### 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?

## 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

structureor property-likestuff? 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???