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Forming the direct limit of this direct system yields the ring of symmetric functions. 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 set 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.
Other objects that can be defined by universal properties include: all free objects, direct products and direct sums, free groups, free lattices, Grothendieck group, completion of a metric space, completion of a ring, Dedekind–MacNeille completion, product topologies, Stone–Čech compactification, tensor products, inverse limit and direct ...
In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces.
An exact replica of Kobe Bryant's 2000 NBA championship ring owned by his parents that was removed from auction in 2013. The ring is currently being auctioned again by Joe Bryant. (Associated Press)
Examples of limits and colimits in Ring include: The ring of integers Z is an initial object in Ring. The zero ring is a terminal object in Ring. The product in Ring is given by the direct product of rings. This is just the cartesian product of the underlying sets with addition and multiplication defined component-wise.
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The limit L of F is called a pullback or a fiber product. It can nicely be visualized as a commutative square: Inverse limits. Let J be a directed set (considered as a small category by adding arrows i → j if and only if i ≥ j) and let F : J op → C be a diagram. The limit of F is called an inverse limit or projective limit.