Direct Limit - Examples

Examples

• A collection of subsets of a set M can be partially ordered by inclusion. If the collection is directed, its direct limit is the union .
• Let I be any directed set with a greatest element m. The direct limit of any corresponding direct system is isomorphic to X_m and the canonical morphism φ_m: X_m → X is an isomorphism.
• Let p be a prime number. Consider the direct system composed of the groups Z/pnZ and the homomorphisms Z/pnZ → Z/pn+1Z which are induced by multiplication by p. The direct limit of this system consists of all the roots of unity of order some power of p, and is called the Prüfer group Z(p∞).
• Let F be a C-valued sheaf on a topological space X. Fix a point x in X. The open neighborhoods of x form a directed poset ordered by inclusion (U ≤ V if and only if U contains V). The corresponding direct system is (F(U), r_U,V) where r is the restriction map. The direct limit of this system is called the stalk of F at x, denoted F_x. For each neighborhood U of x, the canonical morphism F(U) → F_x associates to a section s of F over U an element s_x of the stalk F_x called the germ of s at x.
• Direct limits in the category of topological spaces are given by placing the final topology on the underlying set-theoretic direct limit.
• Inductive limits are linked to projective ones via

• Consider a sequence {A_n, φ_n} where A_n is a C*-algebra and φ_n : A_n → A_{n + 1} is a *-homomorphism. The C*-analog of the direct limit construction gives a C*-algebra satisfying the universal property above.