Category Seminar: Proceedings Sydney Category Theory Seminar by Brian Day (auth.), Prof. Dr. Gregory M. Kelly (eds.)

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

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X, y E ObJ A} of all V-valued functors on A. ~kmbeddin~ of promonoidal catesories. First consider a given small promonoidal category over V. Let F = [A,V] be the convolution of A A with V. ) That is, F is monoidal biclosed with respect to the following operations : F , G = fAA'FA @ GA' @ P(AA'-) F \ G = f [P(A-A') ® FA, GA'] AA' ! F = f fP(-AA') ® FA, GA'I AA' The detailed description of this structure is given in Let E : A B c F be a small monoidal subcategory of F -~ t B,V] denote the evaluation embedding.

Remark. 4) and K(AD) in C in C* to the exponentials ED/ENA and ENA\ED respectively. 30 §4 Let that is, T = (T,~,q) a monad where ( T , T , T °) w i t h r e s p e c t natural MONADS be a m o n o i d a l T: monad on a m o n o i d a l B ~ B has a m o n o i d a l to w h i c h p: T 2 ~ T a n d functor q: category B; structure i ~ T are m o n o i d a l transformations. Let egory MONOIDAL B(T) exists denote over the the b a s e category category B(T)(CD) ~ of T - a l g e b r a s V when B(CD) over V has t h e ....

Moreover, c o n s i d e r e d in §5 exists on [~°P,B] admits the i n t e r n a l structure and the e m b e d d i n g [~°P,BIz C [~°P,B 1 has a left adjoint then this a d j u n c t i o n admits m o n o i d a l e n r i c ~ e n t the r e f l e c t i o n theorem of §i. 1, Z may be r e p l a c e d by an~ class of m o r p h i s m s in [A°P,s] w h i c h defines Cat as its class of orthogonal objects. Remark. The general q u e s t i o n of the e x i s t e n c e of a left-adj- oint functor to an i n c l u s i o n of the form |A,B] Z C [A,B] studied in some de

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