Limit (category theory)

In algebra, or more generally the category theory of projective limit ( or inverse limit or just limit ) is a construction with which one can connect in some way belong together different structures. The result of this connection process is mainly determined by mappings between these structures.

Projective Limites for quantities and simple algebraic structures

The following construction defines the limit for quantities or any algebraic structures using Limites (products, leaf objects, difference kernels ) are defined. As an example, groups are treated.

Consider the following situation: Given a semi- ordered set, for each group and for each pair of indices with a group homomorphism

These homomorphisms should be compatible in the sense that it covers'

( " To come from to, you can also take a detour over ").

The projective limit is the set of all families with the property

With component-wise multiplication is defined as a group.

The universal property

The projective limit together with the homomorphisms

Canonical projections has the following universal property:

Projective Limites in any of the categories

Using the concept of projective limit for quantities one can define projective Limites in any of the categories: Are objects Xi of a category C and Übergangsmorphismen fi, j is given as the limit of this projective system characterized by a natural equivalence

Of functors in T; while the limit is on the right side of the previously defined limit term for quantities. The limit defined in this way satisfies the analogous universal property.

For "simple" algebraic structures such as vector spaces, groups or rings true this limit concept with the above-defined set-based match.

However, there are categories in which projective Limites do not exist, for example, the category of finite abelian groups: Let ( Xi, fi, j ) is the projective system

With the projection on the first factors as the transition pictures. For T = Z/2Z

Infinity, that is not the same

For any finite abelian group L.

Examples

• In the category of topological spaces there are Limites: The set-based limit was constructed as a subset of the Cartesian product. Provides you the product with the product topology and the quantity limes with the subspace topology, we obtain the categorical limit. If all Ai compact and Hausdorff, then the projective limit A is also compact and Hausdorff.
• Every compact topological group is projective limit of compact Lie groups.
• The ring Zp of p- adic integers is the projective limit of the residue class rings Ai = Z / pi, where the partially ordered index set I = N is equipped with the natural order and the morphisms are the residue class pictures. The natural topology on Zp is induced by the discrete topology on the Ai product topology.
• For an arbitrary Galois field extension E / K is the Galois group G is the semi-ordered index set (E / K) is isomorphic to the projective limit of the Galois groups G (L / K ), where L runs through all finite and Galois intermediate extensions of E / K, the amount of these intermediate body is the inclusion order, and the morphism for M / L is given by fM, L: G (M / K) → G (L / K), s → s | L (ie, the restriction of an automorphism on the smaller body ). Considering all of G (L / K) as discrete topological groups, then G (E / K) is induced product topology Krull topology is known.

Limites with index categories

As a generalization of the limes for some ordered index sets can be considered Limites for any index categories:

Let I be a small category, C an arbitrary category and X: I → C a functor. Then a limit of X is a representative object for the functor

Here denote the constant functor constT I → C with value T. The Limes is therefore an object L together with a natural equivalence

Of functors in T.

From this natural equivalence is obtained for T = L, the canonical projections ( as the equivalent of idL on the left ).

The natural equivalence is essentially just a compact notation of the universal property: morphisms in a limit object corresponding to compatible systems of morphisms in the individual objects, just as in the special case of partially ordered index sets.

This limit term includes some other universal constructions as special cases:

If the index category, an initial object A, so the limit is equal to X (A).