Floer's idea of obtaining homological invariants in symplectic and low-dimensional topology via a construction of a Morse-Smale-Witten chain complex, computing the "middle-dimensional homology" of some infinite-dimensional spaces, led to the construction of groundbreaking invariants known collectively under the name of Floer homology. Soon after the emergence of these ideas Atiyah suggested thinking about Floer theories as describing the homology of what he called "semi-infinite dimensional cycles". I will describe the construction of Floer homology facilitating such cycles following the work of Lipyanskiy, and based on earlier work of Mrowka and Ozsváth. The construction is appealing since it does not require considering the compactification of moduli spaces of trajectories, which is a subtle analytical procedure, or perturbing the Floer functional, which is usually a serious obstacle for defining equivariant Floer homologies.