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Stream: theory: category theory

Topic: enriched directed colimit

Matteo Capucci (he/him) (Dec 19 2023 at 18:50):

I'm interested in understanding what a 'small' object is in an enriched setting. To do so, I have to first understand what '($\kappa$-)directed colimit' means in this setting.
One can extend the definition of filtered colimit as those weighted enriched colimits which commute with finite limits (as suggested in this MO question).
I'm stuck here: can I just replace directed by filtered in the general definition? I understand that having directed and filtered colimits is the same for plain categories, that doesn't seem enough to conclude I can just use preservation of filtered colimits as a definition of enriched small object.

Kevin Arlin (Dec 19 2023 at 19:12):

I'm confused about the implication that you see "small" as related to directed but not to filtered colimits.

Nathanael Arkor (Dec 19 2023 at 19:12):

What "small" means depends on what your objective is in considering a notion of size. For the theory of enriched local presentability, the most appropriate definition is $\kappa$-presentability: an object is $\kappa$-presentable if homming out of it preserves $\kappa$-flat colimits (see Kelly's Structures defined by finite limits in the enriched context, I, for instance). "Directedness" is a concept particular to Set-enrichment; there's no notion of directedness for arbitrary enriched categories.

Matteo Capucci (he/him) (Dec 19 2023 at 19:14):

Kevin Arlin said:

I'm confused about the implication that you see "small" as related to directed but not to filtered colimits.

I'm following the definition at [[small object]]. But I'm a quite out of my depth so I don't know if I'm just cargo culting it.

Kevin Arlin (Dec 19 2023 at 19:15):

Oh, yeah, I think using directed colimits there is a borrowing from Adamek and Rosicky but one can also say filtered here, because every kappa-filtered category has a cofinal kappa-directed poset.

Matteo Capucci (he/him) (Dec 19 2023 at 19:17):

Neat!

Matteo Capucci (he/him) (Dec 19 2023 at 19:17):

Nathanael Arkor said:

What "small" means depends on what your objective is in considering a notion of size. For the theory of enriched local presentability, the most appropriate definition is $\kappa$-presentability: an object is $\kappa$-presentable if homming out of it preserves $\kappa$-flat colimits (see Kelly's Structures defined by finite limits in the enriched context, I, for instance). "Directedness" is a concept particular to Set-enrichment; there's no notion of directedness for arbitrary enriched categories.

Thanks for the pointer and the suggestion!