Concrete category

A concrete category is a category in mathematics together with a faithful functor from her in the category of sets ( " forgetful "). A category to which there is such a forgetful, ie konkretisierbare category. By virtue of this Vergissfunktors can imagine the objects of the abstract se category as quantities with an additional mathematical structure, where the morphisms are exactly the compatible with this structure mappings between the corresponding quantities.

Motivation

Many important categories are already given in the following form:

• Objects have a lot of underlying or are quantities with additional structures
• Morphisms are compatible with the additional structure maps between these quantities,
• The composition of morphisms is simply the concatenation Running pictures,
• Identitätsmorphismus of an object is given by the identity map.

Due to the obvious functor in the category of sets such categories are determinable for. This is particularly the case for the category Top of topological spaces ( with continuous maps as morphisms ) to the category Grp of groups and trivially also for the category Set of sets itself If you can speak in this manner of elements of an object, allowing this example simple and clear definitions of terms such as core and image of a morphism and the proof procedures of the diagram chase. An important statement in this direction provides about the embedding theorem of Mitchell

Definition

Let X be a category, called the base category. A concrete category over X is a pair ( C, U ) from a category C and a faithful functor U: C → X in the base category. A category C is called determinable for over X if there is a concrete category over X (C, U), that is a faithful functor U: C → X is.

If X is the category Set of sets and figures, ie (C, U) also plain concrete category C and determinable for. Some authors refer to a specific category as a construct.

The functor U is also called forgetful that every object of C of its underlying X object (or underlying amount ) and any morphism in C assigns its underlying X - morphism (or underlying figure). [ Acc 1 ]

Comments

• The investigation relative concreteness (ie with a different base category as a set ) is especially common in the theory of topoi and you can view for example models of a theory with N species as objects of a concrete category over SETn. Subsequently, however, consistently set is considered as a base category.
• 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 have quite a number of different faithful functors on the set, and thus there are various concrete categories (C, U) for a given category C. In practice, however, is usually clear what is meant forgetful, and one speaks then reduced from the "concrete category C". For example, " concrete category Set, strictly speaking, the concrete category ( Set I), where I: Set → Set is Identitätsfunktor.
• The requirement that U is faithful, means that U different morphisms between two given objects assigns different pictures. However, it may well be that different objects U assigns the same amount. In the case of U assigns quite different morphisms (with different sources and / or targets) the same figure to. [Acc 2] As an example, think in topological spaces to the same amount once again provided with the lump topology with the discrete topology.

Examples

• Any small category is determinable for: first is the set of all morphisms For an object after. For a morphism, one can define the mapping by. The fact that in this way a faithful functor U: C → Set is defined, can be verified directly.
• If a group, we can define this, a category C with only one object and. Operates faithfully on a set, then ( C, U ) with the given functor and by a specific category.
• A partially ordered set can be interpreted as category whose objects are the elements of and with an arrow if and only if. By defining and each arrow assigns the appropriate inclusion mapping, you get a concrete category.
• Together with the contravariant Potenzmengenfunktor Setop → Set, the amount of the power set of each and every picture the picture assigns, is Setop to a specific category.
• From the above example it follows that the dual category is a category konkretisierbaren also determinable for With concrete is [acc 3].
• In the category Ban of Banach spaces and linear contractions you do not use the most " obvious " forgetful, but assigns a space just its (closed ) unit ball in order to make this a Rechtsadjunktion. [Acc 4]

Counter-examples

• The homotopy category htop whose objects are topological spaces and whose morphisms are the homotopy classes of continuous maps, is a non konkretisierbare category. Although the objects are already amounts ( with additional structure), but the morphisms are just no pictures between them, but equivalence classes of such maps. The first evidence that this deficiency is not remedied, so that there is absolutely no faithful functor from htop on the set, written by Peter Freyd.
• The category of small categories with natural equivalence classes of functors as morphisms is also not determinable for.