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In their paper Bicategories of processes, Katis, Sabadini and Walters describe the 'suspension' of a monoidal category (which coincides with what I usually call delooping) and the 'looping' of a monoidal category, which I never heard of.
They consider the looping of a monoidal cat the space of functors, lax nat transformations and modifications .
Is this a standard thing? In particular, is this the literal looping in some model category?
Doesn't seem to be--besides being directed, it's also unbased. in homotopy means based loops, so the loop has to start and end at the basepoint and, more importantly in this context, that basepoint has to be held fixed through a homotopy.
So their is like a free loop space (but directed somehow).
I think both and are seen as 2-categories with a single 0-cell.
The latter is free on the graph with a single vertex and one loop, so it is a kind of "directed loop" object in the category of 2-categories.
(The loops are based at the single 0-cell of , which is already "delooped")
So yes, I would see that as a sensible definition of a directed loop space...
Mmmh I see
Amar Hadzihasanovic said:
The latter is free on the graph with a single vertex and one loop, so it is a kind of "directed loop" object in the category of 2-categories.
That's very sensible, indeed
Is 'directed loop object' a thing? Like the name you give to whatever represents ?