nLab
concrete category

Contents

Contents

Idea

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

Definition

With one collection of elements

Definition

Given a category CC with a type of objects Ob(C)Ob(C) and a set of morphisms Mor(C)Mor(C) with source and target functions s:Mor(C)Ob(C)s:Mor(C) \to Ob(C) and t:Mor(C)Ob(C)t:Mor(C) \to Ob(C), CC is a concrete category if there is a set of elements U(C)U(C) with a function o:U(C)Ob(C)o:U(C) \to Ob(C) with an injection i:Mor(C)Func(Set)i:Mor(C) \to Func(Set) (and functions s Set:Func(Set)U(C)s_{Set}:Func(Set) \to U(C) and t Set:Func(Set)U(C)t_{Set}:Func(Set) \to U(C)) such that for every term f:Mor(C)f:Mor(C), s(f)=o(s Set(i(f)))s(f) = o(s_{Set}(i(f))) and t(f)=o(t Set(i(f)))t(f) = o(t_{Set}(i(f))).

With a family of collection of elements

Definition

Given a category CC with a type of objects Ob(C)Ob(C) and for every object a:Ob(C)a:Ob(C) and b:Ob(C)b:Ob(C) a set of morphisms Mor C(a,b)Mor_C(a, b), CC is a concrete category if for every object a:Ob(C)a:Ob(C) there is a set of elements U(a)U(a) and for every object a:Ob(C)a:Ob(C) and b:Ob(C)b:Ob(C), there is an injection i a,b:Mor C(a,b)(U(a)U(b))i_{a,b} \colon Mor_C(a,b) \to (U(a) \to U(b)).

With a functor into Set

Definition

A concrete category is a category CC equipped with a faithful functor

U:CSet U : C \to Set

to the large category Set. We say a category CC is concretizable if and only if it admits a faithful functor U:CSetU: C \to Set.

Remark

Very often it is useful to consider the case where UU is representable by some object c 0Cc_0 \in C, in that UC(c 0,)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:CSet)(C, U: C \to Set) is representably concrete. By definition, the object c 0c_0 is then a separator of the category.

We remark that the existence of a left adjoint FF to U:CSetU: C \to Set implies that UU is representable by F(1)F(1). Conversely, if CC has coproducts or even just copowers, then representability of UU implies that UU has a left adjoint.

Remark

One can also consider concrete categories over any base category XX instead of necessarily over SetSet. This is the approach taken in The Joy of Cats. Then the (small) categories concrete over XX form a 2-category Cat(X)Cat(X).

Examples

  • Any small category CC can be made into a concrete category by setting U(X)U(X) to be the set of all arrows with codomain XX, and U(XY)U(X \to Y) given by composition with XYX \to Y. This can be generalized to any locally small category with a small separator.

The following furnish examples of concrete categories, with the first three representably concrete:

  • C=SetC = Set itself with separator c 0={}c_0 = \{\bullet\} the singleton set.

  • C=TopC = Top with the separator c 0c_0 taken to be the one-point space.

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

  • Insofar as categories such as SetSet, TopTop, Vect kVect_k, etc. admit many separators, these categories may be rendered representably concrete in a variety of ways. Indeed, the category Vect kVect_k may be monadic over SetSet in many different ways. For example, if VV is nn-dimensional, the functor hom(V,):Vect kSet\hom(V, -): Vect_k \to Set is monadic and realizes Vect kVect_k as equivalent to the category of modules over the matrix algebra hom(V,V)\hom(V, V).

  • Any Grothendieck topos is concretizable, but not necessarily (and typically not) representably concretizable. If E=Sh(C,J)E = Sh(C, J) is the category of sheaves on a small site (C,J)(C, J), we have a familiar string of faithful functors

    Sh(C,J)Set C opmonadicSet/C 0Σ C 0Set.Sh(C, J) \hookrightarrow Set^{C^{op}} \stackrel{monadic}{\to} Set/C_0 \stackrel{\Sigma_{C_0}}{\to} Set.

    But if for example EE is the category of sheaves over \mathbb{R}, then no object XX can serve as a single separator of EE, since it cannot detect differences between arrows YZY \stackrel{\to}{\to} Z whenever the support of YY is strictly contained in the support of XX.

  • 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 UU is the usual ‘underlying set’:

  • The category MonMon of monoids and monoid homomorphisms is a concrete category.

  • The category AbAb of abelian groups and abelian group homomorphisms is a concrete category.

  • Given a commutative ring RR, the category RModR Mod of RR-modules and RR-linear maps is a concrete category.

  • Given a commutative ring RR, the category RAlgR Alg of RR-algebras and RR-algebra homomorphisms is a concrete category.

  • The category CRingCRing of commutative rings and commutative ring homomorphisms is a concrete category.

  • The category FieldField of fields and field homomorphisms is a concrete category.

  • The category HeytAlgHeytAlg of Heyting algebras and Heyting algebra homomorphisms is a concrete category.

  • The category FrmFrm of frames and frame homomorphisms is a concrete category.

  • The category ConvConv of convergence spaces and continuous functions is a concrete category.

  • The category TopTop of topological spaces and continuous functions is a concrete category.

  • The category MetMet of metric spaces and isometries is a concrete category.

  • The category StrictCatStrictCat 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 UU satisfying U(X)=XU(X) = X

  • The category Set Set_\bot of sets and partial functions is a concrete category when equipped with the functor U(X)=X⨿{*}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 RelRel 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)U(X) = PowerSet(X), and U(XY)U(X \to Y) given by composition of relations (viewing a subset of XX as a relation on XX). This is faithful since for any relation ΦRel(X,Y)\Phi \in Rel(X, Y) we have (x,y)Φ(x,y) \in \Phi iff yΦ{x}y \in \Phi \circ \{x\}.

Non-examples

Properties

Proposition

Every small category CC is concretizable (since it fully and faithfully embeds in the concrete category Set C opSet^{C^{op}}).

Proposition

If CC is concretizable, so is C opC^{op}.

Proof

By assumption, there is a faithful functor U op:C opSet opU^{op}: C^{op} \to Set^{op}, and hom(,2):Set opSet\hom(-, \mathbf{2}): Set^{op} \to Set is monadic.

Remark

Of course, since a category CC may possess a separator but no coseparator, it does not follow that C opC^{op} is representably concrete if CC is.

Morphism evaluation and extensionality

Proposition

In any concrete category (C,U:CSet)(C, U:C \to Set), there is an evaluation map

()(()):Hom(a,b)×U(a)U(b)(-)((-))\colon Hom(a,b) \times U(a) \to U(b)

such that for every morphism f:Hom(a,b)f \colon Hom(a,b) and g:Mor C(b,c)g \colon Mor_C(b,c) and every element x:U(a)x:U(a), (gf)(x)=g(f(x))(g \circ f)(x) = g(f(x)) and id A(x)=xid_A(x) = x.

Proof

Because Set is a cartesian closed category, currying the injective function f a,bf_{a,b} of the functor FF in Set means that there is an evaluation map ()(()):Hom(a,b)×U(a)U(b)(-)((-))\colon Hom(a,b) \times U(a) \to U(b) which satisfies the above axioms.

Proposition

In any concrete category (C,U:CSet)(C, U:C \to Set), the morphisms satisfy function extensionality with respect to the evaluation map: for all morphisms f:Hom(A,B)f \colon Hom(A,B) and g:Hom(A,B)g \colon Hom(A,B), if f(x)=g(x)f(x) = g(x) for all elements x:El(A)x \colon El(A), then f=gf = g.

Proof

Since Set is a well-pointed category, and there is a bijection between 𝟙U(a)\mathbb{1} \to U(a) and U(a)U(a), function extensionality follows.

Relationship with well-pointedness

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 FieldField 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.

Characterization

Theorem

A finitely complete category is concretizable, i.e., admits a faithful functor to SetSet, 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:CDF: C\to D is a faithful functor, then it is injective on equivalence classes of regular subobjects. For suppose that m:axm\colon a \to x is the equalizer of f,g:xyf,g\colon x\rightrightarrows y, and n:bxn\colon b\to x is the equalizer of h,k:xzh,k\colon x\rightrightarrows z. If F(a)F(b)F(a) \cong F(b) as subobjects of F(x)F(x), then since fm=gmf m = g m and so F(f)F(m)=F(g)F(m)F(f)\circ F(m) = F(g)\circ F(m), we must also have F(f)F(n)=F(g)F(n)F(f)\circ F(n) = F(g)\circ F(n); hence (since FF is faithful) fn=gnf n = g n, so that nn factors through mm. Conversely, nn factors through mm, so we have aba\cong b as subobjects of xx. Since SetSet is regularly well-powered, it follows that any category admitting a faithful functor to SetSet 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.

Remark

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 TopTop 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

References

  • John Isbell, Two set-theoretical theorems in categories, Fund. Math 53 (1963)
  • 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 May 18, 2022 at 23:42:29. See the history of this page for a list of all contributions to it.