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Péter Ferenczy (Sep 09 2021 at 10:55):hello everyone, I'm new here. I was looking for a CT themed chat because I couldn't find the answer to a question online. What is it called when in a category, you take an object and all the objects and morphisms that are reachable from it? I'd call it the closure of that object but apparently that's not what it's called.
Zhen Lin Low (Sep 09 2021 at 11:09):You might call it the connected component, if you ignore the direction of arrows.
Péter Ferenczy (Sep 09 2021 at 11:11):interesting, although I am looking for the directed reachable subcategory (it does form a category, because all identities and compositions are preserved). maybe could be called the forward connected component?
Péter Ferenczy (Sep 09 2021 at 11:22):In the language of graph theory, it would be the maximal reachable subgraph or the directed induced subgraph
Péter Ferenczy (Sep 09 2021 at 11:24):So maybe it is the maximal reachable subcategory or the induced subcategory of an object. But as always, there must be a dual - and the name should reflect that. I'd then call the dual of this concept the maximal observable subcategory or the coinduced subcategory
Spencer Breiner (Sep 09 2021 at 12:03):It sounds to me like you might be reaching for the notion of a co-sieve. A sieve on is a set of maps into that is closed under precomposition, i.e., if belongs to a sieve and , then belongs to the sieve as well.
For a co-sieve, we switch the directions of the arrows, which sounds (close?) to what you're looking for.
Mike Shulman (Sep 09 2021 at 14:37):Or perhaps the slice category?
Spencer Breiner (Sep 09 2021 at 19:36):Good point! The OP didn't say anything about restricting what goes in, and keeping everything gives the (co)slice.