1.8. Group and Cogroup Objects 21 Comparing Large Sums and Products. If we have a collection {Xi | i ∈ I} of objects of a pointed category C, then we may construct a compar- ison map w : I Xi −→ I Xi just as in Problem 1.46(b). 1.8. Group and Cogroup Objects A group is a set G with a multiplication (which can be thought of as the map μ : G × G → G given by (x, y) → x · y) and, for each element, an inverse (which can be thought of as the map ν : G → G given by x → s−1), which are required to satisfy various properties. The main observation of this section is that all of these properties can be formulated abstractly in terms of diagrams. Let C be a pointed category, and let G ∈ C. Then G is a group object if there are maps μ : G × G → G (multiplication) and ν : G → G (inverse) which satisfy the following properties: (1) (Identity) The following diagram commutes: G (∗,id G ) idG G × G μ G (id G ,∗) idG G. (2) (Inverse) The following diagram commutes: G ∗ (ν,idG) G × G μ G (idG,ν) ∗ G. (3) (Associativity) The following diagram commutes: G × G × G μ×idG idG×μ G × G μ G × G μ G. It is sometimes useful to study objects with are not quite group objects. For example, if we drop the inverse conditions, we obtain a monoid object.

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