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 with a type of objects and for every object and a set of morphisms , is a concrete category if for every object there is a set of elements and for every object and , there is an injection .
Given a category with a type of objects and a set of morphisms with source and target functions and , is a concrete category if there is a set of elements with a function with an injection (and functions and ) such that for every term , and .
A concrete category is a category equipped with a faithful functor
to the large category Set. We say a category is concretizable if and only if it admits a faithful functor .
Very often it is useful to consider the case where is representable by some object , in that . For example, this is important for the statement of various concrete dualities induced by dual adjunctions. We say in this case that is representably concrete. By definition, the object is then a separator of the category.
We remark that the existence of a left adjoint to implies that is representable by . Conversely, if has coproducts or even just copowers, then representability of implies that has a left adjoint.
One can also consider concrete categories over any base category instead of necessarily over . This is the approach taken in The Joy of Cats. Then the (small) categories concrete over form a 2-category .
A concrete category is univalent if its underlying category (by forgetting the functor into ) is a univalent category.
The following furnish examples of concrete categories, with the first three representably concrete:
itself with separator the singleton set.
with the separator taken to be the one-point space.
Any monadic functor 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 to be the category of Banach spaces with morphisms those (everywhere-defined) linear transformations with norm bounded (above) by (so for all in the source). Then there are two versions of that one may use: one where (for a Banach space) consists of every vector in , and one where consists of those vectors bounded by (so the closed unit ball in ). The first may seem more obvious at first, but only the second is representable (by a -dimensional Banach space).
Insofar as categories such as , , , etc. admit many separators, these categories may be rendered representably concrete in a variety of ways. Indeed, the category may be monadic over in many different ways. For example, if is -dimensional, the functor is monadic and realizes as equivalent to the category of modules over the matrix algebra .
Any Grothendieck topos is concretizable, but not necessarily (and typically not) representably concretizable. If is the category of sheaves on a small site , we have a familiar string of faithful functors
But if for example is the category of sheaves over , then no object can serve as a single separator of , since it cannot detect differences between arrows whenever the support of is strictly contained in the support of .
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 is the usual ‘underlying set’:
The category of monoids and monoid homomorphisms is a concrete category.
The category of abelian groups and abelian group homomorphisms is a concrete category.
Given a commutative ring , the category of -modules and -linear maps is a concrete category.
Given a commutative ring , the category of -algebras and -algebra homomorphisms is a concrete category.
The category of commutative rings and commutative ring homomorphisms is a concrete category.
The category of fields and field homomorphisms is a concrete category.
The category of Heyting algebras and Heyting algebra homomorphisms is a concrete category.
The category of frames and frame homomorphisms is a concrete category.
The category of convergence spaces and continuous functions is a concrete category.
The category of topological spaces and continuous functions is a concrete category.
The category of metric spaces and isometries is a concrete category.
The category 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 satisfying
The category of sets and partial functions is a concrete category when equipped with the functor that adds a disjoint point, and sends a partial function to the total function whose undefined values are set to the point.
The category of sets and relations has a separator given by the singleton set. Thus, it is a concrete category when equipped with the functor , and given by composition of relations (viewing a subset of as a relation on ). This is faithful since for any relation we have iff .
The classical homotopy category Ho(Top) of topological spaces is not concretizable
The opposite category of commutative rings equipped with the prime spectrum functor 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 of sets and prefunctions is not a concrete category.
Every small category is concretizable (since it fully and faithfully embeds in the concrete category ).
If is concretizable, so is .
By assumption, there is a faithful functor , and is monadic.
Of course, since a category may possess a separator but no coseparator, it does not follow that is representably concrete if is.
In any concrete category , there is an evaluation map
such that for every morphism and and every element , and .
Because Set is a cartesian closed category, currying the injective function of the functor in Set means that there is an evaluation map which satisfies the above axioms.
In any concrete category , the morphisms satisfy function extensionality with respect to the evaluation map: for all morphisms and , if for all elements , then .
Since Set is a well-pointed category, and there is a bijection between and , 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 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 , 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 is a faithful functor, then it is injective on equivalence classes of regular subobjects. For suppose that is the equalizer of , and is the equalizer of . If as subobjects of , then since and so , we must also have ; hence (since is faithful) , so that factors through . Conversely, factors through , so we have as subobjects of . Since is regularly well-powered, it follows that any category admitting a faithful functor to 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 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)
Peter Freyd, On the concreteness of certain categories, in Symposia Mathematica, vol. 4, 1969, pp. 431–456.
Ivan di Liberti, Fosco Loregian “Homotopical algebra is not concrete.” Journal of Homotopy and Related Structures (2017): 1-15.
Last revised on September 18, 2024 at 15:52:30. See the history of this page for a list of all contributions to it.