homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
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A cyclic space is a cyclic object in a category of spaces, usually in one of the standard presentations of homotopy types, hence in TopologicalSpaces or SimplicialSets.
D. Burghelea Z. Fiedorowicz, Section 1 of: Cyclic homology and algebraic K-theory of spaces—II, Topology Volume 25, Issue 3, 1986, Pages 303-317 (doi:10.1016/0040-9383(86)90046-7)
Jean-Louis Loday, Cyclic Spaces and -Equivariant Homology (doi:10.1007/978-3-662-21739-9_7)
Chapter 7 in: Cyclic Homology, Grundlehren 301, Springer 1992 (doi:10.1007/978-3-662-21739-9)
Jean-Louis Loday, Section 3 of: Free loop space and homology, Chapter 4 in: Janko Latchev, Alexandru Oancea (eds.): Free Loop Spaces in Geometry and Topology, IRMA Lectures in Mathematics and Theoretical Physics 24, EMS 2015 (arXiv:1110.0405, ISBN:978-3-03719-153-8)
See also:
In the context of geometric realization of loop stacks and inertia stacks:
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