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<p dir="ltr">We develop a cyclic theory for categorical coalgebras and show that, when applied to the categorical coalgebra of singular chains on a space, this provides an algebraic model for its free loop space as an S¹-space. In other words, the natural circle action on loop spaces, given by rotation of loops, is encoded in the algebraic structure. In particular, the cyclic homology of the categorical coalgebra of singular chains on a topological space X is isomorphic to the S¹-equivariant homology of the free loop space. This extends known results relating cyclic theories for the algebra of chains on the based loop space and the equivariant homology of its free loop space. In fact, our statements do not require X to be simply connected, and we work over an arbitrary commutative ring. Along the way, we introduce a family of polytopes, coined as Goodwillie polytopes, that control the combinatorics behind the relationship of the coHochschild complex of a categorical coalgebra and the Hochschild complex of its associated differential graded category.</p>

In this thesis, we develop a framework that generalizes the previously known notions of theta-summable Fredholm modules to the setting of locally convex dg algebras. By introducing an additional action of the Clifford algebra, we may treat the even and odd cases simultaneously. In particular, we recover the theory developed by Güneysu/Ludewig and extend the definition of odd theta-summable Fredholm modules to the differential graded category. We then construct a Chern character, which serves as a differential graded refinement of the JLO cocycle, and prove that it has all the expected analytical and homological properties. As an application, we prove an odd noncommutative index theorem relating the spectral flow of a theta-summable Fredholm module to the pairing of the Chern character with the odd Bismut-Chern character in entire (differential graded) cyclic homology, thereby extending results obtained by Güneysu/Cacciatori and Getzler.