Factorization homology theories of topological manifolds, after Beilinson, Drinfeld, and Lurie, are homology-type theories for topological n-manifolds whose coefficient systems are n-disk algebras or n-disk stacks. In this work, we prove a precise formulation of this idea, giving an axiomatic characterization of factorization homology with coefficients in n-disk algebras in terms of a generalization of the Eilenberg-Steenrod axioms for singular homology. Each such theory gives rise to a kind of topological quantum field theory, for which observables can be defined on general n-manifolds and not only closed n-manifolds. For n-disk algebra coefficients, these field theories are characterized by the condition that global observables are determined by local observables in a strong sense. Our axiomatic point of view has a number of applications. In particular, we give a concise proof of the non-abelian Poincaré duality of Salvatore, Segal, and Lurie. We present some essential classes of calculations of factorization homology, such as for free n-disk algebras and enveloping algebras of Lie algebras, several of which have a conceptual meaning in terms of Koszul duality.

Abstract. This work forms a foundational study of factorization homology, or topological chiral homology, at the generality of stratified spaces with tangential structures. Examples of such factorization homology theories include intersection homology, compactly supported stratified mapping spaces, and Hochschild homology with coefficients. Our main theorem characterizes factorization homology theories by a generalization of the Eilenberg-Steenrod axioms; it can also be viewed as an analogue of the Baez-Dolan cobordism hypothesis formulated for the observables, rather than state spaces, of a topological quantum field theory. Using these axioms, we extend the nonabelian Poincaré duality of Salvatore and Lurie to the setting of stratified spaces -this is a nonabelian version of the Poincaré duality given by intersection homology. We pay special attention to the simple case of singular manifolds whose singularity datum is a properly embedded submanifold and give a further simplified algebraic characterization of these homology theories. In the case of 3-manifolds with 1-dimensional submanifolds, these structure gives rise to knot and link homology theories.

Abstract. We develop a theory of conically smooth stratified spaces and their smooth moduli, including a notion of classifying maps for tangential structures. We characterize continuous spacevalued sheaves on these conically smooth stratified spaces in terms of tangential data, and we similarly characterize 1-excisive invariants of stratified spaces. These results are based on the existence of open handlebody decompositions for conically smooth stratified spaces, an inverse function theorem, a tubular neighborhood theorem, an isotopy extension theorem, and functorial resolutions of singularities to smooth manifolds with corners.

For an arbitrary symmetric monoidal ∞-category V, we define the factorization homology of V-enriched ∞-categories over (possibly stratified) 1-manifolds and study its basic properties.In the case that V is cartesian symmetric monoidal, by considering the circle and its self-covering maps we obtain a notion of unstable topological cyclic homology, which we endow with an unstable cyclotomic trace map. As we show in [AMGRa], these induce their stable counterparts through linearization (in the sense of Goodwillie calculus). Date: October 18, 2017. 1 Appendix B. Factorization systems 65 References 67 0. Introduction 0.1. Categorified integration in quantum field theory. A fundamental link between manifolds and higher algebraic structures is factorization homology : in its most primitive form, this takes a framed n-manifold M and an E n -algebra A -in chain complexes, say -and returns a chain complex M A obtained by "integrating" the algebra A over configurations of n-disks in M [Lura, AF15]. 1 Besides being intimately related to the study of mapping spaces and proximate notions in manifold topology [Sal01, Kal01, Böd87, McD75, May72, Seg73, Lura, AF15], factorization homology is a close cousin of Beilinson-Drinfeld's algebro-geometric theory of chiral homology [BD04], which computes conformal blocks in conformal field theory. These ideas have since spawned much subsequent activity in mathematical physics -notably Costello-Gwilliam's work [CG17, CG] on perturbative quantum field theory, wherein they recover global observables from local ones via factorization homology. In order to accommodate higher-codimensional defects, in [AFRa] Ayala-Francis-Rozenblyum generalized the algebraic input of factorization homology from E n -algebras to (∞, n)-categories: 2 for a framed n-manifold M and an (∞, n)-category C, they defined the factorization homology M C of C over M as the space of labelings by C of disk-stratifications of M . This construction allows for field theories that are not necessarily determined by their point-local observables.However, this construction is still one step removed from the production of TQFTs of physical interest, which are linear in nature: where the framework of [AFRa] yields a space, one would like to obtain a vector space or chain complex. In this paper, we provide a blueprint for the appropriate generalization: we construct the factorization homology of enriched (∞, 1)-categories. We expect that our construction contains all the essential features of a full theory of factorization homology of enriched (∞, n)-categories. The key idea that drives our approach can be summarized as follows.Slogan 0.1. Enriched factorization homology arises from categorified factorization homology.We will explain Slogan 0.1 in §0.3. 0.2. Hochschild homology, cyclic homology, and the cyclotomic trace. In fact, our primary motivation for constructing enriched factorization homology comes from a different direction, namely topological Hochschild homology and its connection with algebraic K-theory.Recall that the Hochs...

We prove a duality for factorization homology which generalizes both usual Poincaré duality for manifolds and Koszul duality for En-algebras. The duality has application to the Hochschild homology of associative algebras and enveloping algebras of Lie algebras. We interpret our result at the level of topological quantum field theory.