By Jiri Adamek

This updated introductory therapy employs type concept to discover the idea of constructions. Its new angle stresses concrete different types and provides a scientific view of factorization constructions, providing a unifying viewpoint on prior paintings and summarizing fresh advancements. a number of examples, starting from common to precise, remove darkness from the textual content. 1990 variation, up-to-date 2004.

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Quasicategories as Objects Show that one may not form the “quasicategory of all quasicategories”. [Hint: Russell’s paradox]. 3M. Small Categories Let A be a category whose object-class is a set. Show that (a) – (d) below are true. ] 18th January 2005 Sec. 3] Categories and functors (a) M or(A) = 47 {hom(A, B) | (A, B) ∈ Ob(A) × Ob(A)} is a set. (b) hom : Ob(A) × Ob(A) → P(M or(A)) is a set. (c) ◦ ⊆ M or(A) × M or(A) × M or(A) is a set. (d) A is a set. 3N. Decompositions of Functors F Let A −−→ B be a functor.

C is easily seen to be a category. Now define functors E2 : A → C and G : C → D by f Ff E2 (A −→ A ) = A −−→ A g g and G(C −→ C ) = F C −→ F C . Then E2 is an embedding, G is an equivalence, and F = E1 ◦ G ◦ E2 . 6 DEFINITION A category A is said to be fully embeddable into B provided that there exists a full embedding A → B, or, equivalently, provided that A is isomorphic to a full subcategory of B. , the construct of unary algebras on two operations. Under an additional set-theoretical hypothesis (the non-existence of measurable cardinals), every construct is fully embeddable into Sgr (or Rel or Alg(1, 1)).

3(2)(3)] there is the forgetful functor (or underlying functor) U : A → Set, where in each case U (A) is the underlying set of A, and U (f ) = f is the underlying function of the morphism f . (4) For any category A and any A-object A, there is the covariant hom-functor hom(A, −) : A → Set, defined by f hom(A,f ) hom(A, −)(B −→ C) = hom(A, B) −−−−−−−→ hom(A, C) where hom(A, f )(g) = f ◦ g. 18th January 2005 Sec. 3] Categories and functors 31 (5) For any category A and any A-object A, there is the contravariant13 hom-functor f hom(−, A) : Aop → Set defined on any Aop -morphism14 B −→ C by hom(f,A) f hom(−, A)(B −→ C) = homA (B, A) −−−−−−→ homA (C, A) with hom(f, A)(g) = g ◦ f , where the composition is the one in A.

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