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Atrium > Mathematics, Physics & Philosophy: What is a monoidal category with one (or two) object(s)?

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- CommentRowNumber1.
- CommentAuthorKevin Lin
- CommentTimeJul 24th 2010
- (edited Jul 24th 2010)
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Author: Kevin Lin
Format: MarkdownItexThis question might be really easy. So a ($k$-)linear category with a single object is the same as an algebra -- namely the endomorphism algebra of the single object. I want to figure out what a (($k$-)linear) [[monoidal category]] with a single object is. However, the definition of monoidal category requires a unit object $1$, and moreover we require $\operatorname{End}(1) = k$ (at least according to Definition 1.1.7(v) on page 12 of Bakalov-Kirillov), which I think makes the situation rather trivial and uninteresting. So, maybe it's better to ask, what is a monoidal category with two objects? Or, what is a monoidal category with a single object when we *don't* require $\operatorname{End}(1) = k$? We get the composition $\operatorname{End}(1) \otimes \operatorname{End}(1) \stackrel{\circ}{\to} \operatorname{End}(1)$ and we get the tensor product $\operatorname{End}(1) \otimes \operatorname{End}(1) \stackrel{\otimes}{\to} \operatorname{End}(1)$. Both are associative. Are they the same, by some kind of Eckmann-Hilton argument? Maybe I am being stupid but I don't see it. In particular, for the Eckmann-Hilton argument to work I need the identity endomorphism of the unit object $1$ to be the unit for the tensor product $\otimes$, but I don't see how or whether this is true.

This question might be really easy.

So a ( $k$ -)linear category with a single object is the same as an algebra – namely the endomorphism algebra of the single object.

I want to figure out what a (( $k$ -)linear) monoidal category with a single object is. However, the definition of monoidal category requires a unit object $1$ , and moreover we require $\operatorname{End}(1) = k$ (at least according to Definition 1.1.7(v) on page 12 of Bakalov-Kirillov), which I think makes the situation rather trivial and uninteresting.

So, maybe it’s better to ask, what is a monoidal category with two objects?

Or, what is a monoidal category with a single object when we don’t require $\operatorname{End}(1) = k$ ? We get the composition $\operatorname{End}(1) \otimes \operatorname{End}(1) \stackrel{\circ}{\to} \operatorname{End}(1)$ and we get the tensor product $\operatorname{End}(1) \otimes \operatorname{End}(1) \stackrel{\otimes}{\to} \operatorname{End}(1)$ . Both are associative. Are they the same, by some kind of Eckmann-Hilton argument? Maybe I am being stupid but I don’t see it. In particular, for the Eckmann-Hilton argument to work I need the identity endomorphism of the unit object $1$ to be the unit for the tensor product $\otimes$ , but I don’t see how or whether this is true.
- CommentRowNumber2.
- CommentAuthorHarry Gindi
- CommentTimeJul 24th 2010
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Author: Harry Gindi
Format: MarkdownItex@Kevin: What do you need a monoidal category with a single object for? It seems like a rather unnatural construction.

@Kevin: What do you need a monoidal category with a single object for? It seems like a rather unnatural construction.
- CommentRowNumber3.
- CommentAuthorKevin Lin
- CommentTimeJul 24th 2010
- (edited Jul 24th 2010)
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Author: Kevin Lin
Format: MarkdownItexI've sometimes found it helpful, when trying to study something about categories, to first understand the situation for algebras. Right now I want to understand a few things about monoidal categories, so I'd like to start by looking at monoidal-categories-with-a-single-object.

I’ve sometimes found it helpful, when trying to study something about categories, to first understand the situation for algebras. Right now I want to understand a few things about monoidal categories, so I’d like to start by looking at monoidal-categories-with-a-single-object.
- CommentRowNumber4.
- CommentAuthorHarry Gindi
- CommentTimeJul 24th 2010
- (edited Jul 24th 2010)
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Author: Harry Gindi
Format: MarkdownItexMonoidal categories with only a single object or even with two objects is still going to be degenerate. It's like asking about groups with one or two elements. A better idea is to think in terms of monoidal categories generated by some class of objects.

Monoidal categories with only a single object or even with two objects is still going to be degenerate. It’s like asking about groups with one or two elements.

A better idea is to think in terms of monoidal categories generated by some class of objects.
- CommentRowNumber5.
- CommentAuthorMike Shulman
- CommentTimeJul 24th 2010
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Author: Mike Shulman
Format: MarkdownItexI think that's kind of an odd point of view around here, Harry, in view of the [[delooping hypothesis]] and [[stabilization hypothesis]] and [[horizontal categorification]]. One-object things can be very interesting.

I think that’s kind of an odd point of view around here, Harry, in view of the delooping hypothesis and stabilization hypothesis and horizontal categorification. One-object things can be very interesting.
- CommentRowNumber6.
- CommentAuthorKevin Lin
- CommentTimeJul 24th 2010
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Author: Kevin Lin
Format: MarkdownItex@Harry: What do you mean by "degenerate"? Algebras (= categories with one object) can certainly be pretty nontrivial things.

@Harry: What do you mean by “degenerate”?

Algebras (= categories with one object) can certainly be pretty nontrivial things.
- CommentRowNumber7.
- CommentAuthorDavidRoberts
- CommentTimeJul 24th 2010
- (edited Jul 24th 2010)
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Author: DavidRoberts
Format: MarkdownItexA monoidal category with only one object is the same as a bicategory with only one object and only one 1-arrow. By Eckmann-Hilton, we get that this is the same as a commutative monoid (and yes, the two compositions agree, and the unit of the monoid is the identity arrow of the given object, which is necessary the tensor unit). The coherence conditions on $\otimes$ take care of the interaction between ). Actually, by Cheng-Gurski's <a href="http://arxiv.org/abs/0708.1178">result</a> on the low end of the periodic tabke, this monoid comes with a chosen invertible element (actually they attribute it to Leinster (math.CT/9901139)), so it is a bit more than just a monoid. As far as two objects ($I$ and $x$, say) go, it depends on what $Hom(I,x)$ looks like, and if they are isomorphic or not. In this case there may be some interesting interplay for the $k$-linear version, but I haven't thought about it before.

A monoidal category with only one object is the same as a bicategory with only one object and only one 1-arrow. By Eckmann-Hilton, we get that this is the same as a commutative monoid (and yes, the two compositions agree, and the unit of the monoid is the identity arrow of the given object, which is necessary the tensor unit). The coherence conditions on $\otimes$ take care of the interaction between ). Actually, by Cheng-Gurski’s result on the low end of the periodic tabke, this monoid comes with a chosen invertible element (actually they attribute it to Leinster (math.CT/9901139)), so it is a bit more than just a monoid.

As far as two objects ( $I$ and $x$ , say) go, it depends on what $Hom(I,x)$ looks like, and if they are isomorphic or not. In this case there may be some interesting interplay for the $k$ -linear version, but I haven’t thought about it before.
- CommentRowNumber8.
- CommentAuthorHarry Gindi
- CommentTimeJul 24th 2010
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Author: Harry Gindi
Format: MarkdownItexMy mistake. I feel like I'm off my game tonight (look at the question I just asked!).

My mistake. I feel like I’m off my game tonight (look at the question I just asked!).
- CommentRowNumber9.
- CommentAuthorKevin Lin
- CommentTimeJul 24th 2010
- (edited Jul 24th 2010)
- PermaLink
Author: Kevin Lin
Format: MarkdownItex@DavidRoberts: Thank you. And yeah, I think I figured out why the unit of the tensor product is the identity morphism of the unique object. It's because of the unitors. Follow-up question: What is a *braided* monoidal category with one object?

@DavidRoberts: Thank you. And yeah, I think I figured out why the unit of the tensor product is the identity morphism of the unique object. It’s because of the unitors.

Follow-up question: What is a braided monoidal category with one object?
- CommentRowNumber10.
- CommentAuthorDavidRoberts
- CommentTimeJul 24th 2010
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Author: DavidRoberts
Format: MarkdownItexIt should again give a commutative monoid, again with some distinguished invertible elements. Intuitively this comes from a tricategory with only one k-cell for k=0,1,2, and in Cheng-Gurski's arXiv:0706.2307 they show this is indeed a commutative monoid with eight (count them, eight!) distinguished invertible elements. But, and this is important, a braided monoidal category with one object is perhaps not quite a tricategory with only one k-cell for k=0,1,2, because there is likely some coherence information required by the tricategory definition not present in the braided monoidal category.

It should again give a commutative monoid, again with some distinguished invertible elements. Intuitively this comes from a tricategory with only one k-cell for k=0,1,2, and in Cheng-Gurski’s arXiv:0706.2307 they show this is indeed a commutative monoid with eight (count them, eight!) distinguished invertible elements. But, and this is important, a braided monoidal category with one object is perhaps not quite a tricategory with only one k-cell for k=0,1,2, because there is likely some coherence information required by the tricategory definition not present in the braided monoidal category.
- CommentRowNumber11.
- CommentAuthorDavidRoberts
- CommentTimeJul 24th 2010
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Author: DavidRoberts
Format: MarkdownItexI might add that this is clear, because a braided monoidal category is a monoidal category after all, just with more structure morphisms. This give a couple more distinguished elements in the one-object case.

I might add that this is clear, because a braided monoidal category is a monoidal category after all, just with more structure morphisms. This give a couple more distinguished elements in the one-object case.
- CommentRowNumber12.
- CommentAuthorMike Shulman
- CommentTimeJul 24th 2010
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Author: Mike Shulman
Format: MarkdownItexIn my opinion, the distinguished elements should not really be taken seriously here -- they're just an artifact of the particular way we've chosen to present a (braided) monoidal category, and they go away when you consider things up to the right notion of equivalence. E.g. the 2-category of *pointed* one-object monoidal categories is equivalent to the category of commutative monoids, as are the 2-category of pointed one-object braided monoidal categories and the tricategory of pointed one-object one-morphism bicategories.

In my opinion, the distinguished elements should not really be taken seriously here – they’re just an artifact of the particular way we’ve chosen to present a (braided) monoidal category, and they go away when you consider things up to the right notion of equivalence. E.g. the 2-category of pointed one-object monoidal categories is equivalent to the category of commutative monoids, as are the 2-category of pointed one-object braided monoidal categories and the tricategory of pointed one-object one-morphism bicategories.
- CommentRowNumber13.
- CommentAuthorMike Shulman
- CommentTimeJul 24th 2010
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Author: Mike Shulman
Format: MarkdownItexAlso, the tetracategory of pointed one-object one-morphism tricategories should be equivalent to that of pointed one-object one-morphism Gray-categories, by coherence for tricategories, and unless I'm mistaken a one-object one-morphism Gray-category is precisely a category with two strict monoidal structures satisfying an interchange law up to coherent isomorphism -- and these are known to be equivalent to braided monoidal categories. So I think it really does work at all known levels, when you consider things up to the right kind of equivalence.

Also, the tetracategory of pointed one-object one-morphism tricategories should be equivalent to that of pointed one-object one-morphism Gray-categories, by coherence for tricategories, and unless I’m mistaken a one-object one-morphism Gray-category is precisely a category with two strict monoidal structures satisfying an interchange law up to coherent isomorphism – and these are known to be equivalent to braided monoidal categories. So I think it really does work at all known levels, when you consider things up to the right kind of equivalence.
- CommentRowNumber14.
- CommentAuthorKevin Lin
- CommentTimeJul 24th 2010
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Author: Kevin Lin
Format: MarkdownItexWhat does "pointed" mean here?

What does “pointed” mean here?
- CommentRowNumber15.
- CommentAuthorTobyBartels
- CommentTimeJul 25th 2010
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Author: TobyBartels
Format: MarkdownItexThe adjective ‘pointed’ means that it is equipped with a point (in this case, equipped with an object); see [[pointed object]]. But it's also important that the morphisms (and higher morphisms) preserve the chosen point in an appropriate way; that's what makes the extra distinguished elements end up irrelevant. There is discussion at [[k-tuply monoidal n-category]].

The adjective ‘pointed’ means that it is equipped with a point (in this case, equipped with an object); see pointed object.

But it’s also important that the morphisms (and higher morphisms) preserve the chosen point in an appropriate way; that’s what makes the extra distinguished elements end up irrelevant.

There is discussion at k-tuply monoidal n-category.
- CommentRowNumber16.
- CommentAuthorTodd_Trimble
- CommentTimeAug 16th 2010
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Author: Todd_Trimble
Format: MarkdownItex(From Mike #5) > One-object things can be very interesting. Indeed. Certainly they can encode a lot of complexity: representations of one-object things are equivalent to representations of their Cauchy completions, and such Cauchy completions can be fairlyy complicated. For example, the Cauchy completion of the monoid of endomorphisms on $X$ is equivalent to the category of nonempty sets whose cardinality is that of $X$ or less. In fact, I wonder which categories arise as Cauchy completions of monoids?

(From Mike #5)

One-object things can be very interesting.

Indeed. Certainly they can encode a lot of complexity: representations of one-object things are equivalent to representations of their Cauchy completions, and such Cauchy completions can be fairlyy complicated. For example, the Cauchy completion of the monoid of endomorphisms on $X$ is equivalent to the category of nonempty sets whose cardinality is that of $X$ or less. In fact, I wonder which categories arise as Cauchy completions of monoids?
- CommentRowNumber17.
- CommentAuthorMike Shulman
- CommentTimeAug 17th 2010
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Author: Mike Shulman
Format: MarkdownItex> I wonder which categories arise as Cauchy completions of monoids? Those which are Cauchy complete and contain an object of which every other object is a retract?

I wonder which categories arise as Cauchy completions of monoids?

Those which are Cauchy complete and contain an object of which every other object is a retract?
- CommentRowNumber18.
- CommentAuthorTodd_Trimble
- CommentTimeAug 17th 2010
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Author: Todd_Trimble
Format: MarkdownItexWell, yes. I guess you're really suggesting there's probably no good answer besides "it is what it is"?

Well, yes. I guess you’re really suggesting there’s probably no good answer besides “it is what it is”?
- CommentRowNumber19.
- CommentAuthorMike Shulman
- CommentTimeAug 18th 2010
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Author: Mike Shulman
Format: MarkdownItexI suppose there might be a better answer, but it seems unlikely to me.

I suppose there might be a better answer, but it seems unlikely to me.

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Atrium > Mathematics, Physics & Philosophy: What is a monoidal category with one (or two) object(s)?