stuff, structure, property

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.

This is a generalization of parts of traditional Postnikov tower/(n-connected, n-truncated) factorization system theory from groupoids to general categories.

See also at *structure type* and *stuff type*.

To begin with, 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.

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 worth noting that this formalism captures the intuition of how “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 intuition so well, and in fact the ‘properties’ (and ‘structure’) remembered by a functor that forgets purely structure (or purely stuff) may not match what one expects.

See also the examples below.

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.

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 cartesian monoidal categories (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.

Note that property-like structure is known in traditional logic as **categorical** structure. Obviously, this term can be confusing in categorial logic!

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 ternary 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** $1 im F$ 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** $2 im F$ 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 $2 im F$ 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 |
---|---|---|---|

$C$ | $C$ | $C$ | $C$ |

$2 im F$ | $C$ | $C$ | $D$ |

$1 im F$ | $C$ | $D$ | $D$ |

$D$ | $D$ | $D$ | $D$ |

Then $F$ can be factored into functors

$C \stackrel{F_2}\to 2 im F \stackrel{F_1}\to 1 im F \stackrel{F_0}\to D ,$

where $F_2$ forgets purely stuff, $F_1$ forgets purely structure, and $F_0$ forgets only properties. Conversely, this ternary factorization suffices to determine the notions of faithful, full, and essentially surjective functors: see ternary factorization system.

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).

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

$Ab \to Grp \to Set \to pt$

(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 Eckmann-Hilton argument).

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

$Ab \to \pt \to \pt \to \pt$

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 $Ab \to Set$:

$Ab \to Ab \to Set \setminus \{\empty\} \to Set$

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:

$Ab \times Ab \to Ab \to Set \setminus \{\empty\} \to Set$

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.

In logic a property $P$ is given by a predicate, which we may think of as an operation that takes a thing $x$ to the truth value of the statement ”$x$ has property $P$”. Note that a truth value is a $(-1)$-groupoid; we get structure and stuff by replacing this with a set (a $0$-groupoid) or a groupoid (a $1$-groupoid) and we get $n$-stuff by replacing this with an $n$-groupoid.

Now, if a non-evil property of objects of a category $C$ holds for some object $x$, then it must hold for any object isomorphic to $x$. That is, the predicate defining that property is actually a functor from the core $C_iso$ of $C$ to the groupoid $TV_iso$ of truth values. Given such a predicate functor $P$, it's immediate how to define a full subcategory $C_P$ of $C$ consisting of those objects with the property; the inclusion functor $C_P \hookrightarrow C$ is fully faithful, as it should be for extra property. Conversely, given a fully faithful functor $F: D \to C$, define a non-evil property of objects of $C$ as follows: an object $x$ of $C$ has the property if there is some object $a$ of $D$ such that $x \cong F(a)$. If you apply this to $C_P \hookrightarrow C$, then you get the predicate $P$ back; if you start with an arbitrary fully faithful $F: D \to C$, define a predicate $P$, and then form $C_F$, you'll find that $C_F$ and $D$ are equivalent, even as bundles over $C$.

Diagrammatically this may be phrased as saying that every fully faithful functor $D \to C$ arises as a weak pullback of the $1$-subobject classifier

$\array{
D_{iso} &\to& *
\\
\downarrow &\Downarrow& \downarrow^{true}
\\
C_{iso} &\stackrel{P}{\to}& \{\true, false \}
}$

Similarly, any non-evil structure on objects of $C$ is given by a functor from $C_iso$ to the groupoid $Set_iso$ of sets. Given such a functor $P$, let $C_P$ be the category of elements of $P$, which comes with a faithful functor from $C_P$ to $C$. Conversely, given any faithful functor $F: D \to C$ and an object $x$ of $C$, let $P(x)$ be the essential fiber of $F$ over $x$, which (because $F$ is faithful) is a discrete category and hence (equivalent to) a set. These operations are also invertible, up to equivalence.

Diagrammatically this may be phrased as saying that every faithful functor $D \to C$ arises as a weak pullback of the $2$-subobject classifier (as described at generalized universal bundle and at category of elements)

$\array{
D_{iso} &\to& (Set_*)_{iso}
\\
\downarrow &\Downarrow& \downarrow
\\
C_{iso} &\stackrel{}{\to}& Set_{iso}
}
\,.$

Next, any non-evil stuff on objects of $C$ is given by a functor from $C_iso$ to $Grpd_iso$. Here $Grpd_iso$ should be taken to be the $2$-groupoid whose objects are groupoids, whose morphisms are equivalences, and whose $2$-morphisms are natural isomorphisms; similarly, the functor $P: C_iso \to Grpd_iso$ should be taken in the weakest sense (often called a pseudofunctor). Then the Grothendieck construction turns $P$ into a category $C_P$ equipped with a functor to $C$; again, the essential fiber converts any functor $F: D \to C$ into such a $P$ (although really you must take the core of the essential fiber to get a groupoid).

Diagrammatically this may be phrased as saying that every functor $D \to C$ arises as a weak pullback of the $3$-subobject classifier (as described at generalized universal bundle )

$\array{
D_{iso} &\to& (Grpd_*)_{iso}
\\
\downarrow &\Downarrow& \downarrow
\\
C_{iso} &\stackrel{}{\to}& Grpd_{iso}
}
\,.$

If $C$ is a mere $1$-category, then any $P: C_iso \to 2 Grpd_iso$ is equivalent to some $P: C_iso \to Grpd_iso$, but in general we need to consider $P: C_iso \to n Grpd_iso$ or $P: C_iso \to \infty Grpd_iso$ to study higher forms of $n$-stuff.

Lest we forget, to be even more simple than an extra property, the groupoid of $(-2)$-groupoids is the point $pt$, and there is exactly one functor $P$ from any $C_iso$ to $pt$, corresponding to the unique (up to equivalence) category equivalent to $C$.

For the (∞,1)-topos ∞-Grpd of ∞-groupoids the analog of the subobject classifier is the universal fibration of (∞,1)-categories $Z|_{\infty Grpd} \to \infty Grpd$. See also section 6.1.6 *$\infty$-Topoi and Classifying objects* of HTT.

- 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).

Revised on November 22, 2012 13:43:22
by Urs Schreiber
(82.169.65.155)