Abstract: The humble interval is a gateway to both homotopy theory and higher-dimensional geometry. The interval allows us to define homotopies between continuous functions, while taking products of the interval with itself gives rise to the square, cube, tesseract and beyond. Abstracting away from topological spaces, one may speak of interval objects, cylinder objects or even cylinder functors on other categories, and use these to define homotopies there. However, the connection to higher-dimensional cubes is lost in the process. In this talk, I will show that in categories with algebraic weak factorization systems (AWFS) -- which provide a conducive setting for abstract homotopy theory -- we can recover cylinder functors that share both the homotopical and "cubical" aspects of the interval. More precisely, for any object $X$ in a category $\mathcal{C}$ equipped with coproducts and an AWFS, there is a functor from the category of cubes-with-connections to $\mathcal{C}$ that sends the 0-cube to $X$, the 1-cube to the cylinder on $X$, and so on. As a corollary, any such category is enriched in cubical sets with connections. (Joint work with Chris Kapulkin.)