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In category theory, a branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object (also called terminal element): T is terminal if for every object X in C there exists exactly one morphism X → T.
The terminal object is the terminal category or trivial category 1 with a single object and morphism. [2] The category Cat is itself a large category, and therefore not an object of itself. In order to avoid problems analogous to Russell's paradox one cannot form the “category of all categories”.
The typical diagram of the definition of a universal morphism. In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently from the method chosen for constructing them.
Examples of limits and colimits in Top include: The empty set (considered as a topological space) is the initial object of Top; any singleton topological space is a terminal object. There are thus no zero objects in Top. The product in Top is given by the product topology on the Cartesian product.
The Riemann hypothesis catastrophe thought experiment provides one example of instrumental convergence. Marvin Minsky, the co-founder of MIT's AI laboratory, suggested that an artificial intelligence designed to solve the Riemann hypothesis might decide to take over all of Earth's resources to build supercomputers to help achieve its goal. [2]
Another example: An empty product (that is, is the empty set) is the same as a terminal object, and some categories, such as the category of infinite groups, do not have a terminal object: given any infinite group there are infinitely many morphisms , so cannot be terminal.
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Figure 1: Example two-port network with symbol definitions. Notice the port condition is satisfied: the same current flows into each port as leaves that port.. In electronics, a two-port network (a kind of four-terminal network or quadripole) is an electrical network (i.e. a circuit) or device with two pairs of terminals to connect to external circuits.