nLab cylinder factorisation system

Idea

A cylinder factorisation system on a category is a generalisation of a factorisation system in which we have a pair $(E, M)$, where $E$ is a class of small cocones, and $M$ is a class of small cones, such that every cylinder factorises uniquely as a cocone in $E$ followed by a cone in $M$.

Cylinder factorisation systems are algebras for a pseudomonad structure given by sending each category to its Isbell envelope: see Garner 2015.

Related concepts

• factorisation system, orthogonal factorisation system

• Isbell envelope, Isbell adjunction

References

Cylinder factorisation systems were introduced in:

• Richard Garner, The Isbell monad, Advances in Mathematics 274 (2015) pp.516-537. (draft)

The special case in which $E$ and $M$ comprise classes of discrete cones was introduced in:

• Horst Herrlich, and Walter Meyer, Factorization of flows and completeness of categories, Quaestiones Mathematicae 17.1 (1994): 1-11.

See also:

• George E. Strecker, Flows with respect to a functor, Applied Categorical Structures 8 (2000): 559-578.

The special case in which $E$ is a class of epimorphisms and $M$ is a class of cones was studied in the following papers, including an application to colimits in categories of algebras:

• Rudolf-E. Hoffmann, Factorization of cones, Mathematische Nachrichten 87 1 (1979) 221-238. [doi:10.1002/mana.19790870120]

• Rudolf-E. Hoffmann, Factorization of cones II, with applications to weak Hausdorff spaces, in: Categorical Aspects of Topology and Analysis: Proceedings of an International Conference Held at Carleton University, Ottawa, August 11–15, 1981, Springer (1981) [doi:10.1007/BFb0092878]

Earlier still, the following considered the special case in which the right class furthermore is required to comprise discrete cones:

• Horst Herrlich. Topological functors, General Topology and its Applications 4.2 (1974): 125-142.

See also:

• Horst Herrlich, G. Salicrup, and R. Vazquez, Dispersed factorization structures, Canadian Journal of Mathematics 31.5 (1979): 1059-1071.

• A. Melton, and G. E. Strecker, On the structure of factorization structures, Category Theory: Applications to Algebra, Logic and Topology Proceedings of the International Conference Held at Gummersbach, July 6–10, 1981. Springer Berlin Heidelberg, 1982.

• Abstract and Concrete Categories