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Stream: theory: category theory

Topic: Effective descent morphisms are descent equivalences

Brendan Murphy (Jul 25 2024 at 20:54):

The paper "Composites of effective descent maps" by Reiterman, Sobral, and Tholen considers a commutative triangle $p = \varphi \circ q$ and constructs from this a comparison map $\Pi^q$ from the category of descent data for $\varphi$ to the category of descent data for $p$ (descent data with respect to the self-indexing). It proves that $\Pi^q$ is fully faithful if $q$ is an descent morphism and is an equivalence if $q$ is an effective descent morphism. The proof there is very explicit but also ~5 full pages of calculations. I would like to apply this theorem to an $(\infty, 1)$-category and the explicitness actually makes it harder to figure out how to generalize. Does anyone know a simpler or more abstract proof of this theorem? I don't feel like I understand the idea behind the RST proof

Brendan Murphy (Jul 25 2024 at 20:56):

If the square in their diagram (4) were a 2-pullback then the theorem would be easy. Is this true? It seems like it is if both q and φ satisfy descent, but in the general setting I'm not sure

Brendan Murphy (Jul 25 2024 at 20:56):

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