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Stream: learning: questions

Topic: subcategories

Kalan Kucera (Jul 12 2021 at 21:20):

If you have a category, is the composition map included in that category an endomap of that category? Or put another way, if it is assumed that you have some sort of hierarchical structure to a certain category, are the domain of the parts and the codomain of the whole that those parts form in a structuring function the same?

John Baez (Jul 12 2021 at 21:35):

I don't understand your second version of this question, but the first version I understand, so I'll answer that.

In a category $\mathsf{C}$, the composition map

$\circ : \mathrm{hom}(y,z) \times \mathrm{hom}(x,y) \to \mathrm{hom}(x,y)$

is a function between sets. So, it is not a morphism in the category $\mathsf{C}$ unless $\mathsf{C} = \mathsf{Set}$.

There is, however, something called a closed category, which has a different kind of composition map that is actually a morphism in the category.

There is more to say about this, but my first comment - $\circ$ is a function between sets, not a morphism in $\mathsf{C}$ - is the most important thing to learn at first.

Kalan Kucera (Jul 12 2021 at 21:59):

thank you! I will read the section on Closed categories and wrangle that. The second version of the question was trying to imagine an example that perhaps I misinterpreted, like... if a collection of carbon and hydrogen atoms, their maps, and the composition of them all form a molecule (which here is the category) would there be a map of the collection of the "lower" structures to the "higher" structure that would be considered an endomap, since in language we would say that the atoms and their structure are equivalent to that molecule.

John Baez (Jul 12 2021 at 22:10):

To make that example precise, you'd have to define a category in which mapping a bunch of atoms into a molecule (or something like that) is a morphism. It's actually easier to answer your question in abstract, where the precise definitions already exist.