关于函子的有理对
On Rational Pairings of Functors
摘要：
在环$R$上的余代数$C$的理论中，有理函子将代数$C^*$（具有卷积积）上的模范畴与$C$上的余模范畴联系起来。它是基于代数$C^*$与由求值映射$\ev:C^*\ot_R C\to R$提供的余代数$C$的配对。我们通过定义任何范畴$\A$上的内函子$T$和$G$之间的{ \em配对}作为映射来推广这种情况，在$a,b\in \A$，$$β_{a,b}:\A(a, G(b)) \to \A(T(a),b),$$中是自然的，如果这些都是内射的，我们称之为{ \em有理数}。在$\bT=(T,m_T,e_T)$是单子并且$\bG=(G,δ_G,\ve_G)$是$\A$上的共单子的情况下，对$\bT$和$\bG$之间的配对施加附加的相容性条件。如果这样的配对是给定的，是合理的，和$\bT$有一个右伴随单子$\bT^\di$，我们构造了一个{ \em有理函子}作为一个幂等余单子上的$\bT$ -模$\A_{\rT}$的函子的一部分，推广了有理函子的重要性质的余代数。作为一种特殊情况，我们考虑monoidal范畴上的配对。
英文摘要：
In the theory of coalgebras $C$ over a ring $R$, the rational functor relates the category of modules over the algebra $C^*$ (with convolution product) with the category of comodules over $C$. It is based on the pairing of the algebra $C^*$ with the coalgebra $C$ provided by the evaluation map $\ev:C^*\ot_R C\to R$. We generalise this situation by defining a {\em pairing} between endofunctors $T$ and $G$ on any category $\A$ as a map, natural in $a,b\in \A$, $$β_{a,b}:\A(a, G(b)) \to \A(T(a),b),$$ and we call it {\em rational} if these all are injective. In case $\bT=(T,m_T,e_T)$ is a monad and $\bG=(G,δ_G,\ve_G)$ is a comonad on $\A$, additional compatibility conditions are imposed on a pairing between $\bT$ and $\bG$. If such a pairing is given and is rational, and $\bT$ has a right adjoint monad $\bT^\di$, we construct a {\em rational functor} as the functor-part of an idempotent comonad on the $\bT$-modules $\A_{\rT}$ which generalises the crucial properties of the rational functor for coalgebras. As a special case we consider pairings on monoidal categories.
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