Marco Grandis has been working to collect and formalize the ideas of directed homotopy theory (his main work on the subject has been listed in the references at the nLab page on the subject: directed homotopy theory). I am working with the diagram relating the adjunctions forming free categories on quivers and free monoids on sets; the two adjunctions are connected by the functors taking a monoid to its categorical delooping and a set to its delooping (as a quiver with one object). Both left adjoints commute with these delooping funtors, as do both right adjoints, so the commuting diagram looks like a cylinder. This diagram is then "closed" at the ends by the units and counits of the adjunctions. In terms of directed homotopy theory, what is the "geometric realization" of this diagram? What do adjunctions look like as directed homotopy types? What would be the fundamental monoid of the geometric realization of a single adjunction?
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