unit, this also gives the Quillen equivalence between Segal categories and simplicial categories proved by J. Bergner, as well as the Quillen equivalence between quasi-categories and simplicial categories proved by …
We construct a topological cellular operad such that the algebras over its cellular chains are the homotopy unital A-infinity algebras of Fukaya-Oh-Ohta-Ono.
The present article exploits the fact that permutads (aka shuffle algebras) are algebras over a terminal operad in a certain operadic category Per. In the first, classical part we formulate and prove a claim envisaged by Loday and Ronco that the cellular chains of the permutohedra form the minimal model of the terminal permutad which is moreover, in the sense we define, self-dual and Koszul. In the second part we study Koszulity of Per-operads. Among other things we prove that the terminal Per-operad is Koszul self-dual. We then describe strongly homotopy permutads as algebras of its minimal m…
Let $G$ be a compact Lie group and $\mathbf{k}$ be a field of characteristic $p \geq 0$ such that $H^* (G)$ does not have $p$-torsion. We show that a free Lagrangian orbit of a Hamiltonian $G$-action on a compact, monotone, symplectic manifold $X$ split-generates an idempotent summand of the monotone Fukaya category $\mathcal{F}(X; \mathbf{k})$ if and only if it represents a non-zero object of that summand (slightly more general results are also provided). Our result is based on: an explicit understanding of the wrapped Fukaya category $\mathcal{W}(T^*G; \mathbf{k})$ through Koszul twisted com…
In this paper, we describe a general theory of modules over an algebra over an operad. We also study functors between categories of modules. Specializing to the operad E_d of little d-dimensional disks, we show that each (d-1)-manifold gives rise to a theory of modules over E_d-algebras and each bordism gives rise to a functor from the category defined by its incoming boundary to the category defined by its outgoing boundary. We describe how to assemble these categories into a map from a certain operad to the operad of (infinity)-categories.
In this paper, we study conditions for extending Quillen model category properties , between two symmetric monoidal categories, to their associated category of symmetric sequences and of operads. Given a Quillen equivalence $λ: \mathcal{C}=Ch_{\Bbbk,t} \leftrightarrows \mathcal{D}: R,$ so that $\mathcal{D}$ is any symmetric monoidal category and the adjoint pair $(λ, R)$ is weak monoidal, we prove that the categories of connected operads $Op_\mathcal{C}$ and $Op_\mathcal{D}$ are Quillen equivalent. This expands an analogous result of Schwede-Shipley(\cite{SS03}) when we replace these categorie…
We study a certain type of action of categories on categories and on operads. Using the structure of the categories Δ and Ω governing category and operad structures, respectively, we define categories which instead encode the structure of a category acting on a category, or a category acting on an operad. We prove that the former has the structure of an elegant Reedy category, whereas the latter has the structure of a generalized Reedy category. In particular, this approach gives a new way to regard group actions on categories and on operads.
The structure of a $k$-fold monoidal category as introduced by Balteanu, Fiedorowicz, Schwänzl and Vogt can be seen as a weaker structure than a symmetric or even braided monoidal category. In this paper we show that it is still sufficient to permit a good definition of ($n$-fold) operads in a $k$-fold monoidal category which generalizes the definition of operads in a braided category. Furthermore, the inheritance of structure by the category of operads is actually an inheritance of iterated monoidal structure, decremented by at least two iterations. We prove that the category of $n$-fold oper…
This is an expository article about operads in homotopy theory written as a chapter for an upcoming book. It concentrates on what the author views as the basic topics in the homotopy theory of operadic algebras: the definition of operads, the definition of algebras over operads, structural aspects of categories of algebras over operads, model structures on algebra categories, and comparison of algebra categories when changing operad or underlying category. In addition, it includes two applications of the theory: The original application to $n$-fold loop spaces, and an application to algebraic …
We prove that a pairing between the Fukaya category and the oo-category of Lagrangian cobordisms respects mapping cones. This is another step toward constructing a lift of Fukaya categories to the level of spectra (in the sense of stable homotopy theory). As corollaries, we show that the map in our previous work from cobordism groups to Floer cohomology lifts to the level of spectra, and one also recovers some results of Biran and Cornea for what we call "vertically collared" cobordisms.
This short note reports on joint work with Michael Batanin towards a general machine for proving Baez-Dolan Stabilization Theorems for various models of higher categories, based on substitudes, Bousfield localization, and homotopical Beck-Chevalley squares. I provide a road map to our recent papers, and include new results proving Baez-Dolan Stabilization Theorems for Tamsamani weak $n$-categories, higher Segal categories, Ara's $n$-quasi-categories, and cartesian models of Segal and complete Segal objects due to Bergner and Rezk. I also attempt to clarify the connection to higher braided oper…
We introduce the dendroidal analogs of the notions of complete Segal space and of Segal category, and construct two appropriate model categories for which each of these notions corresponds to the property of being fibrant. We prove that these two model categories are Quillen equivalent to each other, and to the monoidal model category for infinity-operads which we constructed in an earlier paper. By slicing over the monoidal unit objects in these model categories, we derive as immediate corollaries the known comparison results between Joyal's quasi-categories, Rezk's complete Segal spaces, and…
One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we continue the work of [7] to adapt the machinery of globular operads [4] to this task. The resulting theory includes the Gray tensor product of 2-categories and the Crans tensor product [12] of Gray categories. Moreover much of the previous work on the globular approach to higher category theory is simplified by our new foundations, and we illustrate this by giving an expedited account of many aspects of Cheng's analysis [11] of Trimb…
The goal of the present paper is to compare, in a precise way, two notions of operads up to homotopy which appear in the literature. Namely, we construct a functor from the category of strict unital homotopy colored operads to the category of infinity-operads. The former notion, that we make precise, is the operadic generalization of the notion of A-infinity-categories and the latter notion was defined by Moerdijk--Weiss in order to generalize the simplicial notion of infinity-category of Joyal--Lurie. This functor extends in two directions the simplicial nerve of Faonte--Lurie for A-infinity-…
A moment category is endowed with a distinguished set of split idempotents, called moments, which can be transported along morphisms. Equivalently, a moment category is a category with an active/inert factorisation system fulfilling two simple axioms. These axioms imply that the moments of a fixed object form a monoid, actually a left regular band. Each locally finite unital moment category defines a specific type of operad which records the combinatorics of partitioning moments into elementary ones. In this way the notions of symmetric, non-symmetric and $n$-operad correspond to unital moment…
We set up a general theory of weak or homotopy-coherent enrichment in an arbitrary monoidal $\infty$-category $\mathcal{V}$. Our theory of enriched $\infty$-categories has many desirable properties; for instance, if the enriching $\infty$-category $\mathcal{V}$ is presentably symmetric monoidal then $\mathrm{Cat}^\mathcal{V}_\infty$ is as well. These features render the theory useful even when an $\infty$-category of enriched $\infty$-categories comes from a model category (as is often the case in examples of interest, e.g. dg-categories, spectral categories, and $(\infty,n)$-categories). This…
One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product of 2-categories. In this paper we continue the developments of [3] and [2] by understanding the natural generalisations of Gray's little brother, the funny tensor product of categories. In fact we exhibit for any higher categorical structure definable by an n-operad in the sense of Batanin [1], an analogous tensor product which forms a symmetric monoidal closed structure on the category of algebras of the operad.
The homotopy theory of infinity-operads is defined by extending Joyal's homotopy theory of infinity-categories to the category of dendroidal sets. We prove that the category of dendroidal sets is endowed with a model category structure whose fibrant objects are the infinity-operads (i.e. dendroidal inner Kan complexes). This extends the theory of infinity-categories in the sense that the Joyal model category structure on simplicial sets whose fibrant objects are the infinity-categories is recovered from the model category structure on dendroidal sets by simply slicing over the point.
We establish, by elementary means, the existence of a cofibrantly generated monoidal model structure on the category of operads. By slicing over a suitable operad the classical Rezk model structure on the category of small categories is recovered.
We construct and study projective and Reedy model category structures for bimodules and infinitesimal bimodules over topological operads. Both model structures produce the same homotopy categories. For the model categories in question, we build explicit cofibrant and fibrant replacements. We show that these categories are right proper and under some conditions left proper. We also study the extension/restriction adjunctions.
Generalizing a result of Dwyer and Kan for simplicial categories, we characterize the morphisms of multi-sorted simplicial algebraic theories and simplicial coloured operads which induce a Quillen equivalence between the corresponding categories of algebras.
We extend Thomason's homotopy colimit construction in the category of permutative categories to categories of algebras over an arbitrary $\Cat$ operad and analyze its properties. We then use this homotopy colimit to prove that the classifying space functor induces an equivalence between the category of $n$-fold monoidal categories and the category of $\mathcal{C}_n$-spaces after formally inverting certain classes of weak equivalences, where $\mathcal{C}_n$ is the little $n$-cubes operad. As a consequence we obtain an equivalence of the categories of $n$-fold monoidal categories and the categor…
