By C. Menini, F. van Oystaeyen

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An arrow k: h-h0 from h to h0 is deﬁned by an arrow k: C-C0 in K such that (h, h0 , k) be a commutative triangle. Its composite with the arrow k0 : h0 -h00 is the arrow kk0 : h-h00 . In the left ﬁgure, the elements of the ﬁeld of A are seen in the category K; the ﬁeld as a category is drawn on the right. 36 Chapter 1 complete view of the system) only if A is a final object, that is, an object to which any other object is linked by one and only one link. For example, in an ecosystem an animal recognizes only its Umwelt (in the terminology of von Uexku¨ll, 1956) in which he cannot see a predator which is not near enough.

Translated into the categorical setting it has led to the important notion of adjoint functors (Kan, 1958). In this book, we consider several universal problems, in particular the construction of colimits, the complexiﬁcation process, and the construction of concepts. Though it will not always be done explicitly, they can be brought back to the following important situation. Definition. Let p be a functor from K to K0 and A0 an object of K0 . We say that A0 generates a free object A with respect to p if A is an object of K and if there exists an arrow g0 from A0 to pA in K0 satisfying the ‘universal’ condition: (UC) If B is an object of K and f 0 an arrow from A0 to pB, then there exists a unique arrow f from A to B in K such that f 0 ¼ g0 p( f ) (Fig.

In fact, the most usual definition of a category is a collection of objects (say A, B,y) and of sets Hom(A, B) of morphisms between them satisfying some rules. g. the Bernays–von Neumann theory [Bernays, 1937, 1941]), so that the collection of objects can be a proper class, thus allowing one to speak of large categories, and in particular of the large category of sets. This category, denoted by Set, has for objects all the sets and for arrows from A to B the maps from A to B. The composite of a map p from A to B with p0 from B0 to C is deﬁned if and only if B0 ¼ B, and then it is the map from A to C which is their usual composite: it associates to an element a of A the element p0 (p(a)) of C.

### Abstract Algebra - A Comprehensive Trtmt by C. Menini, F. van Oystaeyen

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