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Stream: theory: category theory

Topic: Wreath products

Reuben Stern (they/them) (Mar 10 2023 at 20:22):

The categorical wreath product construction $\Delta \wr (-)$ allows one to construct the categories $\Theta_n$ inductively. In what generality does the wreath product make sense, i.e. for which categories other than $\Delta$? I'm particularly interested in if one can make sense of $\mathbb{F}_* \wr (-)$, where $\mathbb{F}_*$ is a skeleton of the category of finite pointed sets.

Tim Campion (Mar 10 2023 at 21:31):

I'm sure I've seen some slightly-more-general-wreath-product-including-the-case-of-finite-sets before -- I want to say either in a Barwick paper or in an Ayala-Francis paper.

Tim Campion (Mar 10 2023 at 21:32):

But the "ultimate" level of generality does seem rather mysterious...

Reuben Stern (they/them) (Mar 10 2023 at 22:41):

ah, I found it in Ayala-Francis Factorization Homology I: Higher Categories. Thanks! This is less straightforward than I imagined :rofl:

Tim Campion (Mar 10 2023 at 23:30):

Awesome! I was pretty hazy on the recollection there, glad it worked out!

Reuben Stern (they/them) (Mar 10 2023 at 23:32):

it's quite inexplicit, though... i don't immediately see how to describe even $\mathsf{Fin}_* \wr \mathsf{Fin}_*$

Fernando Yamauti (Mar 11 2023 at 02:27):

Not as general as possible, but there's an explicit description in https://www.sciencedirect.com/science/article/pii/S002240491000157X

Heiko Braun (Jun 26 2023 at 14:18):

According to Remark 3.4 in https://arxiv.org/abs/math/0512575, the construction even yields a monoidal structure on $\mathsf{Cat}/\Gamma$ and I think it goes as follows:
Let $\# \colon \mathcal{C} \to \Gamma$ and $\# \colon \mathcal{D} \to \Gamma = \mathsf{Fin}_{\ast}^{op}$ be two functors (conveniently denoted with the same symbol). Here $\# c$ as an index denotes the cardinality (excluding the basepoint) of the finite set corresponding to this as an object of $\Gamma$.
We define the objects of $\mathcal{C} \wr \mathcal{D}$ to be $(c; d_1, \dots, d_{\# c})$, where $c \in \mathcal{C}$ and $d_k \in \mathcal{D}$.
Morphisms $(c; d_1, \dots, d_{\# c}) \to (c'; d'_1, \dots, d'_{\# c'})$ consist of a morphism $f \colon \# c \to \# c'$ in $\mathcal{C}$ together with morphisms $f_{ij} \colon d_{j} \to d_{i}$ in $\mathcal{D}$ for all $i, j$ such that $i \mapsto j$ under the map of finite sets dually corresponding to $\# f$.
Finally, the functor $\mathcal{C} \wr \mathcal{D}$ sends $(c; d_1, \dots, d_{\# c})$ to $\bigvee_{1 \leq k \leq \# c} \# d_{k}$ and a morphism $(f; (f_{ij}))$ to $\bigvee_{1 \leq k \leq \# c} f^\#_{ij}$, where $f^\#_{ij}$ denotes the dual of $\# f$.