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Stream: theory: category theory

Topic: looping of monoidal cats


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

view this post on Zulip Reid Barton (Jun 11 2021 at 14:45):

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

view this post on Zulip Reid Barton (Jun 11 2021 at 14:46):

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

view this post on Zulip Amar Hadzihasanovic (Jun 11 2021 at 15:08):

I think both CC and BN\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.

view this post on Zulip Amar Hadzihasanovic (Jun 11 2021 at 15:09):

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

view this post on Zulip Amar Hadzihasanovic (Jun 11 2021 at 15:10):

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

view this post on Zulip Matteo Capucci (he/him) (Jun 14 2021 at 08:45):

Mmmh I see

view this post on Zulip 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

view this post on Zulip Matteo Capucci (he/him) (Jun 14 2021 at 08:46):

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