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Stream: learning: reading & references

Topic: Bicategory(/double category) of spans with lax 2-cells

Harrison Grodin (Jan 02 2025 at 23:31):

For a category $\mathcal{C}$, the bicategory of spans has

• objects from $\mathcal{C}$,
• morphisms $X \to Y$ are spans $X \xleftarrow{f} Z \xrightarrow{g} Y$
• a 2-cell from $X \xleftarrow{f} Z \xrightarrow{g} Y$ to $X \xleftarrow{f'} Z' \xrightarrow{g'} Y$ is a map $Z \xrightarrow{h} Z'$ such that $f = h \mathbin{;} f'$ and $g = h \mathbin{;} g'$

Replacing category $\mathcal{C}$ with a bicategory $\mathcal{B}$, we can get a similar construction with "(co?)lax" 2-cells:

• a 2-cell is a map $Z \xrightarrow{h} Z'$ equipped with $\mathcal{B}$-2-cells $f \Rightarrow h \mathbin{;} f'$ and $g \Rightarrow h \mathbin{;} g'$

(An analogous story should work for double categories, too.)

This seems like a rather natural construction (and I have a use case for it!), but I haven't been able to find a reference. (I expected to see it near discussions of spans and bi/double categories, but nothing so far afaict.) Does anyone know where I might look?

Harrison Grodin (Jan 03 2025 at 01:29):

diagram

Roland Baumann (Jan 03 2025 at 02:52):

@Harrison Grodin This is actually the main subject of my thesis (soon to be defended). There is a nice three-dimensional structure involving "lax" maps of spans: I've taken to calling it Strata(B). I'm happy to chat about it over email.

Matteo Capucci (he/him) (Jan 03 2025 at 09:13):

Unless your legs enjoy adequate lifting properties, you cannot horizontally compose these 2-cells, so they don't form a bicategory

Matteo Capucci (he/him) (Jan 03 2025 at 09:14):

In my last paper with David Jaz Myers we make good use of left-fibrant, right-lax spans, check https://arxiv.org/abs/2410.21889