The notation is supposed to be suggestive of a product with an object $I$. While this is the motivating example, the interval functor need not be of that form.

Cylindrical model structures

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 his thesis and also in the ArXiv article listed below.

References

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

K. H. Kamps and T. Porter, Abstract Homotopy and Simple Homotopy Theory (GoogleBooks)

Cylinder functors also form one of the key elements in Baues’ approach to algebraic homotopy:

H. J. Baues: Algebraic Homotopy, Cambridge studies in advanced mathematics 15, Cambridge University Press, (1989).

H. J. Baues: Combinatorial Homotopy and 4-Dimensional Complexes, de Gruyter Expositions in Mathematics 2, Walter de Gruyter, (1991).

H. J. Baues: Homotopy Types, in I.M.James, ed., Handbook of Algebraic Topology, 1–72, Elsevier, (1995).