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of the codiagonal X∐X→XX \coprod X \to X out of the coproduct of XX with itself, such that Cyl(X)→XCyl(X) \to X is a weak equivalence and such that the morphism X∐X→Cyl(X)X \coprod X \to Cyl(X) is “nice” in some way.

If CC has the structure of a model category then “nice” means that X∐X↪Cyl(X)X \coprod X \hookrightarrow Cyl(X) is a cofibration. The factorization axiom of a model category ensures that for each object there is a cylinder object with this property; in fact, one with the additional property that Cyl(X)→XCyl(X) \to X is an acyclic fibration. Cylinder objects such that X∐X↪Cyl(X)X \coprod X \hookrightarrow Cyl(X) is a cofibration are sometimes called good, and those for which moreover Cyl(X)→XCyl(X) \to X is an acyclic fibration are then called very good.

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The notion dual to cylinder object is path space object, which is thus sometimes alternatively called a cocylinder. Cylinder objects and path space objects are used to define left homotopies and right homotopies, respectively.

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There are several views on the role of cylinders / cocylinders in homotopy theory. If there is a natural notion of weak equivalence or quasi-isomorphism then the cylinder is used to encode a notion of homotopy equivalence compatible with the weak equivalences. In some other situations, a ‘cylinder’ , often functorially given and well structured in some way, may be the primitive notion that allows a notion of ‘homotopy equivalence’ to be put forward. Below we give a definition optimised for the former situation. Some indication of the second context is given in the entry cylinder functor.

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This standard cylinder is generally a “good cylinder” in the above sense only for Top StronTop_{Stron} (in which case it is in fact a “very good cylinder”).

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In sSet equipped with the standard model structure on simplicial sets the standard cylinder object for any S∈sSetS \in sSet is S×Δ[1]S \times \Delta[1].

In Top QuillenTop_{Quillen} a sufficient condition for the standard cylinder X×IX\times I to be good is that XX is a CW-complex.

The concept of a cylinder object in a category is an abstraction of the construction in Top which associates to any topological space XX the cylinder X×[0,1]X \times [0,1] over XX, where [0,1][0,1] is the standard topological interval. It is notably used to define the concept of left homotopy, say in a model category.

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The composite of these two maps is the codiagonal (Id,Id):X∐X→X(Id,Id) : X \coprod X \to X. Moreover, the cylinder X×[0,1]X \times [0,1] is homotopy equivalent to XX.

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In some situations the assignment of cylinder objects may exist functorially, in which case one speaks of a cylinder functor.

These properties are the characterizing properties of the cylinder that can be abstracted and realized in other categories.

The precise argument that for XX a cell complex then also the standard cyclinder X×IX\times I is a cell complex is spelled out in

In Top, the standard cylinder X×[0,1]X\times [0,1] is a cylinder object for both the classical model structure on topological spaces Top QuillenTop_{Quillen} (the one with Serre fibrations) as well as for the Strøm model structure Top StromTop_{Strom} (the one with Hurewicz fibrations).

In a category with weak equivalences CC that has coproducts a cylinder object Cyl(X)Cyl(X) for an object XX is a factorization