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Stream: learning: questions

Topic: Finitely presentable objects

John Baez (May 22 2020 at 01:11):

Let $\widehat{C}$ be the category of presheaves on a small category $C$.

I know all representable presheaves are finitely presented objects of $\widehat{C}$.

In any category, any finite colimit of finitely presented objects is again finitely presented.

So, any finite colimit of representables is a finitely presented object of $\widehat{C}$.

Question: if $x \in \widehat{C}$ is a finitely presented object, is it a finite colimit of representables?

Jens Hemelaer (May 22 2020 at 06:16):

@Morgan Rogers and I just used this in our paper, but only in the case where $C$ has only one object.
As reference we used:
Adámek and Rosicky, "Locally Presentable and Accessible Categories", Corollary 3.13.
The result holds for arbitrary varieties of finitary algebras.
After your helpful comments in "basic questions : finitely presentable objects", I think they include many-sorted finitary algebras in their definition.
So I would guess this includes the case you are interested in?

Jens Hemelaer (May 22 2020 at 09:05):

Some details (mostly for myself):
Corollary 3.13 shows that finitely presentable objects (in the topos-theoretic sense) are finitely presentable in the sense of universal algebra, i.e. they are a finite colimit of finitely generated free objects.
So it remains to show that the free objects are themselves finite colimits of representables. In fact, as maybe suspected, they are finite coproducts of representables.
To show this, it is enough to show that finite coproducts of representables satisfy the universal property of the free algebra, as given in Corollary 3.3 of loc.cit. (it says the free algebra construction is left adjoint to the functor that sends a many-sorted algebra to its set of elements, seen as object of $\mathbf{Sets}^S$, with $S$ the set of sorts).

John Baez (May 25 2020 at 20:36):

Aurielen Damien answered my question on MathOverflow:

• Are compact objects in presheaf categories finite colimits of representables?.

John Baez (May 25 2020 at 20:38):

But the proof goes by a bit quick for me. How do you get that split mono, exactly?

Reid Barton (May 25 2020 at 20:57):

The split mono comes from factoring the identity map from $X$ to the colimit of the filtered diagram (also $X$) through some object of the diagram, as provided by the finite presentability hypothesis.

Reid Barton (May 25 2020 at 20:58):

However, the proof ends a bit abruptly since you asked for a finite colimit of representables, not a finite colimit of finite colimits of representables.

Reid Barton (May 25 2020 at 20:59):

It's not hard to rectify this, but this uses the fact that you're specifically dealing with a category of presheaves and not an arbitrary locally finitely presentable category

John Baez (May 25 2020 at 21:13):

Thanks! Are you saying we can't in general write a finite colimit of finite colimits of X's as a finite colimit of X's (where "X" is some class of objects)?

Reid Barton (May 25 2020 at 21:14):

Yes, for example: https://mathoverflow.net/questions/204792/is-every-abelian-group-a-colimit-of-copies-of-z

Reid Barton (May 25 2020 at 21:17):

I guess that's only an answer to your question with "finite" removed, but I assume the answer would be the same...

sarahzrf (May 25 2020 at 22:20):

ooh, this is a nice fact to keep in my pocket

Reid Barton (May 25 2020 at 22:23):

The rest of the proof goes like this: we've represented our object as a coequalizer $B \rightrightarrows A$ of two maps between objects that are each finite colimits of representables. We may replace $B$ with an object that surjects onto $B$, say the coproduct of all the objects in the diagram which constructs it as a colimit of representables. So, we may assume that $B$ is a finite coproduct of representables. I'll just treat one representable at a time.

Reid Barton (May 25 2020 at 22:26):

A map from a representable to the colimit of a diagram (here, $A$) factors through one of the objects of the diagram (basically because colimits are computed objectwise). This is where we use the fact that we are working in a presheaf category. Then, we can replace each map from a representable in $B$ to $A$ with an additional map in the diagram constructing $A$, and this expresses our original object as the colimit of a finite diagram of representables.

John Baez (May 25 2020 at 22:28):

Yay! It'll take me a while to fully absorb this - I'm thinking about something very different today - but I'll definitely put in the time needed to do it. I need this fact for a paper I'm writing! I'd better remember to acknowledge you.

John Baez (Jun 15 2020 at 21:48):

Todd Trimble has given a different proof that finitely presentable (=compact) objects in a presheaf category are precisely the finite colimits of representables, here:

• Todd Trimble, Are compact objects in presheaf categories finite colimits of representables?

He uses Gabriel-Ulmer duality in a nice way.

Chaitanya Leena Subramaniam (Jun 22 2020 at 11:47):

John Baez said:

Todd Trimble has given a different proof that finitely presentable (=compact) objects in a presheaf category are precisely the finite colimits of representables, here:

• Todd Trimble, Are compact objects in presheaf categories finite colimits of representables?

He uses Gabriel-Ulmer duality in a nice way.

The corresponding (more general) statement for any small strongly generating subcategory of f.p. objects of a l.f.p. 1-category is theorem 7.2 (i) of Kelly's Structures defined by finite limits in the enriched context, I (Cahiers 1982).

John Baez (Jun 22 2020 at 17:47):

Great, thanks! This generalization may be what I'm really looking for. I was thinking about the case of presheaf categories just because I couldn't make any progress on more general l.f.p categories!

Reid Barton (Jun 22 2020 at 18:09):

But the closure of G under finite colimits isn't the same as the class of objects which can be written as a finite colimit of objects of G, at least not a priori.

John Baez (Jun 22 2020 at 19:13):

Whoops, you're right. Kelly talks about the "closure under finite colimits", but I guess that allows iterated finite colimits.

Reid Barton (Jun 22 2020 at 19:17):

I think Todd Trimble's argument is really the same proof, but packaged more neatly into known results. Notably, the argument in section 5.9 of Kelly's Basic Concepts of Enriched Category Theory uses the same computation about coequalizers of colimits of representables to show that in the case of a presheaf category, the generation under finite colimits finishes after a single stage.