关于索引动作
On indexed actions
摘要：
我们提出了一些与$\Cat$ -索引范畴的左、右和双作用有关的定律：通过定义$(A\comp M)x = Mx^{Ax}$，人们得到$\Set^{X\op}$对$(\Set^X)\op$的双闭幺半群作用，而$\B X$和$\Cat/X$通过指数作用于它们的对立面（部分地）;两个包含$(\B X,\B X)\to (\Set^{X\op},\Set^X) \to (\Cat/X,\Cat/X)$都保持了（carnival）幺半群结构和作用，同样的情况也适用于沿着函子的替换。这些强态射的强索引monoidal行动实际上有更广泛的应用，特别是，取代PDMATHAMT与任何（共）完全对称monoidal封闭范畴$\V$，我们认为对索引范畴$(\V_0^{X\op},\V_0^X ; X\in\Cat)$与双闭索引monoidal行动的每一个对另一个和它的形式关系的双作用和常数行动。一些由此产生的定律也在双闭双范畴的一个片段中成立（有一个支持对称么半群范畴的对象），并在第二部分中作为发展某些抽象范畴论的基础。最后，我们将$\Set^{X\op\tm X}$添加到图片中，并给出理解附加语的对称版本。
英文摘要：
We present some laws relating the $\Cat$-indexed categories of left, right and bi-actions: by defining $(A\comp M)x = Mx^{Ax}$ one gets a biclosed monoidal action of $\Set^{X\op}$ on $(\Set^X)\op$, while $\B X$ and $\Cat/X$ act (partially) on their opposites by exponentials; both the inclusions $(\B X,\B X)\to (\Set^{X\op},\Set^X) \to (\Cat/X,\Cat/X)$ preserve the (cartesian) monoidal structures and the actions, and the same holds for substitutions along functors. These strong morphisms of strong indexed monoidal actions have in fact a wider range of applications; in particular, replacing $\Set$ with any (co)complete symmetric monoidal closed category $\V$, we consider the pair of indexed categories $(\V_0^{X\op},\V_0^X ; X\in\Cat)$ with the pair of biclosed indexed monoidal actions of each one on the opposite of the other one and its formal relationships with biactions and constant actions. Some of the resulting laws also hold in a fragment of biclosed bicategory (with an object supporting a symmetric monoidal category) and are taken, in the second part, as the basis for developing some abstract category theory. Finally, we add $\Set^{X\op\tm X}$ to the picture and give a symmetrical version of the comprehension adjunction.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
