Examples of using Functor in English and their translations into Chinese
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In Haskell, for a type to be a monad it has to be also a functor and applicative.
如果某个类型是Monad，它一定是Appliacative和Functor。In this new sideways interpretation of Cat there are two ways of getting from object to object: using a functor or using a natural transformation.
在这个从侧面看Cat的新解释里，从一个对象到另一个对象存在两种方法：使用函子或使用自然变换。A monad is a functor M: C→ C{\displaystyle M: C\to C}, along with two morphisms[2] for every object X in C.
Monad是一个函子：M:C->C，并且对C中的每一个对象x以下两个态射：.These maps are"natural" in the following sense: the double dual operation is a functor, and the maps are the components of a natural transformation from the identity functor to the double dual functor. .
这些映射在以下意义上是“自然”的：二重对偶运算是一个函子，这些映射正好构成了从恒等函子到二重对偶函子的自然变换。More abstractly,"G-bundles over X" is a functor in G: given a map H→ G, one gets a map from H-bundles to G-bundles by inducing(as above).
更抽象地，“X上G-丛”是G的一个函子：给定一个映射H→G，诱导一个从H-丛到G-丛的一个映射（见上）。A simplicial object is a presheaf on Δ{\displaystyle\Delta}, that is a contravariant functor from Δ{\displaystyle\Delta} to another category.
单纯对象是Δ{\displaystyle\Delta}上的一个预层，即从Δ{\displaystyle\Delta}到另一个范畴的反变函子。A functor is essentially a transformation between categories, so given categories C and D, a functor F: C→ D{\displaystyle F: C\to D}.
Functor的实质是范畴之间的转换关系，因此对于范畴C和D，有functorF:C→D{\displaystyleF:C\toD}：.If F and G are contravariant functors one speaks of a duality of categories instead.
如果F与G是反变函子我们则说范畴的对偶。Suppose we have two functors and.
假如我们有两个functor.F is a left adjoint of G and both functors are full and faithful.
F是G的一个右伴随且两个函子都完全且忠实。We have two functors.
假如我们有两个functor.G is a right adjoint of F and both functors are full and faithful.
F是G的一个右伴随且两个函子都完全且忠实。F is a left adjoint of G and both functors are full and faithful.
F是G的一个左伴随且两个函子都完全且忠实。Instances of those classes are called functors or function objects.
这种类型的对象称为functor或者functionobject。Java made you use functors, which is even uglier.
Java让你使用算子对象，一种更丑陋的东西。Functors often describe"natural constructions" and natural transformations then describe"natural homomorphisms" between two such constructions.
函子通常用来描述“自然构造”，而自然变换则用来描述两个构造之间的“自然同态”。These isomorphisms are"natural" in the sense that they define a natural transformation between the two involved functors Abop x Abop x Ab-> Ab.
这些同构是“自然”的，因为它们定义了两个函子间的一种自然变换：Abop×Abop×Ab→Ab。Categories, functors and natural transformations were introduced by Samuel Eilenberg and Saunders MacLane in 1945.
范畴、函子和自然变换是由塞缪尔·艾伦伯格和桑德斯·麦克兰恩在1945年引进的。Since all standard algebraic data types are functors, any polymorphic function between such types is a natural transformation.
因为所有的标准代数数据类型都是函子，在这些类型之间的任何一个多态函数都是自然变换。Then Alexander Grothendieck used derived functors of the global section functor, providing a more definitive solution.
然后格罗滕迪克（AlexanderGrothendieck）使用全局截面函子的导函子，给出了更权威的解决。
