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Stream: learning: questions

Topic: Full but not isomorphism-closed subcategory

Bernd Losert (Aug 11 2024 at 17:41):

Is there an non-artificial example of a full subcategory that is not isomorphism-closed?

Jean-Baptiste Vienney (Aug 11 2024 at 17:47):

I would say the full subcategory of $\mathbf{Vec}_k$ given by all the objects of the form $k^n$

Bernd Losert (Aug 11 2024 at 17:49):

Ah, that's a good one actually. So any subcategory where the objects have to have a specific "shape" would qualify as not isomorphism-closed.

Jean-Baptiste Vienney (Aug 11 2024 at 17:52):

Yes, if you take a full subcategory which is a skeleton it will never be isomorphism closed except if the original category is already skeletal.

Jean-Baptiste Vienney (Aug 11 2024 at 17:55):

By the way the term isomorphism-closed subcategory seems to have another common definition than this but I understood it here as "a subcategory $\mathcal{D}$ of a category $\mathcal{C}$ which contains all the objects in $\mathcal{C}$ isomorphic to any object $A \in \mathcal{D}$."

Bernd Losert (Aug 11 2024 at 18:00):

Yes, that's the definition I'm basically working with. There's also one that says that if f : A → B is an isomorphism where A is in the subcategory and B in the parent category, then f belongs to the subcategory. Not sure if this version is actually stronger.

Ralph Sarkis (Aug 11 2024 at 18:04):

That version is stronger when you don't require fullness. Any subgroup of a group (seen as categories) is isomorphism-closed in the first sense but not the second.

Bernd Losert (Aug 11 2024 at 18:08):

Ah, that's a nice example.

John Baez (Aug 11 2024 at 21:05):

I think one useful term is [[replete subcategory]]: a subcategory is replete if when an object x is in the subcategory and $f:x \stackrel{\sim}{\to} y$ then f and y are in the subcategory too.