nLab
concrete category

Idea

A concrete category is a category that looks like a category of “sets with extra structure”.

Definition

A concrete category is a category $C$

• equipped with a faithful functor

  $$U:C\to \mathrm{Set}$$U : C \to Set

  to the category Set;

• such that $U$ is representable by some object $c_0\in C$, in that $U\simeq C(c_0,-)$

The object $c_0$ is called a generator of the category.

Remark

Often the term “concrete category” is used without implying the second condition of representability. This second condition however is important for the statement of concrete dualities induced by dual adjunctions.

Examples

• $C=\mathrm{Set}$ itself with $c_0=\{•\}$ the singleton set;

• $C$ any category which has an object $c_0$ which is “free on one generator”, i.e which is the image under a functor $F:\mathrm{Set}\to C$ left adjoint to $U$ of the singleton set. Then by definition of left adjoint and the above example, we have $C(F(\{•\}),d)\simeq \mathrm{Set}(\{•\},U(d))\simeq U(d)$, naturally in $d$. This abstract nonsense indicates the usual collection of examples of concrete categories: for instance monoids, groups, rings, algebras, etc. (… many more examples…)

• Take $C$ to be the category of Banach spaces with morphisms those (everywhere-defined) linear transformations with norm bounded (above) by $1$ (so $\parallel Tv\parallel \le \parallel v\parallel$ for all $v$ in the source). Then there are two versions of $U$ that one may use: one where $U(V)$ (for $V$ a Banach space) consists of every vector in $V$, and one where $U(V)$ consists of those vectors bounded by $1$ (so the closed unit ball in $V$). The first may seem more obvious at first, but only the second is representable (by a $1$-dimensional Banach space).

• A concrete category that is equipped with the structure of a site in a compatible way is a concrete site.

Characterization

Theorem

A finitely complete category is concrete, in the sense that it admits a faithful functor to Set, if and only if it is well-powered with respect to regular subobjects.

Proof

“Only if” was proven in (Isbell). To prove it, note that if $F:C\to D$ is a faithful functor, then it is injective on equivalence classes of regular subobjects. For suppose that $m:a\to x$ is the equalizer of $f,g:x\rightrightarrows y$, and $n:b\to x$ is the equalizer of $h,k:x\rightrightarrows z$. If $F(a)\cong F(b)$ as subobjects of $F(x)$, then since $fm=gm$ and so $F(f)\circ F(m)=F(g)\circ F(m)$, we must also have $F(f)\circ F(n)=F(g)\circ F(n)$; hence (since $F$ is faithful) $fn=gn$, so that $n$ factors through $m$. Conversely, $n$ factors through $m$, so we have $a\cong b$ as subobjects of $x$. Since $\mathrm{Set}$ is regularly well-powered, it follows that any category admitting a faithful functor to $\mathrm{Set}$ must also be so.

(Actually, Isbell proved a more general condition which applies to categories that may lack finite limits.)

“If” was proven in (Freyd). The argument is rather more involved, passing through additive categories, and is not reproduced here.

Generalisations

One can consider concrete categories over any base category $X$ instead of necessarily over $\mathrm{Set}$. This is the approach taken in The Joy of Cats. Then the (small) concrete categories over $X$ form a 2-category $\mathrm{Cat}(X)$.

References

• The Joy of Cats

• John Isbell, Two set-theoretical theorems in categories , Fund. Math 53 (1963)

• Peter Freyd, Concreteness , JPAA 3 (1973)