By Brian Day (auth.), Prof. Dr. Gregory M. Kelly (eds.)

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**Extra resources for Category Seminar: Proceedings Sydney Category Theory Seminar 1972/1973**

**Example text**

Subcategory of [c°P,v] which functors as a full strongly c o g e n e r a t i n g This coincides with the left-adjoint f a c t o r i s a t i o n of the c o n j u g a t i o n functor [c°P,v] through a reflection ~ [C,V]°P; followed F ~ [ [FC,C(C,-)], C by an i s o m o r p h i s m - r e f l e c t i n g embedding, denoted: [ c°P,vJ ;- C* [ C,V] op The r e s u l t i n g embedding of C into its c o m p l e t i o n will be denoted by: E ~ E(C): C ~ C*. 1). 6. The c o m p l e t i o n C* can be d e s c r i b e d e x p l i c i t l y as the full subcategory of [c°P,v] of functors F such that ~ [Sc,FC] is an i s o m o r p h ~ m 27 whenever s is a m o r p h i s m in [c°P,v] inverted by conjugation, C* is equivalent to the c a t e g o r y of V-fractions of [c°P,v] As such, with respect to the class of m o r p h i s m s inverted by conjugation.

The second m a i n o b s e r v a t i o n of this section is that the completion process provides a "structural change of V-universe". Consider the case V = S and C = V and let S* be a larger c a r t e s i a n closed category of sets c o n t a i n i n g S and V, and any other categories that we want to regard as "small", as internal category objects. be the c o m p l e t i o n of V with respect to S *. Let W = V* The basic p r o p e r t i e s of V* (verified in [83) are as follows: Property i. Because V °p is a symmetric m o n o i d a l category, the functor category [ v°P,s *] is symmetric m o n o i d a l closed and this structure extends the original structure of V.

The pair has a common right inverse T ( n ~ ) ; that is, it is reflective. Moreover, the coequaliser in B(T) is then the joint c o e q u a l i s e r of the pairs: T(C@TD) Tk ~ T2(C®D) ~ TO T(C®D) T(TC@D) . 3); C ® D = [AB(B(T)(NA,C) @ B(T)(NB,D))-N(A@B) in B(T). this B e c a u s e E: completes the B(T) ~ B ( T ) * p r e s e r v e s and r e f l e c t s colimits, proof. The p r e c e d i n g propositions provide an a l t e r n a t i v e earlier work by Kock [16] and Linton [18]. approach to The use of c o m p l e t i o n leads to a significant r e d u c t i o n in coherence computations.