Abstract. The goal of this paper is to prove an equivalence between the model categorical approach to pro-categories, as studied by Isaksen, Schlank and the first author, and the ∞-categorical approach, as developed by Lurie. Three applications of our main result are described.In the first application we use (a dual version of) our main result to give sufficient conditions on an ω-combinatorial model category, which insure that its underlying ∞-category is ω-presentable. In the second application we consider the pro-category of simplicialétale sheaves and use it to show that the topological realization of any Grothendieck topos coincides with the shape of the hyper-completion of the associated ∞-topos. In the third application we show that several model categories arising in profinite homotopy theory are indeed models for the ∞-category of profinite spaces. As a byproduct we obtain new Quillen equivalences between these models, and also obtain an example which settles negatively a question raised by G. Raptis.

We use factorization homology over manifolds with boundaries in order to construct operations on Hochschild cohomology and Hochschild homology. These operations are parametrized by a colored operad involving disks on the surface of a cylinder defined by Kontsevich and Soibleman. The formalism of the proof extends without difficulties to a higher dimensional situation. More precisely, we can replace associative algebras by algebras over the little disks operad of any dimensions, Hochschild homology by factorization (also called topological chiral) homology and Hochschild cohomology by higher Hochschild cohomology. Note that our result works in categories of chain complexes but also in categories of modules over a commutative ring spectrum giving interesting operations on topological Hochschild homology and cohomology. Contents 1. Colored operad 3 2. Homotopy theory of operads and modules 8 3. The little d-disk operad 11 4. Homotopy pullback in Top W 12 5. Embeddings between structured manifolds 15 6. Homotopy type of spaces of embeddings 17 7. Factorization homology 22 8. KS and its higher versions. 25 9. Action of the higher version of KS 27 References 28Let A be an associative algebra over a field k. A famous theorem by Hochschild Kostant and Rosenberg (see [HKR09]) suggests that the Hochschild homology of A should be interpreted as the graded vector space of differential forms on the non commutative space "SpecA". Similarly, the Hochschild cohomology of A should be interpreted as the space of polyvector fields on SpecA.If M is a smooth manifold, let Ω * (M ) be the (homologically graded) vector space of de Rham differential forms and V * (M ) be the vector space of polyvector fields (i.e. global sections of the exterior algebra on T M ). This pair of graded vector spaces supports the following structure:• The de Rham differential : d : Ω * (M ) → Ω * −1 (M ).• The cup product of vector fields : −.− :

We use mixed Hodge theory to show that the functor of singular chains with rational coefficients is formal as a lax symmetric monoidal functor, when restricted to complex schemes whose weight filtration in cohomology satisfies a certain purity property. This has direct applications to the formality of operads or, more generally, of algebraic structures encoded by a colored operad. We also prove a dual statement, with applications to formality in the context of rational homotopy theory. In the general case of complex schemes with non-pure weight filtration, we relate the singular chains functor to a functor defined via the first term of the weight spectral sequence.

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We use Galois group actions on étale cohomology to prove results of formality for dg-operads and dg-algebras with torsion coefficients. Our theory applies, among other related constructions, to the dg-operad of singular chains on the operad of little disks and to the dg-algebra of singular cochains on the configuration space of points in the complex space. The formality that we obtain is only up to a certain degree, which depends on the cardinality of the field of coefficients.