In this poster, we construct singular homology, and homotopy from a homological point of view. We begin with simplices and other simplicial structures, and upgrade them to their singular counterparts on an arbitrary pointed topological space.
Homology is captured via the singular homology groups, which measure the difference between cycles and boundaries, capturing 'interesting' topological invariants such as holes. The homotopical alternative is to non-Abelieanize cycles (loops) by equipping loop spaces with concatenation and modding out by homotopical equivalence. The resulting homotopy groups include non-contractible classes of loops, which, once again, encode holes.
Finally, we consider holonomies of physical observables (which map configurations on the topological space to states), given a Borel measure on the underlying toplogical space. Surprisingly, a weak set-theoretic form of the first isomorphism theorem from group theory shows up in that since states are images of holonomies generated by loops, state space is (set-theoretically) isomorphic to loop space modded out by its kernel. This represents the philosophy that states are not point-like but loop-like instead, and motivates the classification of observables into holonomic and non-holonomic. In the latter and more general case, states are not functions of configurations, but rather sections of the bundle encoding how loops and the states they generate live on the basepoint of the pointed topological space in question.