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In category theory, a natural number object (NNO) is an object endowed with a recursive structure similar to natural numbers. More precisely, in a category E with a terminal object 1 (alternately, a topos), an NNO ''N'' is given by: # a global element ''z'' : 1 → ''N'', and # an arrow ''s'' : ''N'' → ''N'', such that for any object ''A'' of E, global element ''q'' : 1 → ''A'', and arrow ''f'' : ''A'' → ''A'', there exists a unique arrow ''u'' : ''N'' → ''A'' such that: # ''u'' ∘ ''z'' = ''q'', and # ''u'' ∘ ''s'' = ''f'' ∘ ''u''. In other words, the triangle and square in the following diagram commute. The pair (''q'', ''f'') is sometimes called the ''recursion data'' for ''u'', given in the form of a recursive definition: # ⊢ ''u'' (''z'') = ''q'' # ''y'' ∈E ''N'' ⊢ ''u'' (''s'' ''y'') = ''f'' (''u'' (''y'')) NNOs are defined up to isomorphism. Every NNO is an initial object of the category of diagrams of the form : If the arrow ''u'' as defined above merely has to exist, ''i.e.'' uniqueness is not required, then ''N'' is called a ''weak'' NNO. If a cartesian closed category has weak NNOs, then every slice of it also has a weak NNO. NNOs in CCCs or topoi are sometimes defined in the following equivalent way (due to Lawvere): for every pair of arrows ''g'' : ''A'' → ''B'' and ''f'' : ''B'' → ''B'', there is a unique ''h'' : ''N'' × ''A'' → ''B'' such that the squares in the following diagram commute. This same construction defines weak NNOs in cartesian categories that are not cartesian closed. NNOs can be used for non-standard models of type theory in a way analogous to non-standard models of analysis. Such categories (or topoi) tend to have "infinitely many" non-standard natural numbers. (Like always, there are simple ways to get non-standard NNOs; for example, if ''z'' = ''s z'', in which case the category or topos E is trivial.) Freyd showed that ''z'' and ''s'' form a coproduct diagram for NNOs; also, !''N'' : ''N'' → 1 is a coequalizer of ''s'' and 1''N'', ''i.e.'', every pair of global elements of ''N'' are connected by means of ''s''; furthermore, this pair of facts characterize all NNOs. ==See also== * Peano's axioms of arithmetic 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Natural number object」の詳細全文を読む スポンサード リンク
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