Concrete category

In mathematics, a concrete category is a category together with a faithful functor from it in the category of sets (“forgetful functor”). A category for which such a forget function exists is called a concretizable category . By virtue of this forget function one can imagine the objects of the abstract category also as sets with an additional mathematical structure , the morphisms being exactly the mappings between the corresponding sets that are compatible with this structure.

motivation

Many important categories are already given in the following form:

• Objects have an underlying set or are sets with additional structures,
• Morphisms are mappings compatible with the additional structure between these sets,
• the composition of morphisms is simply the execution of images one after the other,
• the identity morphism of an object is given by the identical mapping .

Such categories can be made concrete through the obvious functor in the category of sets. This is particularly the case for the Top category of topological spaces (with continuous mappings as morphisms), for the Grp category of groups and, trivially, also for the Set category of the sets themselves. If one can speak of elements of an object in this way, this is possible for example, simple and descriptive definitions of terms such as core and image of a morphism and the method of proving diagram hunting . Mitchell's embedding theorem provides an important statement in this direction .

definition

Be a category, the so-called base category . A concrete category above is a pair of a category and a faithful functor in the base category. A category is called over concretizable if there is a over concrete category , i. H. gives a faithful functor . ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle (C,U)}$ ${\displaystyle C}$ ${\displaystyle U\colon C\to X}$ ${\displaystyle C}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle (C,U)}$ ${\displaystyle U\colon C\to X}$

If the category is a set of quantities and images, it simply means a concrete category and can be made concrete . Some authors also refer to a specific category as a construct . ${\displaystyle X}$ ${\displaystyle (C,U)}$ ${\displaystyle C}$

The functor is also referred to as the forget functor , which assigns each object from its underlying - object (or underlying set ) and each morphism in its underlying - morphism (or underlying mapping ). ${\displaystyle U}$ ${\displaystyle C}$ ${\displaystyle X}$ ${\displaystyle C}$ ${\displaystyle X}$

Remarks

• The investigation of relative concreteness (i.e. with a different basic category than ) is particularly common in the theory of topoi and one can, for example, overlook models of a theory with sorts as objects of a concrete category . However, the following is considered to be the basic category throughout . ${\displaystyle {\mathbf {Set} }}$ ${\displaystyle N}$ ${\displaystyle {\mathbf {Set} }^{N}}$ ${\displaystyle {\mathbf {Set} }}$
• Contrary to what one might assume by intuition, concreteness is not a property that either belongs to a category or not. Rather, one and the same category can have several different faithful functors and thus different concrete categories exist for a given category . In practice, however, it is usually clear which forgetful functor is meant, and one speaks in short of the “concrete category ”. For example, “the concrete category ” is, strictly speaking, the concrete category , where is the identity function. ${\displaystyle {\mathbf {Set} }}$ ${\displaystyle (C,U)}$ ${\displaystyle C}$ ${\displaystyle C}$ ${\displaystyle {\mathbf {Set} }}$ ${\displaystyle ({\mathbf {Set} },I)}$ ${\displaystyle I\colon {\mathbf {Set} }\to {\mathbf {Set} }}$
• The premise that is true means that different morphisms between two given objects associate different mappings. However, it may well be that different objects are allocated the same amount. In this case, assigns the same mapping to different morphisms (with different sources and / or destinations). As an example, think of the same set of topological spaces that is provided with the cluster topology and the discrete topology . ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle U}$

Examples

• Every small category can be concretized: For an object, let the set of all morphisms after . For a morphism , the mapping can be defined by. That a true functor U : C → Set is defined in this way can be verified immediately. ${\displaystyle X\in \operatorname {Ob} (C)}$ ${\displaystyle U(X)}$ ${\displaystyle X}$ ${\displaystyle f\colon X\to Y}$ ${\displaystyle U(f)\colon U(X)\to U(Y)}$ ${\displaystyle g\mapsto f\circ g}$
• If there is a group, you can define a category C with just a single object and . If operating faithfully on a set , then ( C , U ) with the functor given by and is a concrete category. ${\displaystyle G}$ ${\displaystyle \ast }$ ${\displaystyle \operatorname {Mor} (\ast ,\ast ):=G}$ ${\displaystyle G}$ ${\displaystyle M}$ ${\displaystyle U(\ast )=M}$ ${\displaystyle U(g)\colon m\mapsto g\cdot m}$
• A partially ordered set can be understood as a category whose objects are the elements of and with an arrow if and only if . By defining and assigning the identical image to each arrow , a specific category is obtained. ${\displaystyle (P,{\leq })}$ ${\displaystyle P}$ ${\displaystyle x\to y}$ ${\displaystyle x\leq y}$ ${\displaystyle U(x):=\emptyset }$ ${\displaystyle \emptyset }$
• Together with the contravariant Potenzmengenfunktor Set ^op → Set which lots of the power set and each picture the picture , assigns, is set ^op to a specific category. ${\displaystyle A}$ ${\displaystyle \operatorname {Pot} (A)}$ ${\displaystyle f\colon B\to A}$ ${\displaystyle \operatorname {Pot} (A)\to \operatorname {Pot} (B)}$ ${\displaystyle S\mapsto f^{-1}(S)=\{x\in B\mid f(x)\in S\}}$
• From the above example it follows that the dual category can also be concretized for a category that can be concretized: With is also concrete. ${\displaystyle (C,U)}$ ${\displaystyle (C^{\operatorname {op} },\operatorname {Pot} \circ U^{\operatorname {op} })}$
• With the category Ban of the Banach spaces and linear contractions , one usually does not use the “obvious” forget function, but only assigns its (closed) unit sphere to a room in order to turn it into a right adjunct .

Counterexamples

• The homotopy category hTop , whose objects are topological spaces and whose morphisms are the homotopy classes of continuous mappings, is a category that cannot be concretized. The objects are already sets (with an additional structure), but the morphisms are not mappings between these, but rather equivalence classes of such mappings. The first proof that this deficiency cannot be remedied, that there is no faithful functor from hTop to Set , comes from Peter Freyd .
• The category of small categories with natural equivalence classes of functors as morphisms cannot be concretized either.

literature

• Jiří Adámek, Horst Herrlich, George E. Strecker: Abstract and Concrete Categories: The Joy of Cats . John Wiley & Sons, New York 1990, ISBN 0-471-60922-6 (English, katmat.math.uni-bremen.de [PDF; accessed October 12, 2010]).
• Jiří Rosický: Concrete categories and infinitary languages . In: Journal of Pure and Applied Algebra . tape 22 , no. 3 , 1981, p. 309-339 , doi : 10.1016 / 0022-4049 (81) 90105-5 (English).

Individual evidence

1. ↑ Saunders MacLane , Garrett Birkhoff : Algebra . 3. Edition. AMS, Chelsea 1999, ISBN 978-0-8218-1646-2 (English).
2. ^ Peter Freyd: Homotopy is not concrete . In: Franklin Peterson (Ed.): The Steenrod Algebra and its Applications: A conference to celebrate NE Steenrod's 60th birthday (=  Springer Lecture Notes in Mathematics . No. 168 ). Springer, 1970, ISBN 3-540-05300-X (English, Reprints in Theory and Applications of Categories . [Accessed October 11, 2010]).
3. ^ Peter Freyd: On the concrete of certain categories . In: Symposia Mathematica . tape IV . Academic Press, London 1970, pp. 431-456 (English).

Adámek, Herrlich, Strecker: Abstract and Concrete Categories: the joy of cats .

1. ↑ p. 61
2. ↑ cf. P. 63
3. ↑ p. 62, (4)
4. ↑ p. 62, (3)