J. Ad'amek, H. Herrlich, and G. Strecker. Abstract and concrete categories: the joy of cats. John Wiley & Sons, 1990. Home/Search Document Not in Database Summary Related Articles Check |
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Category Theory in Coq - Carvalho (1998) (Correct)
....previous axiomatization. Finally we present basic concepts and results related to this notions. In the second part of this work we define adjunction situation in Coq. Adjunction was already defined in Coq by Saibi (see for instance [Sai95] however we choose an alternative definition following [AHS90]. After we present some examples of adjunction situation. We end this chapter showing two results concerning adjunctions. First we prove that a functor G has left adjoint iff for any object X the comma category X # G has initial object. Second we prove that the left adjoint of a functor is unique ....
....c2 c1) Prf iso : IsoLaw IsoMor InvIso) g. 26 End IsoDef. We say that an object ObI is initial in a category c if there is a family of morphisms MorI: c2:c) Hom ObI c2) such that for every c2 in c any morphism g: Hom ObI c2) belongs to the source hObI, c2:c. MorI c2)i. See for instance [AHS90] for more details in sources. Section InitialDef. Variable c: Category. Definition InitialLaw: c1:c] f: c2:c) Hom c1 c2) c2:c) g: Hom c1 c2) f c2) S g. Structure Initial: Type: f ObI : c; MorI : c2:c) Hom ObI c2) Prf initial : InitialLaw MorI) g. End InitialDef. Terminal ....
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J. Ad'amek, H. Herrlich, and G. Strecker. Abstract and concrete categories: the joy of cats. John Wiley & Sons, 1990.
Combining Probabilistic Automata: Categorial Characterization - Mateus, Sernadas (Correct)
..... Prop Definition 2.6 The faithful prefunctor F : S R induced by a material precategory S over R is the pair hF 0 ; i. Therefore, at this stage, we look at materialness as the precategory counterpart of concreteness of categories. For details on concrete categories see for instance [AHS90] We are now ready to relate materialness with uniqueness up to isomorphism of limits. Lemma 2.7 Let F : S R be the prefunctor induced by a material precategory S over R. Then: F (f) ffi R F (g) id F (A) f ffi S g = id A . Proof: By hypothesis, F (f) ffi R F (g) id F (A) Thus, by ....
J. Ad'amek, H. Herrlich, and G. Strecker. Abstract and concrete categories: the joy of cats. John Wiley & Sons, 1990.
Precategories for Combining Probabilistic Automata - Mateus, Sernadas, Sernadas (1999) (2 citations) (Correct)
....of arbitrary length. Prop Definition 4.3 The faithful prefunctor F : S R induced by a material precategory S over R is the pair hF 0 ; i. Therefore we look at materialness as the precategory counterpart of concreteness of categories. For details on concrete categories see for instance [AHS90]. We now state a lemma that will be used several times in the sequel. Lemma 4.4 Let F : S R be the prefunctor induced by a material precategory S over R. Then: F (f) ffi R F (g) id F (A) f ffi S g = id A . Proof: By hypothesis, we have F (f) ffi R F (g) id F (A) Thus, by requirement ....
J. Ad'amek, H. Herrlich, and G. Strecker. Abstract and concrete categories: the joy of cats. John Wiley & Sons, 1990.
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