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Stream: learning: questions

Topic: ✔ Can we get fibrations as algebras for a monad?

Fernando Chu (Jul 18 2025 at 18:46):

Street (?) showed that we can get category of fibrations $p : E \to B$ over $B$ as algebras for the the monad $-/\text{id}_B : \text{Cat}/B \to \text{Cat}/B$ that sends $p : E \to B$ to the projection from the comma category $\pi:p/\text{id}_B \to B$. I'm wondering if instead we can get the (non-based) category of fibrations $\text{Fib}$ as the category of algebras for a monad? That is, an algebra $A$ for this monad $T$ would be a fibration over some base.

Mike Shulman (Jul 18 2025 at 18:49):

Yes, it's basically the same monad.

Mike Shulman (Jul 18 2025 at 18:50):

I mean, the monad on the category $Cat^\to$ that sends $p:E\to B$ to $\pi : p/\mathrm{id}_B \to B$.

Fernando Chu (Jul 18 2025 at 18:57):

Right, and this works for any representable 2-category $K$, thanks! I'm also wondering if with this monad it's somehow easier to see that $\text{Fib}_K$ is fibered over $K$.

Mike Shulman (Jul 18 2025 at 18:59):

That should follow from the fact that the category of algebras for a fibred monad on a fibration is also a fibration.

Fernando Chu (Jul 18 2025 at 19:02):

I didn't know about that result, thank you :)

Notification Bot (Jul 18 2025 at 19:02):

Fernando Chu has marked this topic as resolved.