Idea

Category theory frequently allows to give precise and useful formalized meanings to “everyday” terms, at least terms used everyday by practicing mathematicians.

It was indeed introduced originally in order to formalize the use of the notion “natural” in mathematics. Another frequently recurring pair of terms in math are “extra structure” and “extra properties”, to which we add the more general concept of “extra stuff”. In discussion among Jim Dolan, John Baez and Toby Bartels, the following useful formalization of these concepts in category theoretic terms was established.

Definition (for groupoids)

Let $C$ and $D$ be groupoids, and let $F: C \to D$ be a functor. By fiat, declare $F$ to be a forgetful functor. Then * $F$ forgets nothing if it is an equivalence of categories; * $F$ forgets only properties if $F$ is fully faithful; * $F$ forgets at most structure if $F$ is faithful; * $F$ forgets at most stuff regardless.

Another way to break down the possibilities (used in a $3$-way factorisation system) is as follows: * $F$ forgets only properties if $F$ is full and faithful; * $F$ forgets purely structure if $F$ is essentially surjective (on objects) and faithful; * $F$ forgets purely stuff if $F$ is essentially surjective and full.

Interpretation in terms of k-surjectivity

The pattern here is best understood in terms of the notion of essentially k-surjective functor.

Recall that for a functor between ordinary categories (1-categories)

• essentially surjective $\simeq$ essentially 0-surjective
• full $\simeq$ essentially 1-surjective
• faithful $\simeq$ essentially 2-surjective

and that every 1-functor is essentially $k$-surjective for all $k \geq 3$.

So the above says for a functor $F : C \to D$:

If it is …	then it …	but it …
essentially $(k \geq 0)$-surjective	forgets nothing	remembers everything
essentially $(k \geq 1)$-surjective	forgets only properties	remembers at least stuff and structure
essentially $(k \geq 2)$-surjective	forgets at most structure	remembers at least stuff
essentially $(k \geq 3)$-surjective	may forget everything	may remember nothing

It is noteworthy how this formalism captures the heuristic way in which “stuff”, “structure” and “properties” are expected to be related: * stuff may be equipped with structure; * structure may have (be equipped with) properties.

The $3$-way breakdown looks like this:

If it is …	then it …	but it …
essentially $(k \ne 0)$-surjective	forgets only properties	remembers at least stuff and structure
essentially $(k \ne 1)$-surjective	forgets purely structure	remembers at least stuff and properties
essentially $(k \ne 2)$-surjective	forgets purely stuff	remembers at least structure and properties

This formalism does not capture the heuristic so well, and in fact the ‘properties’ (and ‘structure’) remembered by a functor that forgets purely structure (or purely stuff) may not match one's intuition.

See also the examples below.

Generalization to higher groupoids

The formulation in terms of $k$-surjectivity induces an immediate generalization of the notions of stuff, structure and properties to the context of infinity-groupoids. Baez’s students speak of “2-stuff,” “3-stuff,” and so on. Of course, structure and properties can then be called 0-stuff and $(-1)$-stuff, respectively.

Generalization to categories and higher categories

The theory is easiest when restricted to groupoids as above; for categories, there are several ways to go. One is to keep the definition as phrased above (a functor between categories forgets only properties if it is fully faithful, forgets at most structure if it is faithful, etc.). Another is to apply the above definition instead to the functor between the underlying groupoids of the categories in question.

To tell the difference, ask yourself whether the difference between a monoid and a semigroup is the structure of being equipped with an identity element or only the property that an identity element exists. Note that an identity element, if it exists, must be unique and must be preserved by semigroup isomorphisms and by monoid homomorphisms but not by semigroup homomorphisms.

A third option is to define a new notion: a functor forgets at most property-like structure if it is pseudomonic. This means that (1) the functor is faithful and (2) its induced functor between underlying groupoids is fully faithful. Intuitively, property-like structure can be described as consisting of “properties which need not automatically be preserved by morphisms” or “structure which, if it exists, is uniquely determined.”

Property-like structure becomes much more prevalent for higher categories. For example, the forgetful functor from the 2-category of categories-with-products (and product-preserving functors) to $Cat$ is essentially $(k\ge 2)$-surjective, and its induced functor between 2-groupoids is essentially $(k\ge 1)$-surjective; thus it forgets property-like structure. See also lax-idempotent 2-monad.

A factorisation system

Just as the category Set has the best known $2$-way factorisation system, in which every function is factored into a surjection followed by an injection, so the 2-category Cat has a $3$-way factorisation system, in which every functor is factored into parts which forget ‘purely’ stuff, structure, and properties.

Specifically, given a functor $F: C \to D$, let the $1$-image $1im F$ of $F$ be the category whose objects are objects of $C$ and whose morphisms $x \to y$ are morphisms $F(x) \to F(y)$ in $D$; let the $2$-image $2im F$ of $F$ be the category whose objects are objects of $C$ and whose morphisms $x \to y$ are morphisms $b: F(x) \to F(y)$ in $D$ such that $b = F(a)$ for some $a: x \to y$ in $C$. (So the only difference bewteen $2im F$ and $C$ itself is equality of morphisms.) If you want to be complete, call $C$ itself the $3$-image of $F$ and $D$ the $0$-image.

The situation looks like this:

This category …	gets objects from …	and morphisms from …	and equality of morphisms from …
$C$	$C$	$C$	$C$
$2im F$	$C$	$C$	$D$
$1im F$	$C$	$D$	$D$
$D$	$D$	$D$	$D$

Then $F$ can be factored into functors

where $F_2$ forgets purely stuff, $F_1$ forgets purely structure, and $F_0$ forgets only properties.

Examples

The classical categories of sets with extra structure

The classical examples are the forgetful functors to Set that define the classical categories such as Top, Grp, Vect, etc. All these categories are categories of sets equipped with extra structure (e.g. with a topology, with a group structure, etc). Accordingly the obvious functors to Set are

• faithful
• not full.

Hence indeed, by the above yoga, they forget this extra structure but remember the stuff in question (the underlying set).

More examples

The embedding of abelian groups into all groups, $F :$ Ab $\to$ Grp is faithful and full, but not essentially surjective. Hence it should remember stuff and structure but forget properties. Indeed, the property it forgets is the property “is abelian” which is a property of the group structure sitting on the underlying set of a group. Hence the sequence of functors

(with pt the terminal category) successively forgets first a property (being abelian) then structure (the group structure on a set) then stuff (the underlying set).

Notice that the order here is backwards from the automatic factorisation given by the $3$-way factorisation system described above. (And in fact, the structure forgotten here is not ‘pure’; $Grp \to Set$ is not essentially surjective.) Indeed, the above factorisation is arbitrary; it comes from seeing an abelian group as a group with an extra property and a group as a set with extra structure, but one may view things differently (for example, that an abelian group is a monoid with extra property, or a set with two group structures that are related as in the Eckman-Hilton argument?).

The automatic factorisation is in fact like this (up to equivalence of categories):

because the original functor $Ab \to \pt$ is already essentially surjective and full. In other words, from the perspective of $\pt$, an abelian group is simply extra stuff.

More interestingly, we can factor the forgetful functor $Ab \to Set$:

Here, the first part is trivial because $Ab \to Set$ is faithful. The category $Set \setminus \{\empty\}$ is the category of inhabited sets, that is the category of sets that are capable of being equipped with the structure of an abelian group. So from the point of view of its underlying set, an abelian group is a set with the property that it is inhabited and the structure of an abelian group but no additional stuff.

For something interesting at every level, take the functor $Ab \times Ab \to Set$ that takes the underlying set of the first abelian group. This factors as follows:

So a pair of abelian groups (from the perspective of the underlying set of the first one) consists of the property that the set is inhabited, then the structure of an abelian group on that set, and finally extra stuff consisting of the entire second group.

References

• original UseNet discussion on sci.physics.research in 1998;
• a pedagogical comparison to quadratic, linear, and constant polynomials (PDF) by Toby Bartels in 2004;
• section 2.4, p. 15 and section 3.1, p. 17 of J. Baez and M. Shulman, Lectures on $n$-categories and cohomology (arXiv).