Properties, Structures, and Stuff
Table of Contents
Here we present a general setup in which we can speak of objects and objects equipped with extra properties, structures, or stuff. Let us first give an informal explanation of extra properties, structures, and stuff:
- an object with “extra properties” (a ring being a commutative ring, a ring being a field, …). Intuitively we add extra constraints.
- an object with “extra structure” (a set equipped with a topology, a category equipped with a monoidal product, …). Intuitively we enrich the object with additional data (we will assume that every object can carry additional data).
- an object with “extra stuff” (a ring together with a module over the ring, a cartesian closed category \( \mathcal C \) and a \( \mathcal C \)-enriched category, …). Intuitively we add additional objects (that “live over” or “live in relation to” our base object) to our base object (we will assume that every object can be equipped with additional stuff).
By an enriched object we mean an object that satisfies extra properties, carries extra structure, or is equipped with extra stuff. In contrast a base object is an object without extra properties, structure, or stuff.
1. Definition
Now we give a formal definition of a setup that tries to capture these three notions. For this we take two categories \( \mathcal C \) and \( \mathcal D \) and think of \( \mathcal C \) as the category of enriched objects and of \( \mathcal D \) as the category of base objects. Then we take a functor \( F \colon \mathcal C \rightarrow \mathcal D \) which forgets the enrichment.
Let \( \mathcal C \), \( \mathcal D \) be categories and \( F \colon \mathcal C \rightarrow \mathcal D \) a functor. Then
- \( F \) forgets nothing if \( F \) is an equivalence of categories.
- \( F \) forgets only properties if \( F \) is fully faithful.
- \( F \) forgets only structure if \( F \) is essentially surjective and faithful.
- \( F \) forgets only stuff if \( F \) is essentially surjective and full.
Let’s try to justify this definition using our intuitive understanding. Let \( \mathcal D \) be a category of base objects (for example \( \Set \)). If we now look at the objects that satisfy an additional property (for example finite sets), then we get a new category \( \mathcal C \) of these objects. This is of course the full subcategory of \( \mathcal D \) generated by these objects and the forgetful functor is the inclusion of the subcategory, ie precisely a fully faithful functor. If we define a new notion of structure on our base objects (for example topology or group structure), then we get a new category \( \mathcal C \) of base objects that carry a concrete incarnation of this additional structure (a topological space or a group). The morphisms in \( \mathcal C \) are the morphisms in \( \mathcal D \) that are compatible with the extra structure (continuous maps or morphisms of groups). The forgetful functor sends an enriched object to the underlying base object. Then it has to be faithful because all morphisms in \( \mathcal C \) are morphisms in \( \mathcal D \) but it is not necessarily full because a morphism in \( \mathcal D \) may not be compatible with the additional structure (see topological spaces and groups). Now we assume that for every base object there exists such an additional structure (discrete topology, hmm, what about groups?). Hence the forgetful functor is precisely essentially surjective. If we equip the objects in \( \mathcal D \) with extra stuff (for example we equip every set X \( X \) with a module over the polynomial ring \( \mathbb Z [X] \)), then we get a new category \( \mathcal C \) of enriched objects. A morphism between two objects in \( \mathcal C \) is a morphism between the underlying base objects in \( \mathcal D \) together with morphisms between the additional stuff that may satisfy some compatibility (morphism \( f \) of the sets \( X \) and \( Y \) and a morphism of the modules that is compatible with \( f \)). Hence the forgetful functor is full. We again assume that every base object can be equipped with extra stuff, therefore the forgetful functor is essentially surjective.
2. QUESTION Does every set (maybe without the empty set) admit a group structure?
3. Factorization System in \( \Cat \)
In \( \Set \) every morphism can be factored into two maps (a surjective map followed by an injective map). An analogous statement holds in \( \Cat \) where we get a factorization into three functors.
Let \( F \colon \mathcal C \rightarrow \mathcal D \) be a functor. The \( 1 \)-image \( \oneim F \) of \( F \) is the following category
- objects are the objects of \( \mathcal C \) (\( \Ob (\oneim F) = \Ob \mathcal C \),
- the morphisms between \( x \) and \( y \) are the morphisms from \( Fx \) to \( Fy \) (\( \oneim(x,y) = \mathcal D(Fx, Fy) \).
The \( 2 \)-image \( \twoim F \) of \( F \) is the following category:
- objects are the objects of \( \mathcal C \) (\( \Ob (\oneim F) = \Ob \mathcal C \),
- a morphisms between \( x \) and \( y \) is a morphism \(Ff \colon Fx \rightarrow Fy \) (a morphism from \(Fx \) to \(Fy \) that is in the image of \(F \)). Alternatively we can identify \(Ff \) with the equivalence class \( [f] \) of morphisms \( \mathcal C \) under the equivalence relation \( f \sim g \) if \( Ff = Fg \).
There are canonical functors
\begin{equation*} \xymatrix{ \mathcal C \ar[r] & \twoim F \ar[r] & \oneim F \ar[r] & \mathcal D. } \end{equation*}The first functor is given applying \(F \) to the morphisms:
\begin{equation*} x \mapsto x, f \mapsto Ff \text{ (or } [f] \text{)}. \end{equation*}The second functor is given by embedding morphisms that are in the image of \(F \) into all morphisms in \(\mathcal D \) (or by embedding the subcategory of \(\mathcal D \) into the full subcategory it generates):
\begin{equation*} x \mapsto x, Ff \mapsto Ff \text{ (or } [f] \mapsto Ff \text{)}. \end{equation*}The third functor is given by applying F to the objectives (or by embedding the full subcategory of \(\mathcal D\) into \(\mathcal D \)):
\begin{equation*} x \mapsto Fx, Ff \mapsto Ff. \end{equation*}Let \( F \colon \mathcal C \rightarrow \mathcal D \) be a functor of \( 1 \)-categories (a \( 1 \)-morphism in \( \Cat \)). Then
\begin{equation*} \xymatrix{ \mathcal C \ar[r] & \twoim F \ar[r] & \oneim F \ar[r] & \mathcal D } \end{equation*}is a factorization of \(F\) into an essentially surjective and full functor followed by an essentially surjective and faithful functor followed by a fully faithful functor. Alternatively \(F \) factors into three functors, where the first forgets extra stuff, the second forgets extra structure, and the third forgets extra properties.
Let’s do an example:
Let \( \mathcal C \) be the following category:
- objects are tuples \((G,M) \) where \(G \) is a finite abelian group and \(M \) is a module over the group ring \(\mathbb Z [G] \),
- a morphisms is a tuple \((f,g) \colon (G,M) \rightarrow (H,N) \) such that \( f \colon G \rightarrow H \) is a morphism of groups and \(g \colon M \rightarrow N \) is a morphism of sets with \(g(a.m) = f(a).g(m) \).
Then we have a forgetful functor \(F \colon \mathcal C \rightarrow \Set, (G,M) \mapsto G \). Then we calculate that indeed \( \twoim F \simeq \mathrm{FinAb} \) and \( \oneim F \simeq \mathrm{FinSet} \setminus \emptyset \).
But there is a second forgetful functor \(F' \colon \mathcal C \rightarrow \Set, (G,M) \mapsto M \). In this case it is not clear to me what the images of \(F' \) are \dots