By Berestovskii V. N.

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**Extra info for A. D. Alexandrovs length manifolds with one-sided bounded curvature**

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D , f + g = (f inl ) r (g inr ) : A + C ! B + D f g = (exl f ) (exr g) : A C ! B D . g inr (gb) . The mappings + and are bifunctors: id + id = id and f + g h + j = (f h) + (g j ) , and similarly for . Throughout the text we shall use several properties of product and sum. These are referred to by the hint `product' or `sum'. Here is a list some of these are just the laws presented before. + f f f f g exl g exl g exr g exr f g h exl exr (h exl ) (h exr ) f g h j f g h j f g=h j + = = = = = = = = = exl f f +g f rg f +g f rg f rg h inl inl inr inr f exr g g (f g) (f h) id h (f h) (g j ) (f h) (g j ) f =h^g =j (inl inl r inr r h) (inr h) f +g hrj f +g h+j f rg = hrj = = = = = = = = = f inl f g inr g (f h) r (g h) id h (f h) r (g j ) (f h) + (g j ) f =h^g =j Exercise: identify the laws that we've seen already, and prove the others.

The declaration that a category is the default category means that it is this category, rather than another one, that should be mentioned whenever there is an ambiguity. For example, when A is declared the default category, and several other (auxiliary) categories are discussed in the same context (in particular categories built upon A ), then a formula like f : A ! A B , and `an object' really means `an object in A '. 2a Iso, epic, and monic All of the following de nitions are relative to a category, the default one, which we don't mention explicitly to simplify the formulas.

Then category D , built upon A , is de ned as follows its objects are called cocones for D . DD DA Df- DB DD DA A A A C C C A C B A AC A C AC AC AUCW ? S @ ; S @ S ; @ S; @;S ;@ S ; @S ; @S ; @S ? ; R @ w? S - ::: ::: x C A cocone for D is: a family A: DA ! D B . This condition is called `commutativity of the triangles'. Using naturality and constant functors there is a technically simpler de nition of a cocone. De ne C to be the constant functor, C x = C for each object x , and C f = id C for each morphism f .

### A. D. Alexandrovs length manifolds with one-sided bounded curvature by Berestovskii V. N.

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