Verdier duality for general coefficient systems

Local to global phenomena are omnipresent in mathematics, and since the appearance of the work of Grothendieck and his school it has been settled that the best way to treat such problems formally is via the theory of sheaves. It has been noticed already many years ago that sheaves are natural coefficients for cohomology theories defined on geometric objects of any kind, which means that they show their full power when set within a homotopical context. Therefore, one is somehow forced to move to the world of higher categories to work efficiently on the subject. This thesis essentially revolves around the theory of sheaves with values in infinity-categories, with a particular attention to its manifestations in topology and differential geometry. The initial sparkle that paved the way for the making of the present work was the intuition that Lurie's version of Verdier duality, expressed as an equivalence between sheaves and cosheaves, would have to be the starting point in the setup of the whole theory. The term Verdier duality is also often used in more classical literature to refer to an equivalence between some derived categories of constructible sheaves and their opposite: we will conclude this thesis by providing a generalization of this result. For these two reasons, we felt obliged to pay tribute to Jean-Louis Verdier in the choice of the title of this work. Verdier duality will make its initial appearence in the first chapter, which is devoted to the construction of the six functor formalism for sheaves on locally compact Hausdorff spaces. We will use it to transport the pushforward and pullback of cosheaves to sheaves, thus providing an abstract description of the well-known pushforward with proper support and exceptional pullback. Alongside with this interaction between sheaves and cosheaves, we will consider Lurie's tensor product of cocomplete infinity-categories as a second fundamental tool to build our six operations. This will allow us to work with a surprisingly vast class of stable coefficients, and to formulate Kunneth and projection formulas in a both unusual and convenient way. In the second chapter, Verdier duality will be used to produce an actual duality on certain infinity-categories of constructible sheaves. More precisely, we will consider stratified spaces equipped with conically smooth atlases, an extension of smooth atlases to the setting of singular spaces introduced by Ayala, Francis and Tanaka. The theory of conically smooth structures provides a procedure that allows to functorially resolve singularities to smooth manifolds with corners. We will make use of this procedure to perform some computations of exit path infinity-categories needed to obtain the aforementioned duality theorem. The reader who is not familiar with conically smooth spaces might wonder how our results interplay with more classical approaches to smoothly stratified spaces. To resolve this reasonable doubt, we also provide a proof of a conjecture of Ayala, Francis and Tanaka written in collaboration with Guglielmo Nocera, which asserts that any Whitney stratified space admits a conically smooth structure