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By Julia Goedecke

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Definition: a) An adjunction where G is full and faithful is called a reflection. b) A reflective subcategory is a full subcategory D of C for which the inclusion functor D ÝÑ C has a left adjoint. Examples: a) We have already seen that AbGp is reflective in Gp. Given a group G, the commutator subgroup rG, Gs has the property that G④rG, Gs is abelian and any homomorphism G ÝÑ A with A abelian factors uniquely through G ÝÑ G④rG, Gs. b) Let C denote the full subcategory of AbGp whose objects are torsion groups (those in which every element has finite order).

Definition: We say ♣F n steps. ✪ Gq has monadic length n if this process produces an equivalence after Examples: a) The forgetful functor G : Top ÝÑ Set has a left adjoint D, but has monadic length ✽, since GD ✏ 1Set , η ✏ µ ✏ 11Set and so all categories in the monadic tower are isomorphic to Set. b) An equivalence of categories has monadic length 0, and a monadic adjunction has monadic length 1. c) The composite adjunction of torsion-free abelian groups in sets from Example 33c) above has monadic length 2.

As any mono is normal, it is in particular regular monic. So if f is a mono and an epi, it is a regular mono and an epi and so an iso (Proposition 8 in Section 2C). 41 Lemma: (“Preadditive equalisers via kernels”) Let A be preadditive. Then the pair A f g ,P,P B has an equaliser iff the kernel of f ✁ g exists, and then they coincide. Proof. The equaliser of f and g and the kernel of f given h : C ÝÑ A, we have f h ✏ gh ô ♣f ✁ g qh ✏ 0. ✁ g have the same universal property: Notice that in general normal ñ regular ñ strong ñ mono.

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Category Theory [Lecture notes] by Julia Goedecke

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