where $pr_i \,\colon\, X \times X \xrightarrow{\;} X$ are the two projections out of the Cartesian product and $diag \,\colon\, X \xrightarrow{\;} X \times X$ denotes the diagonal map. So:

A significant fraction of dialectical philosophy can be modeled mathematically through the use of “cylinders” (diagrams of shape $\Delta_1$) in a category, wherein the two identical subobjects (united by the third map in the diagram) are “opposite”. In a bicategory, oppositeness can be very effectively characterized in terms of adjointness, but even in an ordinary category it may sometimes be given a useful definition. For example, an effective basis for teaching calculus is a ringed category satisfying the Hadamard-Marx property. The description in engineering mechanics of continuous bodies that can undergo cracking is clarified by an example involving lattices, raising a new questions about the foundations of topology.

($\Delta_1$ is the category of order-preserving maps between totally-ordered sets of one and two elements, respectively.)