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Stream: theory: category theory

Topic: homotopical delooping and monoidal delooping

Matteo Capucci (he/him) (Oct 29 2020 at 12:11):

Is the delooping defined for objects in a 'categories with homotopies' the same thing as the delooping defined for monoidal categories?
By 'categories with homotopies' I mean either model cats or $\infty$-cats (Lurie-Joyal style).
I would say... yes? Conjecturally, the latter delooping would be the delooping as done in $\infty\mathbf{Cat}$, if this makes sense.
Problem is, I don't know how would I go proving this? It amounts to showing that, given a monoidal ($\infty$-)category $M$, its delooping $BM$ is (equivalent to) its suspension, i.e. the homotopy pushout of $* \leftarrow M \to *$.
Is this related to the delooping hypothesis?

John Baez (Oct 29 2020 at 15:49):

I won't exactly answer your questions, but I'll say some stuff that may help (or maybe you already know it):

In topology looping is adjoint to suspension. Looping only becomes inverse to suspension in the 'stable' context, e.g. for infinite loop spaces: a space of loops in a space of loops in a space of loops in...

Delooping takes a topological monoid $M$, or more generally an $A_\infty$-space (a topological monoid up to coherent homotopy), and creates a space $X$ such that $M$ is equivalent to the $A_\infty$-space of loops in $M$.

In higher category theory we also have operations of looping and suspension. Instead of describing them I'll just describe delooping. Delooping turns a k-tuply monoidal n-category into a (k-1)-tuply monoidal (n+1)-category with one object. For example, take k = 1, n = 1. Then delooping a monoidal category gives a bicategory with one object.

So, delooping moves us diagonally up and to the right in the periodic table:

The Periodic Table of n-Categories

Matteo Capucci (he/him) (Oct 29 2020 at 15:54):

Thank you John, you put the question in context in a much better way.

John Baez (Oct 29 2020 at 15:56):

Thanks! I explained looping, delooping, and suspension a lot more carefully using the periodic table here:

• John Baez and James Dolan, Categorification.

Matteo Capucci (he/him) (Oct 29 2020 at 15:56):

I guess I need to understand some more homotopical algebra to be able to say if my question (a) true and (b) trivially true.

Matteo Capucci (he/him) (Oct 29 2020 at 15:57):

John Baez said:

Thanks! I explained looping, delooping, and suspension a lot more carefully using the periodic table here:

• John Baez and James Dolan, Categorification.

Also this seems to be very relevant! Thanks again

John Baez (Oct 29 2020 at 15:58):

One thing I was trying to say just now is that delooping is only going to be equivalent to suspension if you're working with stable objects: infinite loop spaces, or spectra, or stable $n$-categories, or stable $(\infty,1)$-categories.

John Baez (Oct 29 2020 at 15:59):

Loosely speaking, "stable" means "far down enough in the periodic table that you've reached the place where you see ditto signs".

John Baez (Oct 29 2020 at 16:01):

So, delooping a monoidal $n$-category, or monoidal $(\infty,1)$-category, is really different from suspending it.

John Baez (Oct 29 2020 at 16:10):

But for a symmetric monoidal $n$-category, or a symmetric monoidal $(\infty,1)$-category, they're equivalent.