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Stream: learning: questions

Topic: computing the image of a restriction of a map

Nathan Corbyn (Jan 05 2022 at 13:41):

Sorry if this is a silly question! Given a subobject $s : S \to X$ in an arbitrary category $\mathcal{C}$, and a map $f : X \to Y$ in $\mathcal{C}$, we can understand the composite $f \circ s : S \to Y$ as 'the restriction of $f$ to the subobject' (leaning on the usual subset / subobject intuition). What necessary additional structure does $\mathcal{C}$ need in order for us to construct a minimal subobject $t : T \to Y$ and a map $g : S \to T$ such that $f \circ s = t \circ g$. That is, an object that corresponds to the 'image' of the restriction (again, leaning on the usual intuition). Is this some sort of lifting operation? Can it be described as a limit in $\mathrm{Sub}(Y)$ (the category of subobjects of $Y$)?

Zhen Lin Low (Jan 05 2022 at 13:45):

There is no need to introduce subobjects on the domain side. It suffices to understand images of morphisms in general. One class of categories where images exist and are well behaved is the class of regular categories.

Zhen Lin Low (Jan 05 2022 at 13:49):

The _minimal_ subobject through which a morphism factors is, of course, a limit, as all greatest lower bounds are. So if your category is wellpowered and complete then every morphism has a minimal subobject through which it factors, which you might call the image... but without further hypotheses this may not be well behaved.

Nathan Corbyn (Jan 05 2022 at 13:52):

Thank you very much! This makes a lot of sense!