A Geometric Approach to Duplicial Duality
Philipp Joram
Dresden University of Technology
Abstract: Many algebraic constructs can be viewed as presheaves over some indexing category.
Examples are the simplex category, that indexes simplicial sets,
or (in homological algebra) Connes' cyclic category.
The latter is used to define the cyclic (co)homology of an algebra,
and has a slightly more general cousin, called the "duplicial category".
We give a *visual proof* that both the cyclic and the duplicial category are self-dual by
treating their objects as configurations of subspaces in some ambient space.
We make the "proof by picture" precise by applying methods from homotopy- and 2-category theory.
This generalizes to many more categories of the same geometric flavor.