Abstract. We consider the question of defining interleaving metrics on generalized persistence modules over arbitrary preordered sets. Our constructions are functorial, which implies a form of stability for these metrics. We describe a large class of examples, inverseimage persistence modules, which occur whenever a topological space is mapped to a metric space. Several standard theories of persistence and their stability can be described in this framework. This includes the classical case of sublevelset persistent homology. We introduce a distinction between 'soft' and 'hard' stability theorems. While our treatment is direct and elementary, the approach can be explained abstractly in terms of monoidal functors.

We redevelop persistent homology (topological persistence) from a categorical point of view. The main objects of study are (R, ≤)-indexed diagrams in some target category. A set of such diagrams has an interleaving distance, which we show generalizes the previously-studied bottleneck distance. To illustrate the utility of this approach, we generalize previous stability results for persistence, extended persistence, and kernel, image and cokernel persistence. We give a natural construction of a category of ε-interleavings of (R, ≤)-indexed diagrams in some target category, and show that if the target category is abelian, so is this category of interleavings.2010 Mathematics Subject Classification. 55N99, 68W30, 18A25, 18E10, 54E35. Key words and phrases. Applied topology, persistent topology, topological persistence, diagrams indexed by the poset of real numbers, interleaving distance.The first author gratefully acknowledges support from AFOSR grant # FA9550-13-1-0115.1 Two foundational papers in this subject are [ELZ02] and [ZC05]. In the first, Edelsbrunner, Letscher and Zomorodian define persistent homology for (Z + , ≤)-indexed diagrams of finite dimensional vector spaces, that are obtained from filtered finite simplicial complexes by taking simplicial homology with coefficients in a field. In the second, Zomorodian and Carlsson take a purely algebraic point of view. They define persistent homology for tame (Z + , ≤)-indexed diagrams of finite-dimensional vector spaces, and prove a bijection between isomorphism classes of such tame diagrams and finite barcodes whose endpoints lie in Z + ∪ {∞}. (Z + , ≤)-indexed diagrams of finite dimensional vector spaces are called persistence modules.These papers are rounded out by [CSEH07], where Cohen-Steiner, Edelsbrunner and Harer prove that persistent homology is useful in applications by showing that it is stable in the following sense. Let f, g : X → R be continuous functions on a triangulable space. Define an (R, ≤)-indexed diagram of topological spaces, F , by setting F (a) = f −1 (−∞, a] and letting F (a ≤ b) be given by inclusion. Define G similarly using g. Let H be the singular homology functor with coefficients in a field. Assume that HF and HG are diagrams of finite dimensional vector spaces and that they are tame. Then the bottleneck distance between HF and HG is bounded by the supremum norm between f and g.This stability result is significantly strengthened by Chazal, Cohen-Steiner, Glisse, Guibas and Oudot in [CCSG + 09]. They drop the assumptions that X be triangulable, that f, g be continuous, and that HF and HG be tame. Their approach is crucial to this paper. They explicitly work with (R, ≤)-indexed diagrams, though they consider them from an algebraic, not categorical, point of view. They define the interleaving distance, d, between such diagrams, and define the bottleneck distance, d B , between such diagrams using limits of discretizations, and show that d B ≤ d.The basic idea of persistent homology has been extended in numerous ways. Here we focus o...

For any 1-reduced simplicial set K we define a canonical, coassociative coproduct on ΩC(K), the cobar construction applied to the normalized, integral chains on K, such that any canonical quasi-isomorphism of chain algebras from ΩC(K) to the normalized, integral chains on GK, the loop group of K, is a coalgebra map up to strong homotopy. Our proof relies on the operadic description of the category of chain coalgebras and of strongly homotopy coalgebra maps given in [HPS]. ≃ − → C * ΩX, such that θ X restricts to quasi-isomorphisms (T V ≤n , d) ≃ − → C * ΩX n+1 , where X n+1 denotes the (n + 1)-skeleton of X, T V denotes the free (tensor) algebra on a free, 1991 Mathematics Subject Classification. Primary: 55P35 Secondary: 16W30, 18D50, 18G35, 55U10, 55U35, 57T05, 57T30.

Communicated by C. Kassel MSC:Primary: 16E40 19D55 secondary: 18G60 55M20 55U10 81T30 a b s t r a c t Generalizing the work of Doi and of Idrissi, we define a coHochschild homology theory for chain coalgebras over any commutative ring and prove its naturality with respect to morphisms of chain coalgebras up to strong homotopy. As a consequence we obtain that if the comultiplication of a chain coalgebra C is itself a morphism of chain coalgebras up to strong homotopy, then the coHochschild complex H (C) admits a natural comultiplicative structure. In particular, if K is a reduced simplicial set and C * K is its normalized chain complex, then H (C * K ) is naturally a homotopy-coassociative chain coalgebra. We provide a simple, explicit formula for the comultiplication on H (C * K ) when K is a simplicial suspension.The coHochschild complex construction is topologically relevant. Given two simplicial maps g, h : K → L, where K and L are reduced, the homology of the coHochschild complex of C * L with coefficients in C * K is isomorphic to the homology of the homotopy coincidence space of the geometric realizations of g and h, and this isomorphism respects comultiplicative structure. In particular, there is an isomorphism, respecting comultiplicative structure, from the homology of H (C * K ) to H * L|K |, the homology of the free loops on the geometric realization of K .

Let EK be the simplicial suspension of a pointed simplicial set K. We construct a chain model of the James map, α K : CK → ΩCEK. We compute the cobar diagonal on ΩCEK, not assuming that EK is 1-reduced, and show that α K is comultiplicative. As a result, the natural isomorphism of chain algebras T CK ∼ = ΩCK preserves diagonals.In an appendix, we show that the Milgram map, Ω(A ⊗ B) → ΩA ⊗ ΩB, where A and B are coaugmented coalgebras, forms part of a strong deformation retract of chain complexes. Therefore, it is a chain equivalence even when A and B are not 1-connected.