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Stream: theory: category theory

Topic: looping of monoidal cats

Matteo Capucci (he/him) (Jun 11 2021 at 14:31):

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 $\mathbf C$ the space of functors, lax nat transformations and modifications $B\mathbb N \to \mathbf C$.
Is this a standard thing? In particular, is this the literal looping $\Omega$ in some model category?

Reid Barton (Jun 11 2021 at 14:45):

Doesn't seem to be--besides being directed, it's also unbased. $\Omega$ 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.

Reid Barton (Jun 11 2021 at 14:46):

So their $\Omega$ is like a free loop space (but directed somehow).

Amar Hadzihasanovic (Jun 11 2021 at 15:08):

I think both $C$ and $\mathrm{B}\mathbb{N}$ 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.

Amar Hadzihasanovic (Jun 11 2021 at 15:09):

(The loops are based at the single 0-cell of $C$, which is already "delooped")

Amar Hadzihasanovic (Jun 11 2021 at 15:10):

So yes, I would see that as a sensible definition of a directed loop space...

Matteo Capucci (he/him) (Jun 14 2021 at 08:45):

Mmmh I see

Matteo Capucci (he/him) (Jun 14 2021 at 08:45):

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

Matteo Capucci (he/him) (Jun 14 2021 at 08:46):

Is 'directed loop object' a thing? Like the name you give to whatever represents $\Omega$?