concrete structure

Many concepts from ordinary mathematics were originally developed in the course of a particular application, such that the original definition of the concept is tied to that application. The modern definition (typically some kind of structured set) is then obtained by a process of abstraction. Sometimes this history is remembered in the terminology, often with the original notion being called *concrete* and the later notion called *abstract*.

It then becomes possible to define the concrete notion in terms of the abstract one (so giving an *abstract* definition of the original *concrete* concept); usually, a concrete structure is then an abstract structure equipped with some extra stuff. Sometimes there is also a representation theorem? showing that every abstract structure arises from some concrete structure (but sometimes this is false).

We list here only examples where the concrete meaning has historically been the default meaning for at least some authors or where at least one of the adjectives ‘concrete’ or ‘abstract’ have been used.

Given a set $X$, the permutation group on $X$ is the set of permutations on $X$; consider a subset of this that includes the identity function and is closed under composition and taking inverse functions. This is the original notion of **concrete group** due to Évariste Galois. Abstracting from this, we get the modern notion of **abstract group** as a set equipped with an appropriate operation (taking the place of composition of permutations).

Much of the early work on group theory dealt with symmetry groups in geometry, giving a geometric notion of concrete group in which $X$ is a space instead of a set, although (except perhaps in constructive mathematics without the fan theorem) the relevant spaces form a concrete category and so the elements of the group can still be viewed as permutations of an underlying set.

We can now give an *abstract* definition of the notion of *concrete* group: An abstract group $G$ becomes a **concrete group** on the set $X$ once it is given a free action of $G$ on $X$. The Cayley theorem? shows that every group may be made into a concrete group. Of course, if we generalise from Set to an arbitrary category (whether thought of as a category of spaces or not), then every group is *already* a concrete group, identifying a group with the automorphism group of a pointed connected groupoid.

Unlike groups, categories were first defined in full modern abstraction. (At least, modern for the 20th century; by the end of the 21st century, it may seem old-fashioned not to start with weak $(\infty,1)$-categories.) But in light of the motivating examples and Bourbaki's prior theory of structures, we can anachronistically define a **concrete category** as any category of structured sets (as defined there).

Of course, an **abstract category** is the usual notion of category. Then the abstract definition of a **concrete category** is a category equipped with a faithful functor to Set; see structured set again for the equivalence.

The original vector spaces were Cartesian spaces: $\mathbb{R}^n$, where $\mathbb{R}$ is the real line and $n$ is a natural number. This generalises easily to $K^c$, where $K$ is any field and $c$ is any cardinal number; we may call such a **concrete vector space**. Unlike with groups, there is no need (in classical mathematics) to consider subspaces of $K^c$ closed under linear combinations, since these are all isomorphic to $K^d$ for $d \leq c$.

Then an **abstract vector space**, which came later, is a module over $K$ thought of as a commutative ring. The abstract definition of a **concrete vector space** is an abstract vector space equipped with a basis. Using the axiom of choice, we may prove that every abstract vector space has such a concrete structure.

The original notion of Hilbert space (the one used by David Hilbert) was $L^2(\mathbb{R})$, the Lebesgue space on the real line (with Lebesgue measure) of exponent $2$. This immediately generalises to $L^2(X)$, where $X$ is any measure space. As a special case, if $X$ is a discrete measure space (a set equipped with counting measure), then we have a topological version of the concrete vector space $K^c$.

Either of these (any measure space or only a discrete measure space) may be taken as a **concrete Hilbert space**, while the modern notion is an **abstract Hilbert space**. An orthonormal basis on an abstract Hilbert space gives it the structure of a **concrete Hilbert space** (in the strcter sense).

With operator algebras, we have the curious situation that special names may still be found in the literature to distinguish the concrete and abstract structures.

Let $H$ be a Hilbert space over the complex numbers and consider the $*$-algebra $B(H)$ of bounded linear operators from $H$ to itself. A **$C^*$-algebra** is a sub-$*$-algebra of $B(H)$ that is closed in the norm topology; a **von Neumann algebra** is a sub-$*$-algebra of $B(H)$ that is closed in the weak operator topology (a stronger condition).

On the abstract side, a **$B^*$-algebra** is a Banach $*$-algebra such that ${\|x^* x\|} = {\|x\|}^2$ always holds, while a **$W^*$-algebra** is a $B^*$-algebra with a predual as a Banach space. It is then a theorem that (as defined above) every $C^*$-algebra is a $B^*$-algebra, and every von Neumann algebra is a $W^*$-algebra, so we may abstractly define a **$C^*$-algebra** to be a $B^*$-algebra with a faithful representation on a Hilbert spcace, and similarly define a **von Neumann algebra** to be a $W^*$-algebra with a free action on a Hilbert space.

The representation theorem here is that every $B^*$-algebra may be given the structure of a $C^*$-algebra, and in fact the term ‘$B^*$-algebra’ is nearly obsolete. Similarly, every $W^*$-algebra may be given the structure of a von Neumann algebra, but here both terms may yet be found (and even distinguished such that a von Neumann algebra comes with a representation on a Hilbert space but a $W^*$-algebra does not).

The original algebraic varieties were subspaces of affine spaces or projective spaces given as the zero sets of algebraic functions; these are **concrete varieties**. The modern notion of varieties as certain schemes, the **abstract varieties**, is much more general.

Similarly, manifolds can be viewed as subspaces of Cartesian spaces with locally invertible local parametrisations —**concrete manifolds**— or as abstract sets of points equipped with an atlas of locally invertible local charts —**abstract manifolds**. Both concepts were used from the earliest days; the Whitney embedding theorem shows their equivalence (at least if the abstract manifolds are assumed to be second-countable and Hausdorff, as is common).

One might view the sets of material set theory as **concrete sets** and the sets of structural set theory as **abstract sets** (a term used at least by Lawvere). The abstract definition of a concrete set is then that given at pure set. There is also a more naïve version of a **concrete set** as a subset of a given ambient set?; these were the first sets studied, predating set theory as such.

A **concrete point** in a set $X$ is an element of $X$, an **abstract point** is a singleton, and the abstract definition of **concrete point** in $X$ is a function to $X$ from an abstract point. The concrete definition of concrete point doesn't generalise from Set to arbitrary categories, but the others do: an **abstract point** is a terminal object, and a **concrete point** is a global element.

Since all of these concrete and abstract objects literally are objects in various categories, it would be nice to use the terms ‘concrete object’ and ‘abstract object’ to refer to them collectively. However, there is another meaning of ‘concrete object’, so I have gone for the vaguer term ‘structure’ instead (justified since they are all objects in concrete categories and so are sets with extra structure … with the exception of the concrete and abstract categories themselves!).

There is a relationship: concrete objects generalise concrete sheaves, which are concrete in the sense of having an underlying set of points, similar to how a concrete variety has a set of points from an affine or projective space. However, it's not really an example of the concept on this page, as fair as I can tell.

Revised on October 5, 2014 11:12:19
by Toby Bartels
(98.19.46.50)