Differentiation of higher groupoid objects in tangent categories

The infinitesimal counterpart of a Lie group is its Lie algebra: the space of right-invariant vector fields with their Lie bracket. As a vector space, it is isomorphic to the tangent space to the Lie group at the unit. A Lie group<em>oid</em>, which is a many-unit generalization of a Lie group, can be similarly differentiated to its Lie algebr<em>oid</em>. As a vector bundle, it is the source-vertical tangent bundle restricted to the units. The Lie bracket on the space of sections of this vector bundle is obtained analogously by its identification with the space of right-invariant vector fields on the manifold of arrows. <br> The main goal of this thesis is to give satisfactory answers to the following questions: Given a (higher) groupoid object <em>G</em> in a category <em>C</em>, what are the structures of <em>C</em> and the properties of <em>G</em> needed for its differentiation? How does the differentiation procedure work? What are the generalized infinitesimal objects? <br> The first part of the answer is to identify the categorical structures needed in the differentiation process. I show that the ambient category has to be equipped with an abstract tangent structure so that there is a Lie bracket of vector fields together with some additional properties. For that, we use the categorical generalization of the tangent functor on smooth manifolds developed by Rosický in the 1980s. An abstract tangent structure on a category <em>C</em> consists of an endofunctor <em>T</em> on <em>C</em>, together with the natural transformations of the bundle projection, the zero section, the fiberwise addition, the vertical lift, and the symmetric structure. Rosický's axioms of a tangent structure are the minimal axioms needed to define the Lie bracket of vector fields on an object of <em>C</em>. <br> For the second part of the answer, I introduce differentiable groupoid objects as the analogue of Lie groupoids, as well as abstract Lie algebroids as the analogue of Lie algebroids in the setting of tangent categories. One of the main results of this thesis is the construction of the abstract Lie algebroid of a differentiable groupoid object in a cartesian tangent category with a scalar multiplication by a ring object. Examples include the differentiation of infinite-dimensional Lie groups and elastic diffeological groupoids. <br> In the last part of the thesis, I propose a method of differentiation of differentiable higher groupoid objects in a tangent category equipped with a compatible Grothendieck pretopology. In 2006, Ševera has argued that the L_∞-algebroid of a higher Lie groupoid is given by the enriched hom in the category of simplicial supermanifolds from the nerve of the pair groupoid of R^0|1 to the higher Lie groupoid. I show that this idea can be rigorously implemented by a universal construction given by a categorical end, that works in any tangent category. Dually, the higher generalized Lie algebroid cohomology is given by a coend in differential complexes.