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Stream: learning: questions

Topic: Factorization system for computing Kan extensions

Patrick Nicodemus (Aug 18 2022 at 10:34):

I came across an old notion of factorization system in the literature whose primary purpose seems to be to make the left Kan extension easier to compute. This was used by Borel in his 1953 paper on cohomology of fiber bundles and homogeneous spaces for compact Lie groups. Moore discusses it in his book on simplicial homotopy theory.

Let $M$ be a subcategory of $C$, faithful but not necessarily full. The pair $(C,M)$ is called a 'category with models' (as in the method of acyclic models) if, for each $m \in M, c\in C$, and $u : m\to c$, we can choose a factorization $u = \beta(u)\circ\alpha(u)$, and the functions $\beta,\alpha$ are subject to the following conditions:

1. $\alpha(\operatorname{id}_m)= \beta\operatorname{id}_m)= \operatorname{id}_m$
2. $\alpha(u)$ is a morphism in $M$; for each morphism $u$ in $M$, $\alpha(u)= u$ and $\beta(u) = \operatorname{id}_{\operatorname{cod} u}$
3. $\alpha(\beta(u)) = \operatorname{id}_{\operatorname{dom}{\beta(u)}}$.

Patrick Nicodemus (Aug 18 2022 at 10:34):

4. If $u : m \to c, f : c\to m'$, then $\alpha(fu) = \alpha(f \beta(u))\circ \alpha(u)$ and $\beta(fu) = \beta(f\beta(u))$

Morgan Rogers (he/him) (Aug 18 2022 at 10:38):

Diagrams are tricky to format here, see this thread.