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Friday, April 25, 2014

Powers and copowers

Let A be a category, enriched in a closed category V.
(V closed) means, in particular, that (for all X,YV there is an internal hom [X,Y]V, satisfying certain axioms).
(A enriched in V) implies that (for all A,BA there is a hom object AABV, satisfying certain axioms).
To say that (A admits powers and copowers relative to V) means precisely that,
(for all XV and A,BA there are isomorphisms in V as shown below, V-natural in all their variables).
Shown below the isomorphisms are (the units and counits associated with those natural isomorphisms)
(the identifier Eval is overloaded, relying on context to disambiguate occurrences of it). AAX
The following display shows how (an \objX \xrightarrow{\textstyle \boxed\functionf} \hom \objA \calA \objB in \calV_0) determines (\objA \xrightarrow{\textstyle \boxed{\bar\functionf}} \objX\power\objB and \objX\copower\objA \xrightarrow{\textstyle \boxed{\tilde\functionf}} \objB in \calA_0), and vice versa,
using the units and counits of the isomorphisms (whatever they may be named). \begin{array}{} & \calA_0 & & \mkern4em & && \calV_0 && & \mkern4em & & \calA_0 \\ \\ &&&&&& \objX \\ &&&&& {}\rlap{\mkern-.5em \raise4pt\hbox{$\boxed\pi$}} \swarrow && \searrow \rlap{\mkern-1ex \raise4pt\hbox{$\boxed\iota$}} \\ \objX\power\objB && && \hom {\objX\power\objB} \calA \objB && \Bigg\downarrow \rlap{\mkern-13mu\boxed\functionf} && \hom \objA \calA {\objX\copower\objA} &&& & \objX\copower\objA \\ \llap{\functionf\power\objB} \Bigg\uparrow & \nwarrow \rlap{\mkern-1ex \raise4pt\hbox{$\boxed{\bar\functionf}$}} & && & \llap{\hom {\bar\functionf} \calA \functionB} \searrow && \swarrow \rlap{\hom \objA \calA {\tilde\functionf}} & && & {}\rlap{\mkern-.5em \raise4pt\hbox{$\boxed{\tilde\functionf}$}} \swarrow & \Bigg\downarrow \rlap{\functionf\copower\objA} \\ {\hom \objA \calA \objB} \power \objB & \xleftarrow[\textstyle \boxed\Eval]{} & \objA && && \hom \objA \calA \objB && && \objB & \xleftarrow[\textstyle \boxed\Eval]{} & {\hom \objA \calA \objB} \copower \objA \\ \end{array}

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