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Stream: practice: terminology & notation

Topic: categories with all objects compact

Matteo Capucci (he/him) (Jul 22 2025 at 09:27):

Is there a name for these things?
Notable examples: FinSet, fpGrp, Op(X) for X a Noetherian space.

Indeed given the last example, I'd be inclined to call such categories Noetherian (warning: this is ragebait).

Kevin Carlson (Jul 22 2025 at 15:43):

I don’t get it, which sense of compact do you mean?

fosco (Jul 22 2025 at 16:13):

...for every object, $\hom(X,-)$ commutes with $\omega$-filtered colimits?

Kevin Carlson (Jul 22 2025 at 16:28):

Indeed...But it's not very meaningful in a category where basically no filtered colimits exist, is it?

Matteo Capucci (he/him) (Jul 23 2025 at 11:36):

What do you mean?

Matteo Capucci (he/him) (Jul 23 2025 at 11:37):

FinSet certainly admits all finite filtered colimits, what am I missing?

Matteo Capucci (he/him) (Jul 23 2025 at 11:38):

Ah, you need omega filtered colimits for them to be interesting... uhm.

John Baez (Jul 23 2025 at 13:34):

You need a bit of infinity to say which objects are finite (compact)?

Kevin Carlson (Jul 23 2025 at 17:10):

Matteo Capucci (he/him) said:

FinSet certainly admits all finite filtered colimits, what am I missing?

A category has all finite filtered colimits if and only if it’s Cauchy complete, because a finite filtered category is just one with a homomorphic weak terminal object (the maps from each object form a cocone) and for such a category the inclusion of the weak terminal and its canonical idempotent is final.

Kevin Carlson (Jul 23 2025 at 17:12):

A different way of saying roughly the same thing is that every finite filtered colimit is absolute, so every object is compact with respect to every finite filtered colimit.

Mike Shulman (Jul 23 2025 at 17:32):

There's a potential confusion because there are two "sizes" that could be related to filteredness. On one hand we could talk about the actual size of the domain category, which is what Kevin is talking about.

On the other hand there is a more general notion of $\kappa$-filtered category for a cardinal $\kappa$, which means that every diagram of size $<\kappa$ admits a cocone. Usually when we say "filtered" we mean "$\omega$-filtered", which is to say that every finite diagram admits a cocone. So Kevin's comment is about "finite $\omega$-filtered colimits": when the entire category is also finite, there must exist a cocone under the entire diagram, i.e. a homomorphic weak terminal object.

Maybe this is obvious, but I thought it would be good to make explicit.