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\section*{concrete structure}

\hypertarget{concrete_and_abstract_structures}{}\section*{{Concrete and abstract structures}}\label{concrete_and_abstract_structures}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{groups}{Groups}\dotfill \pageref*{groups} \linebreak
\noindent\hyperlink{categories}{Categories}\dotfill \pageref*{categories} \linebreak
\noindent\hyperlink{vector_spaces}{Vector spaces}\dotfill \pageref*{vector_spaces} \linebreak
\noindent\hyperlink{hilbert_spaces}{Hilbert spaces}\dotfill \pageref*{hilbert_spaces} \linebreak
\noindent\hyperlink{operator_algebras}{Operator algebras}\dotfill \pageref*{operator_algebras} \linebreak
\noindent\hyperlink{manifolds_and_varieties}{Manifolds and varieties}\dotfill \pageref*{manifolds_and_varieties} \linebreak
\noindent\hyperlink{sets}{Sets}\dotfill \pageref*{sets} \linebreak
\noindent\hyperlink{points}{Points}\dotfill \pageref*{points} \linebreak
\noindent\hyperlink{concrete_objects}{Concrete objects}\dotfill \pageref*{concrete_objects} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

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 \emph{concrete} and the later notion called \emph{abstract}.

It then becomes possible to define the concrete notion in terms of the abstract one (so giving an \emph{abstract} definition of the original \emph{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).

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

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.

\hypertarget{groups}{}\subsubsection*{{Groups}}\label{groups}

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 \textbf{concrete group} due to [[Évariste Galois]]. Abstracting from this, we get the modern notion of \textbf{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 \emph{abstract} definition of the notion of \emph{concrete} group: An abstract group $G$ becomes a \textbf{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 \emph{already} a concrete group, identifying a group with the [[automorphism group]] of a [[pointed category|pointed]] [[connected category|connected]] [[groupoid]].

\hypertarget{categories}{}\subsubsection*{{Categories}}\label{categories}

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 $(\infty,1)$-[[(infinity,1)-category|categories]].) But in light of the motivating examples and [[Bourbaki]]'s prior theory of structures, we can anachronistically define a \textbf{concrete category} as any category of [[structured sets]] (as defined there).

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

\hypertarget{vector_spaces}{}\subsubsection*{{Vector spaces}}\label{vector_spaces}

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 \textbf{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 \textbf{abstract vector space}, which came later, is a [[module]] over $K$ thought of as a [[commutative ring]]. The abstract definition of a \textbf{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.

\hypertarget{hilbert_spaces}{}\subsubsection*{{Hilbert spaces}}\label{hilbert_spaces}

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 space|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 \textbf{concrete Hilbert space}, while the modern notion is an \textbf{abstract Hilbert space}. An [[orthonormal basis]] on an abstract Hilbert space gives it the structure of a \textbf{concrete Hilbert space} (in the stricter sense).

\hypertarget{operator_algebras}{}\subsubsection*{{Operator algebras}}\label{operator_algebras}

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 $*$-[[star-algebra|algebra]] $B(H)$ of [[bounded linear operators]] from $H$ to itself. A \textbf{$C^*$-[[C-star-algebra|algebra]]} is a sub-$*$-algebra of $B(H)$ that is [[closed subspace|closed]] in the [[norm topology]]; a \textbf{[[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 \textbf{$B^*$-[[B-star-algebra|algebra]]} is a [[Banach algebra|Banach]] $*$-algebra such that ${\|x^* x\|} = {\|x\|}^2$ always holds, while a \textbf{$W^*$-[[W-star-algebra|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 \textbf{$C^*$-algebra} to be a $B^*$-algebra with a [[faithful representation|faithful]] [[representation]] on a Hilbert space, and similarly define a \textbf{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).

\hypertarget{manifolds_and_varieties}{}\subsubsection*{{Manifolds and varieties}}\label{manifolds_and_varieties}

The original [[algebraic varieties]] were [[subspaces]] of [[affine spaces]] or [[projective spaces]] given as the [[zero set]]s of [[algebraic function]]s; these are \textbf{concrete varieties}. The modern notion of varieties as certain [[schemes]], the \textbf{abstract varieties}, is much more general.

Similarly, [[manifolds]] can be viewed as [[subspaces]] of [[Cartesian spaces]] with locally invertible local parametrisations ---\textbf{concrete manifolds}--- or as abstract sets of points equipped with an [[atlas]] of locally invertible local charts ---\textbf{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 space|second-countable]] and [[Hausdorff space|Hausdorff]], as is common).

\hypertarget{sets}{}\subsubsection*{{Sets}}\label{sets}

One might view the [[sets]] of [[material set theory]] as \textbf{concrete sets} and the sets of [[structural set theory]] as \textbf{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 na\"i{}ve version of a \textbf{concrete set} as a [[subset]] of a given [[ambient set]]; these were the first sets studied, predating [[set theory]] as such.

\hypertarget{points}{}\subsubsection*{{Points}}\label{points}

A \textbf{concrete [[point]]} in a [[set]] $X$ is an [[element]] of $X$, an \textbf{abstract point} is a [[singleton]], and the abstract definition of \textbf{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 \textbf{abstract point} is a [[terminal object]], and a \textbf{concrete point} is a [[global element]].

\hypertarget{concrete_objects}{}\subsection*{{Concrete objects}}\label{concrete_objects}

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]] \ldots{} 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.

[[!redirects concrete structure]] [[!redirects concrete structures]] [[!redirects abstract structure]] [[!redirects abstract structures]] [[!redirects concrete or abstract structure]] [[!redirects concrete or abstract structures]] [[!redirects concrete and abstract structure]] [[!redirects concrete and abstract structures]]

[[!redirects concrete group]] [[!redirects concrete groups]]



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