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Stream: learning: questions

Topic: is every X in the category of Ys a Y in the category of Xs?

Sam Speight (Aug 03 2021 at 08:20):

Slight aside (maybe this needs a different topic), but your answer reminded me of a question:

Is an X object in the category Y always the same as a Y object in the category X (as far as both notions make sense)?

Morgan Rogers (he/him) (Aug 03 2021 at 08:45):

That's not a clear-cut question! For example, a monoid in the category of abelian groups equipped with the tensor product can be identified with a ring, but a monoid in the category of abelian groups equipped with a different monoidal product will be something else, and there's similar ambiguity about which monoidal product to take on the category of monoids (although I can't say I know many monoidal products on that category, I'm sure someone else does).

Kenji Maillard (Aug 03 2021 at 08:54):

One way of precising the question so that there is a positive answer is to ask for the notions X and Y to be presented by a theory (for instance a Lawvere theory) or a sketch. I think Proposition 4.6 of that second link together with the symmetry of the monoidal product gives a partial answer to your question.

Jens Hemelaer (Aug 03 2021 at 09:09):

Here is a counterexample: topological groups are group objects in the category of topological spaces. But a topological space in the category of groups would be a group together with a specified family of subgroups (the "open subgroups").

However, specifying the collection of open subgroups is not enough to unambiguously turn a group into a topological group. For example, the real line under addition, with the standard Euclidean topology, does not have any nontrivial open subgroups. So if you just look at the open subgroups you wouldn't be able to distinguish it from the real line under addition with the indiscrete topology (which is also a topological group).

Fawzi Hreiki (Aug 03 2021 at 09:18):

Except that your definition is not really what it means to be a ‘topological space object’ since the topology should be an internal object which cannot be done in the category of groups.

Fawzi Hreiki (Aug 03 2021 at 09:18):

For example, a topological space internal to a topos is an object with a subobject of the power object satisfying the usual axioms.

Fawzi Hreiki (Aug 03 2021 at 09:19):

But a counter example which (I think) works is a monoid object internal to categories versus a category object internal to monoids.

Kenji Maillard (Aug 03 2021 at 09:22):

Fawzi Hreiki said:

But a counter example which (I think) works is a monoid object internal to categories versus a category object internal to monoids.

Shouldn't both be models for isomorphic sketchable theories ? :thinking:

Fawzi Hreiki (Aug 03 2021 at 09:23):

Ah you’re right actually. I know that algebraic theories commute but I haven’t read in detail about finite limit theories.

Morgan Rogers (he/him) (Aug 03 2021 at 09:33):

Fawzi Hreiki said:

Except that your definition is not really what it means to be a ‘topological space object’ since the topology should be an internal object which cannot be done in the category of groups.

I think "object equipped with collection of subobjects" is a reasonable abstraction of topological space which is applicable more broadly than the power object version, and is equivalent in the category of sets.

Sam Speight (Aug 03 2021 at 09:35):

@Kenji Maillard yes, that seems like a good way to make the question precise (and give an answer!).

Sam Speight (Aug 03 2021 at 09:48):

So the theory of categories is sketchable then?

Fawzi Hreiki (Aug 03 2021 at 09:48):

Yes, as is any finite limit theory.

Jens Hemelaer (Aug 03 2021 at 10:45):

Fawzi Hreiki said:

For example, a topological space internal to a topos is an object with a subobject of the power object satisfying the usual axioms.

I agree that this is the standard definition of topological space object and that it is different from the one that I mentioned (if both exist).

Fawzi Hreiki (Aug 03 2021 at 11:28):

Perhaps a way to salvage that example is to consider a diagrammatic rather than algebraic theory which would then give a topos. So, is a topological digraph (i.e. a parallel pair of arrows in $\text{Top}$) the same thing as a topological space object in the presheaf topos of directed graphs?

Fawzi Hreiki (Aug 03 2021 at 11:36):

But the question is more interesting when restricted to the first order case (since no one really ever considers 'topological space objects' in categories).

Morgan Rogers (he/him) (Aug 03 2021 at 12:56):

Fawzi Hreiki said:

Perhaps a way to salvage that example is to consider a diagrammatic rather than algebraic theory which would then give a topos. So, is a topological digraph (i.e. a parallel pair of arrows in $\text{Top}$) the same thing as a topological space object in the presheaf topos of directed graphs?

Carefully examining the descriptions of exponentials and the subobject classifier in https://arxiv.org/ftp/math/papers/0306/0306394.pdf,
it seems that for a digraph $G$, the power object $P(G)$ has $P(N(G))$ (that is, subsets of the collection of nodes of $G$) as its nodes, and a morphism from one subset to another is a subgraph of $((0 \to 1) \times G)$ whose collection of nodes is the disjoint union of those subsets.

For example, if I take $G$ to be a digraph with no edges, $P(G)$ is the set of subsets of the nodes of $G$ equipped with one edge between each pair of subsets, corresponding to the unique subgraph on those nodes with no edges. This has a lot of subobjects, but which are valid topologies?

For a subset of $P(G)$ to be a valid topology, we need it to contain the minimal and maximal elements $1 \rightrightarrows P(G)$ (which correspond to the empty and maximal subsets of the nodes of $G$, respectively). We also need it to be closed under arbitrary unions and finite intersections, which is trickier: we need to take into account the order relation (another graph!) on the subobject classifier and how that lifts to an ordering on $P(G)$. At this stage it seems plausible that the correspondence could hold up, but I'm not certain yet..!

Peter Arndt (Aug 03 2021 at 13:20):

Kenji Maillard said:

One way of precising the question so that there is a positive answer is to ask for the notions X and Y to be presented by a theory (for instance a Lawvere theory) or a sketch. I think Proposition 4.6 of that second link together with the symmetry of the monoidal product gives a partial answer to your question.

Right, the question is then when the factors in the tensor product of sketches can be swapped. This can be done, roughly, when the colimits and limits specified in the sketches commute. For example, since limits always commute, you can swap the sketches if both are limit sketches. And you can swap finite-product+sifted-colimit sketches etc. For more precise statements see here.

Fawzi Hreiki (Aug 03 2021 at 13:22):

Peter Arndt said:

Kenji Maillard said:

One way of precising the question so that there is a positive answer is to ask for the notions X and Y to be presented by a theory (for instance a Lawvere theory) or a sketch. I think Proposition 4.6 of that second link together with the symmetry of the monoidal product gives a partial answer to your question.

Right, the question is then when the factors in the tensor product of sketches can be swapped. This can be done, roughly, when the colimits and limits specified in the sketches commute. For example, since limits always commute, you can swap the sketches if both are limit sketches. And you swap finite-product+sifted-colimit sketches etc. For more precise statements see here.

The nLab page for sketches seems to imply that the tensor product is symmetric meaning we can always swap them, but I can't find the reference for this fact.

Peter Arndt (Aug 03 2021 at 13:40):

I don't think it's true that you can always swap. For example take the (terminal obj.+ coproduct)-sketch $S$ whose models are maps $X \to X \coprod 1$ and the limit sketch $M$ whose models are monoids. Look at models of these in $Set$:
Since in the category of monoids the terminal object is also initial, the coproduct of a monoid with the terminal object is isomorphic to that monoid again. Therefore $S$-models in $M$-Mod are monoids with an endomorphism.
On the other hand $M$-models in $S$-Mod are pairs of monoids one of which has one element more, plus a homomorphism between them. Those don't look like equivalent categories.

Zhen Lin Low (Aug 03 2021 at 13:46):

If you stick to limits only then there's no problem. It boils down to "limits commute with limits".

Spencer Breiner (Aug 03 2021 at 14:48):

Does anyone know what happens if the commutation of (co)limits is weakened to a comparison map?

Fawzi Hreiki (Aug 03 2021 at 18:46):

What do you mean? There is always a map from the colimit of the limit to the limit of the colimit.

Spencer Breiner (Aug 03 2021 at 19:36):

I was hoping that weakening the commutation to a comparison map might analogously weaken the equivalence of X's in Y's and Y's in X's to an inclusion or comparison functor.

Peter Arndt (Aug 03 2021 at 20:02):

(Whoever reads this later and wonders what the above confusion was about: The nLab page on sketches used to claim that the tensor product of sketches was part of a symmetric monoidal structure. I just corrected that nLab page and added my remarks from above. )