typical contexts
A concrete category is a category that looks like a category of “sets equipped with extra structure”, hence like a category of structured sets.
Given a category $C$ with a type of objects $Ob(C)$ and for every object $a:Ob(C)$ and $b:Ob(C)$ a set of morphisms $Mor_C(a, b)$, $C$ is a concrete category if for every object $a:Ob(C)$ there is a set of elements $U(a)$ and for every object $a:Ob(C)$ and $b:Ob(C)$, there is an injection $i_{a,b} \colon Mor_C(a,b) \to (U(a) \to U(b))$.
Given a category $C$ with a type of objects $Ob(C)$ and a set of morphisms $Mor(C)$ with source and target functions $s:Mor(C) \to Ob(C)$ and $t:Mor(C) \to Ob(C)$, $C$ is a concrete category if there is a set of elements $U(C)$ with a function $o:U(C) \to Ob(C)$ with an injection $i:Mor(C) \to Func(Set)$ (and functions $s_{Set}:Func(Set) \to U(C)$ and $t_{Set}:Func(Set) \to U(C)$) such that for every term $f:Mor(C)$, $s(f) = o(s_{Set}(i(f)))$ and $t(f) = o(t_{Set}(i(f)))$.
A concrete category is a category $C$ equipped with a faithful functor
to the large category Set. We say a category $C$ is concretizable if and only if it admits a faithful functor $U: C \to Set$.
Very often it is useful to consider the case where $U$ is representable by some object $c_0 \in C$, in that $U \simeq C(c_0,-)$. For example, this is important for the statement of various concrete dualities induced by dual adjunctions. We say in this case that $(C, U: C \to Set)$ is representably concrete. By definition, the object $c_0$ is then a separator of the category.
We remark that the existence of a left adjoint $F$ to $U: C \to Set$ implies that $U$ is representable by $F(1)$. Conversely, if $C$ has coproducts or even just copowers, then representability of $U$ implies that $U$ has a left adjoint.
One can also consider concrete categories over any base category $X$ instead of necessarily over $Set$. This is the approach taken in The Joy of Cats. Then the (small) categories concrete over $X$ form a 2-category $Cat(X)$.
A concrete category is univalent if its underlying category (by forgetting the functor into $Set$) is a univalent category.
The following furnish examples of concrete categories, with the first three representably concrete:
$C = Set$ itself with separator $c_0 = \{\bullet\}$ the singleton set.
$C = Top$ with the separator $c_0$ taken to be the one-point space.
Any monadic functor $U: C \to Set$ is faithful (because it preserves equalizers and reflects isomorphisms) and has a left adjoint. As special cases, we have the usual collection of examples of concrete categories: monoids, groups, rings, algebras, etc.
A category may be concretizable in more than one way:
Take $C$ to be the category of Banach spaces with morphisms those (everywhere-defined) linear transformations with norm bounded (above) by $1$ (so $\| T v \| \leq \| v \|$ 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).
Insofar as categories such as $Set$, $Top$, $Vect_k$, etc. admit many separators, these categories may be rendered representably concrete in a variety of ways. Indeed, the category $Vect_k$ may be monadic over $Set$ in many different ways. For example, if $V$ is $n$-dimensional, the functor $\hom(V, -): Vect_k \to Set$ is monadic and realizes $Vect_k$ as equivalent to the category of modules over the matrix algebra $\hom(V, V)$.
Any Grothendieck topos is concretizable, but not necessarily (and typically not) representably concretizable. If $E = Sh(C, J)$ is the category of sheaves on a small site $(C, J)$, we have a familiar string of faithful functors
But if for example $E$ is the category of sheaves over $\mathbb{R}$, then no object $X$ can serve as a single separator of $E$, since it cannot detect differences between arrows $Y \stackrel{\to}{\to} Z$ whenever the support of $Y$ is strictly contained in the support of $X$.
A concrete category that is equipped with the structure of a site in a compatible way is a concrete site. The category of concrete sheaves on a concrete site is concrete.
Many familiar examples of “sets with additional structure” provide examples of concrete categories where $U$ is the usual ‘underlying set’:
The category $Mon$ of monoids and monoid homomorphisms is a concrete category.
The category $Ab$ of abelian groups and abelian group homomorphisms is a concrete category.
Given a commutative ring $R$, the category $R Mod$ of $R$-modules and $R$-linear maps is a concrete category.
Given a commutative ring $R$, the category $R Alg$ of $R$-algebras and $R$-algebra homomorphisms is a concrete category.
The category $CRing$ of commutative rings and commutative ring homomorphisms is a concrete category.
The category $Field$ of fields and field homomorphisms is a concrete category.
The category $HeytAlg$ of Heyting algebras and Heyting algebra homomorphisms is a concrete category.
The category $Frm$ of frames and frame homomorphisms is a concrete category.
The category $Conv$ of convergence spaces and continuous functions is a concrete category.
The category $Top$ of topological spaces and continuous functions is a concrete category.
The category $Met$ of metric spaces and isometries is a concrete category.
The category $StrictCat$ of strict categories and strict functors is a concrete category.
There are other examples of concretizable categories where the objects are described as sets, but one cannot choose $U$ satisfying $U(X) = X$
The category $Set_\bot$ of sets and partial functions is a concrete category when equipped with the functor $U(X) = X \amalg \{ * \}$ that adds a disjoint point, and sends a partial function to the total function whose undefined values are set to the point.
The category $Rel$ of sets and relations has a separator given by the singleton set. Thus, it is a concrete category when equipped with the functor $U(X) = PowerSet(X)$, and $U(X \to Y)$ given by composition of relations (viewing a subset of $X$ as a relation on $X$). This is faithful since for any relation $\Phi \in Rel(X, Y)$ we have $(x,y) \in \Phi$ iff $y \in \Phi \circ \{x\}$.
The classical homotopy category Ho(Top) of topological spaces is not concretizable
The opposite category of commutative rings $CRing^{op}$ equipped with the prime spectrum functor $CRing^{op} \to Set$ is not concrete, since the prime spectrum is not faithful. This is one of the reasons for the use of schemes in algebraic geometry.
The category $Prefunc$ of sets and prefunctions is not a concrete category.
Every small category $C$ is concretizable (since it fully and faithfully embeds in the concrete category $Set^{C^{op}}$).
If $C$ is concretizable, so is $C^{op}$.
By assumption, there is a faithful functor $U^{op}: C^{op} \to Set^{op}$, and $\hom(-, \mathbf{2}): Set^{op} \to Set$ is monadic.
Of course, since a category $C$ may possess a separator but no coseparator, it does not follow that $C^{op}$ is representably concrete if $C$ is.
In any concrete category $(C, U:C \to Set)$, there is an evaluation map
such that for every morphism $f \colon Hom(a,b)$ and $g \colon Mor_C(b,c)$ and every element $x:U(a)$, $(g \circ f)(x) = g(f(x))$ and $id_A(x) = x$.
Because Set is a cartesian closed category, currying the injective function $f_{a,b}$ of the functor $F$ in Set means that there is an evaluation map $(-)((-))\colon Hom(a,b) \times U(a) \to U(b)$ which satisfies the above axioms.
In any concrete category $(C, U:C \to Set)$, the morphisms satisfy function extensionality with respect to the evaluation map: for all morphisms $f \colon Hom(A,B)$ and $g \colon Hom(A,B)$, if $f(x) = g(x)$ for all elements $x \colon El(A)$, then $f = g$.
Since Set is a well-pointed category, and there is a bijection between $\mathbb{1} \to U(a)$ and $U(a)$, function extensionality follows.
The category Set of sets and functions is both concrete and well-pointed. However, not every well-pointed category is an concrete category, as well-pointed categories are not required to be concrete categories: most models of ETCS aren’t defined to be concrete. Moreover, not every concrete category is a well-pointed category: the category $Field$ of fields and field homomorphisms is concrete, but is not well-pointed because it doesn’t have a terminal object.
The distinction between concreteness and well-pointedness is the distinction between elements and global elements in a concrete category with a terminal object, as it is not true that elements and global elements (if they exist) coincide in general.
A finitely complete category is concretizable, i.e., admits a faithful functor to $Set$, if and only if it is well-powered with respect to regular subobjects.
“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 $m\colon a \to x$ is the equalizer of $f,g\colon x\rightrightarrows y$, and $n\colon b\to x$ is the equalizer of $h,k\colon x\rightrightarrows z$. If $F(a) \cong F(b)$ as subobjects of $F(x)$, then since $f m = g m$ 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) $f n = g n$, so that $n$ factors through $m$. Conversely, $n$ factors through $m$, so we have $a\cong b$ as subobjects of $x$. Since $Set$ is regularly well-powered, it follows that any category admitting a faithful functor to $Set$ 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.
A relatively deep application of Isbell’s result is that the classical homotopy category Ho(Top) of topological spaces is not concretizable, even though it is a quotient of $Top$ which is concretizable. (Freyd 70)
A similar way to use Isbell’s result applies to show that a really vast number of model categories can not have a concrete localization at weak equivalences: see Di Liberti and Loregian, 2017
Jiří Adámek, Horst Herrlich, George Strecker, Abstract and Concrete Categories, Wiley 1990, reprinted as: Reprints in Theory and Applications of Categories, No. 17 (2006) pp. 1-507 (tac:tr17)
Peter Freyd, Concreteness, JPAA 3 (1973)
Peter Freyd, Homotopy is not concrete, in The Steenrod Algebra and its Applications, Springer Lecture Notes in Mathematics Vol. 168, Springer-Verlag, 1970, Republished in: Reprints in Theory and Applications of Categories, No. 6 (2004) pp 1-10 (web)
Ivan di Liberti, Fosco Loregian “Homotopical algebra is not concrete.” Journal of Homotopy and Related Structures (2017): 1-15.
Last revised on August 10, 2023 at 14:37:31. See the history of this page for a list of all contributions to it.