We introduce a notion of complexity of diagrams (and, in particular, of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several examples of this new definition in categories of wide common interest such as finite sets, Boolean functions, topological spaces, vector spaces, semilinear and semialgebraic sets, graded algebras, affine and projective varieties and schemes, and modules over polynomial rings. We show that on one hand categorical complexity recovers in several settings classical notions of nonuniform computational complexity (such as circuit complexity), while on the other hand it has features that make it mathematically more natural. We also postulate that studying functor complexity is the categorical analog of classical questions in complexity theory about separating different complexity classes.

中文翻译：

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类别复杂性

我们引入了任意类别中图的复杂性（尤其是对象和态射）的概念，以及配备复杂函数的类别之间的函子复杂性的概念。我们在广泛共同感兴趣的类别中讨论了这个新定义的几个例子，例如有限集、布尔函数、拓扑空间、向量空间、半线性和半代数集、分级代数、仿射和射影变体和方案，以及多项式环上的模块。我们表明，一方面，分类复杂度在几种设置中恢复了非均匀计算复杂度（例如电路复杂度）的经典概念，而另一方面，它具有使其在数学上更自然的特征。