Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: generating set

Leopold Schlicht (Aug 14 2021 at 14:56):

In a presheaf category of the form $\mathbf{Set}^{C^{\mathrm{op}}}$, the representable presheafs are a generating set (in the sense that two parallel arrows $f,g\colon A\to B$ are equal if and only if for any representable presheaf $F$ and any arrow $a\colon F\to A$, $fa=ga$. They also have the property that every presheaf can be written as a colimit of representable presheafs.
Is there any connection between these two properties? (I don't think they are equivalent, but maybe there's any other connection.)
In my opinion, the second property is the one that deserves the name "generating set" a bit more, since it says that the representable presheafs can generate all other presheafs using colimits.

Fawzi Hreiki (Aug 14 2021 at 15:58):

Yes. Any dense subcategory (the second property) is separating (the first property)

Fawzi Hreiki (Aug 14 2021 at 16:01):

The representables in a presheaf category satisfy the even stronger property of freely generating the category under colimits

Fawzi Hreiki (Aug 14 2021 at 16:04):

I agree with you that the term ‘generate’ isn’t so great for the first property.

Leopold Schlicht (Aug 15 2021 at 11:20):

Thank you very much! :-)

Jens Hemelaer (Aug 16 2021 at 10:54):

Fawzi Hreiki said:

Yes. Any dense subcategory (the second property) is separating (the first property)

Yes, and in a presheaf topos (or more generally any Grothendieck topos) the converse holds as well; in this case any separating subcategory is dense. This takes some work to prove, it is written out here.

Leopold Schlicht (Aug 16 2021 at 11:34):

Fantastic! Thanks.