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In software engineering, a class diagram [1] in the Unified Modeling Language (UML) is a type of static structure diagram that describes the structure of a system by showing the system's classes, their attributes, operations (or methods), and the relationships among objects. The class diagram is the main building block of object-oriented modeling.
The definition of limits is general enough to subsume several constructions useful in practical settings. In the following we will consider the limit (L, φ) of a diagram F : J → C. Terminal objects. If J is the empty category there is only one diagram of shape J: the empty one (similar to the empty function in set theory).
A diagram is a partial graphic representation of a system's model. The set of diagrams need not completely cover the model and deleting a diagram does not change the model. The model may also contain documentation that drives the model elements and diagrams (such as written use cases). UML diagrams represent two different views of a system ...
A cone with vertex N of a diagram D : J → C is a morphism from the constant diagram Δ(N) to D. The constant diagram is the diagram which sends every object of J to an object N of C and every morphism to the identity morphism on N. The limit of a diagram D is a universal cone to D. That is, a cone through which all other cones uniquely factor.
Since the empty category is vacuously a discrete category, a terminal object can be thought of as an empty product (a product is indeed the limit of the discrete diagram {X i}, in general). Dually, an initial object is a colimit of the empty diagram 0 → C and can be thought of as an empty coproduct or categorical sum.
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.
which sends any space X to the diagram which consists of X everywhere (and the identity of X as maps between them). In (ordinary) category theory, the right adjoint to this functor is the limit. The homotopy limit is defined by altering this situation: it is the right adjoint to :
A sample UML class and sequence diagram for the Decorator design pattern. [7] In the above UML class diagram, the abstract Decorator class maintains a reference (component) to the decorated object (Component) and forwards all requests to it (component.operation()). This makes Decorator transparent (invisible) to clients of Component.