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Stream: learning: questions

Topic: Is Vect locally presentable?

Mike Stay (Jan 21 2021 at 18:20):

Vect is bicomplete (https://en.wikipedia.org/wiki/Complete_category) so it has all colimits. It is locally small: there is a set of linear transformations between any two vector spaces. Every object in Vect is the colimit of copies of the one-dimensional vector space isomorphic to the field k: any basis for the vector space picks out a set of copies of k and the coproduct of these is isomorphic to the vector space. So it looks locally presentable to me.

Is that right? If so, is FinVect locally finitely presentable?

Fawzi Hreiki (Jan 21 2021 at 18:25):

Yes. Any category of models of a Lawvere theory is locally finitely presentable.

Reid Barton (Jan 21 2021 at 18:26):

FinVect isn't locally finitely presentable though

Mike Stay (Jan 21 2021 at 18:31):

@Fawzi Hreiki

Yes. Any category of models of a Lawvere theory is locally finitely presentable.

Are vector spaces models of a Lawvere theory? I can see how modules could be, but how do you restrict to modules over a field?

Nathanael Arkor (Jan 21 2021 at 18:37):

According to Adámek–Milius–Moss's On Finitary Functors and Their Presentations, $\mathbf{Vect}_K$ is locally finitely presentable for any field $K$.

Fawzi Hreiki (Jan 21 2021 at 18:39):

Ah actually my mistake. Yes finite dimensional vector spaces aren’t a Lawvere theory.

Nathanael Arkor (Jan 21 2021 at 18:39):

(Accordingly, $\mathbf{FinVect}_K$ is not, because it's not cocomplete.)

Mike Stay (Jan 21 2021 at 18:47):

Thanks!

Fawzi Hreiki (Jan 21 2021 at 20:06):

@Mike Stay Finite dimensional vector spaces aren’t an algebraic theory but general vector spaces are.

Fawzi Hreiki (Jan 21 2021 at 20:06):

In fact, the Lawvere theory for vector spaces is (a skeleton of) the category of finite dimensional vector spaces.

Joe Moeller (Jan 21 2021 at 20:07):

iirc you have a different theory for each field, right?

Fawzi Hreiki (Jan 21 2021 at 20:07):

Yes

Joe Moeller (Jan 21 2021 at 20:07):

You need a scaling operation for each element of the base field?

Fawzi Hreiki (Jan 21 2021 at 20:07):

You need a constant for each element of the field basically

Fawzi Hreiki (Jan 21 2021 at 20:08):

Actually, a scaling operation as you said.

John Baez (Jan 21 2021 at 21:06):

I guess it's been settled by now, but:

FinVect is not locally finitely presentable, because it doesn't have all small colimits.

Vect is locally finitely presentable:

1) a category is locally finitely presentable iff it's equivalent to the category of models of a finite limits theory.

2) every category of models of a Lawvere theory is also the category of a models of a finite limits theory.

3) Vect is the category of models of a Lawvere theory.

John Baez (Jan 21 2021 at 21:07):

And here of course Vect means the category of vector spaces over some arbitrary but fixed field that we've chosen.

Mike Stay (Jan 21 2021 at 21:50):

Vect is the category of models of a Lawvere theory.

So if I'm understanding correctly, the Lawvere theory has

• a sort (say V)

• for each element z of the base field, a function symbol $(z\cdot -)\colon V \to V$ that scales the given vector by z

• and for each pair z, z' of elements of the base field, an equation $(z\cdot (z'\cdot -)) = (zz'\cdot -).$

Is that right? For an infinite field, how is that finitely presentable?

Fawzi Hreiki (Jan 21 2021 at 22:19):

Well you also need to add the operations making $V$ into an abelian group.

Joe Moeller (Jan 21 2021 at 22:21):

I don't think local finite presentability is referring to the presentation of the algebraic theory.

Fawzi Hreiki (Jan 21 2021 at 22:22):

Yeah it doesn't. It's about presentability of the objects of the category itself.

Fawzi Hreiki (Jan 21 2021 at 22:23):

Hence the 'locally'.

Mike Stay (Jan 21 2021 at 22:54):

OK, thanks.

Fawzi Hreiki said:

Well you also need to add the operations making $V$ into an abelian group.

Yes, sorry.

John Baez (Jan 22 2021 at 04:57):

Mike Stay said:

Vect is the category of models of a Lawvere theory.

So if I'm understanding correctly, the Lawvere theory has

• a sort (say V)

• for each element z of the base field, a function symbol $(z\cdot -)\colon V \to V$ that scales the given vector by z

• and for each pair z, z' of elements of the base field, an equation $(z\cdot (z'\cdot -)) = (zz'\cdot -).$

Right, and of course you also get operations coming from addition of vectors. So the n-ary operations in this Lawevere theory are all the operations of taking n-ary linear combinations: they look like this in a model:

$\displaystyle{ (v_1, \dots, v_n) \mapsto \sum_{i=1}^n z_i v_i }$

for some elements $z_1, \dots, z_n$ in the field.

Is that right? For an infinite field, how is that finitely presentable?

I said Vect was locally finitely presentable. Click the link to read a bunch of equivalent definitions of this concept. My favorite is that a category is locally finitely presentable iff it is equivalent to the category of models of a finite limits theory. Vect is, because any category of models of a finite products theory, e.g. a Lawvere theory, is also equivalent to a category of models of a finite limits theory.

John Baez (Jan 22 2021 at 05:05):

There are also other definitions, which might make the term "locally finitely presentable category" more clear.

There's a concept of a finitely presentable object in a category, which is an object $x$ such that the representable functor $\mathrm{hom}(-,x)$ preserves filtered colimits. This takes a while to get used to, but a finitely presentable object in Set is a finite set, a finitely presentable object in Vect is a finite-dimensional vector space, and a finitely presentable group is a group with a finite presentation in the usual sense. So it's a good general concept of what it means for an object to be finitely presentable. I could explain it intuitively if I had a bit more energy - "preserving filtered colimits" really does make sense if you think about it a while!

John Baez (Jan 22 2021 at 05:06):

Then a category is locally finitely presentable iff it's essentially small, it has colimits, and each object is a filtered colimit of finitely presentable objects.

John Baez (Jan 22 2021 at 05:09):

So, not every object needs to be finitely presentable, but it needs to be "built out of finitely presentable objects".

For example, every vector space is a union of finite-dimensional subspaces!

John Baez (Jan 22 2021 at 05:13):

(You should think of a filtered colimit as being like a "union".)

John Baez (Jan 22 2021 at 05:14):

I find all this harder to remember than the other equivalent definition: a category is locally finitely presentable if it's equivalent to the category of models of a finite limits theory (in Set, of course).

Fawzi Hreiki (Jan 22 2021 at 10:06):

Well the other definition is really a quite nontrivial theorem (namely Gabriel-Ulmer duality)

Fawzi Hreiki (Jan 22 2021 at 10:08):

It’s in the same vein as the characterisations of algebraic categories (which are categories of models for finite product theories) and of stone spaces (which are spaces of models for classical propositional theories).

Todd Trimble (Jan 22 2021 at 14:37):

John Baez said:

There are also other definitions, which might make the term "locally finitely presentable category" more clear.

There's a concept of a finitely presentable object in a category, which is an object $x$ such that the representable functor $\mathrm{hom}(-,x)$ preserves filtered colimits. This takes a while to get used to, but a finitely presentable object in Set is a finite set, a finitely presentable object in Vect is a finite-dimensional vector space, and a finitely presentable group is a group with a finite presentation in the usual sense. So it's a good general concept of what it means for an object to be finitely presentable. I could explain it intuitively if I had a bit more energy - "preserving filtered colimits" really does make sense if you think about it a while!

You mean $\hom(x, -)$.

John Baez (Jan 22 2021 at 15:58):

Yes.

Mike Stay (Jan 27 2021 at 15:23):

Thanks!