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Stream: learning: questions

Topic: calculus of fractions

James Deikun (Jan 15 2024 at 16:56):

I pretty much understand the view of localizing using a calculus of fractions and spans/cospans/"roofs". However, it seems like often it's more useful to look at things using an explicit formula for the Hom. The article [[calculus of fractions]] has a formula, but it's rather unclear. The formula, in the case of a calculus of right fractions is:

$$\mathrm{Hom}_{C[W^{-1}]}(X,Y) = \underset{X' \xrightarrow{p \in W} X}{\mathrm{colim}} \mathrm{Hom}_{C}(X',Y)$$

What isn't clear to me here is:

• What are the arrows in the colimit diagram? I would guess it's probably the full subcategory of $C/X$ on morphisms in $W$, but I have no basis for this guess except type tetris.
• How does composition work?

James Deikun (Jan 15 2024 at 18:43):

Okay, so it looks like:

• My guess was right
• The colimit formula has a retrospectively obvious equivalence to the span view
• And that's how composition works.

James Deikun (Jan 15 2024 at 18:51):

Specifically, the colimit consists of a morphism $\hat{f} : X' \to X \in W$ (identifying the component) and a morphism $f : X' \to Y$ (the content). Two elements are identified when there is a morphism (any morphism) from X' to X'' so that the diagram commutes; this is extended to zigzags but you only have to check spans because the Ore condition means any zigzag has at least one corresponding span.