October 4, 2010

Structure-Like Stuff

Posted by Mike Shulman

Regular readers of this blog will be familiar with the notions of property, structure, and stuff. Less well-known is an intermediate notion between property and structure called “property-like structure.” This is structure which is essentially unique when it exists, such as the structure of finite products on a category, or the structure of an identity element in a semigroup (making it into a monoid). It is distinguished from a mere property (which is also unique, when it exists/holds) because it need not be preserved by all morphisms: not every functor between categories with products preserves products, and not every semigroup homomorphism between monoids is a monoid homomorphism.

We can also define, by analogy, a similar intermediate notion between structure and stuff, which it is natural to call “structure-like stuff.” But are there any examples?

To recap the definitions, we say that a functor $F\colon C\to D$

• forgets at most properties if it is fully faithful, such as the forgetful functor from abelian groups to groups.
• forgets at most structure if it is faithful, such as the forgetful functor from groups to sets.
• forgets at most stuff if it is arbitrary.

To this we can add that $F$

• forgets at most property-like structure if it is pseudomonic, such as the forgetful functor from monoids to semigroups.

In other words, structure is property-like if it is preserved by all isomorphisms. In particular, two $C$-structures on the same object $d\in D$ must be isomorphic, since the identity of $d$ must preserve that structure and thus be a $C$-isomorphism between them.

Property-like structure is more common after we categorify one level. Now we say that a 2-functor $F\colon C\to D$

• forgets at most properties if it is 2-fully-faithful (i.e. an equivalence on hom-categories), such as the forgetful functor from symmetric monoidal categories to braided ones.
• forgets at most structure if it is full and faithful on hom-categories, such as the forgetful functor from braided monoidal categories to monoidal categories.
• forgets at most stuff if it is faithful on hom-categories, such as the forgetful functor from monoidal categories to categories.
• forgets at most eka-stuff if it is arbitrary.

and

• forgets at most property-like structure if it is full and faithful on hom-categories, and also “pseudomonic” in the sense that for any $c,c'\in C$, any equivalence $F(c) \simeq F(c')$ in $D$ is in the image of $F$.

The most familiar example of such a 2-functor is the forgetful functor from categories with finite products (or any other sort of limit and/or colimit) to categories. But clearly we should now also say that a 2-functor…

• forgets at most structure-like stuff if it is pseudomonic on hom-categories—i.e. it is faithful on 2-cells, and full onto invertible 2-cells.

Can anyone think of a natural example of such a 2-functor?

Posted at October 4, 2010 9:43 PM UTC

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Re: Structure-Like Stuff

I imagine you thought of this after our little discussion about ‘cartesian closed structures’ for categories, over on the category theory mailing list.

Interesting question! As you note, the easiest kind of property-like structure on categories is ‘having all limits of a specified sort’.

Can we boost this up a notch… in the right way? There’s an obvious wrong way: for a 2-category to have all limits still seems pretty property-like.

Hmm, but is it property-like structure or property-like stuff? You didn’t mention ‘property-like stuff’, so I’m not sure it even makes sense, but I think it does.

Anyway: is there something for 2-categories that comes ‘after’ having limits, something more like a structure???

Posted by: John Baez on October 5, 2010 3:50 AM | Permalink | Reply to this

Re: Structure-Like Stuff

I was being a bit silly here, going from categories to 2-categories. There should be examples of categories with structure-like stuff. But maybe 2-categorical examples are easier to find?

I’m still curious whether ‘property-like stuff’ is a sensible concept.

Posted by: John Baez on October 5, 2010 3:57 AM | Permalink | Reply to this

Re: Structure-Like Stuff

I’ve been thinking in similar directions, but haven’t gotten anywhere yet. I feel like there should be examples of categories with structure-like stuff (meaning, in case anyone is confused, that the forgetful functor from the 2-category of such categories would be locally pseudomonic, as I defined it), but it could indeed be that 2-categorical examples are easier to find. Certainly, categories with property-like structure are easier to find than sets with property-like structure, or at least people seem to have noticed them first.

One thought I had was to think about 2-categories whose hom-categories have some kind of property-like structure. But I haven’t been able to make that work out yet.

I didn’t think of “property-like stuff” before, but if I had to define it, I’d say that the forgetful 2-functor should be faithful on 2-cells, full on invertible 2-cells, and also full on invertible 1-cells (up to isomorphism). If you’re talking about stuff on 2-categories, then the forgetful 3-functor should be fully faithful on 3-cells, full on invertible 2-cells, and full on invertible 1-cells. I would expect the forgetful functor from 2-categories with 2-limits to be full on all 2-cells, though, so that having 2-limits is again a property-like structure on a 2-category. So (assuming you agree with my definition of property-like stuff) we have a second puzzle: are there any natural examples of property-like stuff?

Posted by: Mike Shulman on October 5, 2010 4:28 AM | Permalink | Reply to this

Re: Structure-Like Stuff

I claim that the forgetful functor from monoidal categories to “semigroupal categories” (categories with a binary tensor product functor which is associative up to coherent iso, but no unit object) forgets property-like stuff. (Since property-like stuff is a special case of structure-like stuff, this gives another solution to the original question too.) I’ve convinced myself of this, but I don’t have time to write out a proof right now, so I’ll let everyone have the fun of working it out themselves (and maybe even proving me wrong). (-:

Posted by: Mike Shulman on October 5, 2010 7:39 PM | Permalink | Reply to this

Re: Structure-Like Stuff

The first sort of example that I can feel happy about supposes a category enriched in groups (abelian groups, I suppose, thought of as 1-object groupoids, of course) and the 2-functor leaves objects alone and sends each hom-group to the free abelian monoid on the hom-group’s underlying set — this is naturally a rig with product extending the group multiplication, and naturally you use the product for vertical composition. Then the image of a group in its rig is still the set of units — isomorphisms — in that rig.

Someone else can think about more X-oidal variations.

Posted by: some guy on the street on October 5, 2010 5:42 AM | Permalink | Reply to this

Re: Structure-Like Stuff

sends each hom-group to the free abelian monoid on the hom-group’s underlying set

of course what I really mean is that, on a given hom-group, the 2-functor is the natural map from the group to the rig… you can tell I’m still raw at this…

Posted by: some guy on the street on October 5, 2010 5:49 AM | Permalink | Reply to this

Re: Structure-Like Stuff

Sorry, I’m not following. What are the source and target 2-categories of your 2-functor? It sounds like the source is the 2-category of Ab-enriched categories… but what is the target?

Posted by: Mike Shulman on October 5, 2010 5:52 AM | Permalink | Reply to this

Re: Structure-Like Stuff

Maybe I’m not thinking long enough… Can we enrich in abelian rigs? I should have thought yes… and if so, that is the target’s enrichment.

Posted by: some guy on the street on October 5, 2010 6:00 AM | Permalink | Reply to this

Re: Structure-Like Stuff

But, no, I don’t mean the category of Ab-enriched categories, I mean a single category (any …) enriched in abelian 1-object groupoids. So, maybe it’s a silly example.

Posted by: some guy on the street on October 5, 2010 6:04 AM | Permalink | Reply to this

Re: Structure-Like Stuff

So you mean that the source of your 2-functor is a particular 2-category whose hom-categories are abelian one-object groupoids? Okay, what is its target?

Posted by: Mike Shulman on October 5, 2010 6:10 AM | Permalink | Reply to this

Re: Structure-Like Stuff

The target is a particular 2-category with the same 0-cells, whose hom-cats are derived from those of the original 2-category by the recipe “given an abelian group $A$, return the abelian rig $\mathbb{N}[A]$”. The hom-cats are again abelian monoids (with identity $e\in A$); they no longer enjoy the property of being groups, but they have a new extra monoid structure (with identity $0$) which is preserved by both horizontal and vertical composition. (horiz. comp. should be defined by additively extending the original horiz. comp. …)

Let me find some paper, to make sure if that last claim is true…

Posted by: some guy on the street on October 5, 2010 6:29 AM | Permalink | Reply to this

Re: Structure-Like Stuff

Here’s an example which is somewhat tautological, but perhaps also illuminating. Let $C$ be the 2-category whose objects are pseudomonic functors, whose morphisms are squares commuting up to (specified) isomorphism, and whose 2-morphisms are pairs of natural transformations making the evident cylinder commute. Then I claim that the forgetful 2-functor $C\to Cat$, which takes each pseudomonic functor to its target, forgets structure-like stuff. In other words, one way to put structure-like stuff on a category is to describe a way of putting property-like structure on the objects of that category.

This 2-category $C$ has some interesting sub-2-categories, such as the 2-category of property-like monads (on varying base categories). The corresponding 3-category to $C$, whose objects are pseudomonic 2-functors, also contains a sub-3-category of property-like 2-monads, which contains further subcategories of lax-idempotent 2-monads and colax-idempotent 2-monads.

Posted by: Mike Shulman on October 5, 2010 6:26 PM | Permalink | Reply to this

Re: Structure-Like Stuff

Hi,

I don’t think this is the right place to ask this question. This is a kind of meta-post, where I might be asking where this post actually belongs. I am wondering about approximation and idealization. Specifically, I am wondering if anyone has seen some work on the following. In the semantics of programming languages we find Domains as a place to talk about iteration and approximation. We can define a Scott Topology on the Domain and now our Domain-maps are continuous maps. Next,we can see our Domains as categories and turn the continuous Domain-maps into continuous functors. If we push the idea further, we have continuous functors and a notion of approximation which is now over categories. Lambek ponders the existence of _the_ category of Sets. What about _approximations_ to the category of sets. For that matter, what might it look like to approximate any well-known category like that of manifolds or FDVec?
Naturally, any thoughts on this would be most appreciated.

Posted by: pseudonym on October 12, 2010 9:51 PM | Permalink | Reply to this

Re: Structure-Like Stuff

You might try MathOverflow, although it’s kind of a vague question so they might not like it. Or the nForum.

Posted by: Mike Shulman on October 12, 2010 10:12 PM | Permalink | Reply to this

Re: Structure-Like Stuff

You could try Math.stackexchange, which is a little more forgiving as far as vague questions go.

Posted by: David Roberts on October 13, 2010 12:59 AM | Permalink | Reply to this

Re: Structure-Like Stuff

My opinion is, looking at a mathematical gadget as consisting of stuff equipped with structure such that some properties hold, a mathematical gadget itself may be “property-like”.

For example, we can talk about “THE” Euclidean 3-dimensional space (up to isomorphism). Or more generally, if we fix some positive integer n, we can talk about “THE” Euclidean n-dimensional space (up to some isomorphism).

But isn’t this adjective “Property-like” merely a generalization of Bourbaki’s “Univalent theory”? If so, why can’t we borrow the adjective “Univalent”?

Just my thoughts before I have to run to class…

Posted by: Alex on October 28, 2010 6:52 PM | Permalink | Reply to this

Re: Structure-Like Stuff

It sounds like you’re talking about the generalized the. That’s something kind of different from property-like structure: things like property, structure, and stuff are additional data that you have a way of giving to an underlying object. If we’re talking just about a single object, it just is; with no category in which it lives or “underlying thing” to compare it to, there’s no property, structure, or stuff in sight. Unless I’m misunderstanding what you meant.

Posted by: Mike Shulman on October 28, 2010 9:51 PM | Permalink | Reply to this

Re: Structure-Like Stuff

Interesting point, so “Property-like” stuff/structure is not “unique up to isomorphism” but just unique. Right?

Consider, e.g., the forgetful functor $U:\mathbf{Grp}\to\mathbf{Mon}$ from the category of groups to the category of monoids is pseudomonic. We have a unique inversion operator, not one “unique up to isomorphism.”

I see, that is my mistake. Thank you for the clarification :)

Posted by: Alex on October 28, 2010 11:40 PM | Permalink | Reply to this

Re: Structure-Like Stuff

No, that’s not quite right. For instance, “having finite limits” is a property-like structure on a category, but finite limits are unique only up to unique isomorphism.

(And actually, the forgetful functor $Grp \to Mon$ is fully faithful—any monoid homomorphism between groups automatically preserves inverses. So being a group is really just a property of a monoid.)

Posted by: Mike Shulman on October 29, 2010 12:41 AM | Permalink | Reply to this

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