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Stream: theory: category theory

Topic: weak equivalences given by vertical maps

Matteo Capucci (he/him) (Jan 02 2024 at 09:39):

Let $F:A \to B$ be a functor. A map in $A$ is $F$-vertical is $F$ sends it to an iso.
Clearly: (1) all isos are $F$-vertical and (2) if two out of $f,g, fg$ are $F$-vertical, so is the third.

Let me call $\ker F$ the wide subcategory of $A$ spanned by $F$-vertical maps, then $(A,\ker F)$ is a [[category with weak equivalences]]. I'm now interested in localizing $A$ at $\ker F$, and in particular in studying when this localization is reflective.

This feels like a very elementary definition and I was wondering if anyone is aware of this being studied before?

fosco (Jan 02 2024 at 10:10):

look at:

• Cassidy C, Hébert M, Kelly GM. Reflective subcategories, localizations and factorizationa systems. Journal of the Australian Mathematical Society Series A Pure Mathematics and Statistics. 1985;38(3):287-329. doi:10.1017/S1446788700023624 (available here)
• in particular, the relation between torsion theories and pre-fibrations, explained in Rosicky-Tholen here.

Matteo Capucci (he/him) (Jan 02 2024 at 14:40):

Thanks Fosco

Matteo Capucci (he/him) (Jan 02 2024 at 14:41):

The first reference I read. The second reference seems to be very relevant! As I was saving it in Zotero I realized I already had it there and forgot lol