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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For $C$ a category, a cylinder functor on $C$ is a functor denoted
equipped with three natural transformations
such that $\sigma e_0 = \sigma e_1 = Id_C$.
A cylinder functor functorially provides cylinder objects used for talking about homotopy.
The notation is supposed to be suggestive of a product with an object $I$. While this is the motivating example, the cylinder functor need not be of that form.
Richard Williamson has developed a way to build a model structure from the simple point of departure of a structured interval in a monoidal category - more generally, a structured cylinder and a structured co-cylinder in a category. This is given in (Williamson 2012).
A very brief introduction to cylinder functors is given starting on page 9 of Abstract Homotopy Theory.
A fuller development of their properties is given in
Cylinder functors also form one of the key elements in Baues’ approach to algebraic homotopy:
Hans-Joachim Baues, Algebraic Homotopy, Cambridge studies in advanced mathematics 15, Cambridge University Press, (1989).
Hans-Joachim Baues, Combinatorial Homotopy and 4-Dimensional Complexes, de Gruyter Expositions in Mathematics 2, Walter de Gruyter, (1991).
Hans-Joachim Baues, Homotopy Types, in I.M.James, ed., Handbook of Algebraic Topology, 1–72, Elsevier, (1995).
Cylindrical model structures are discussed in
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