The so-called algebraic approach to the constraint satisfaction problem (CSP) has been a prevalent method of the study of complexity of these problems since early 2000's. The core of this approach is the notion of polymorphisms which determine the complexity of the problem (up to log-space reductions). In the past few years, a new, more general version of the CSP emerged, the promise constraint satisfaction problem (PCSP), and the notion of polymorphisms and most of the core theses of the algebraic approach were generalised to the promise setting. Nevertheless, recent work also suggests that insights from other fields are immensely useful in the study of PCSPs including algebraic topology. In this paper, we provide an entry point for category-theorists into the study of complexity of CSPs and PCSPs. We show that many standard CSP notions have clear and well-known categorical counterparts. For example, the algebraic structure of polymorphisms can be described as a set-functor defined as a right Kan extension. We provide purely categorical proofs of core results of the algebraic approach including a proof that the complexity only depends on the polymorphisms. Our new proofs are substantially shorter and, from the categorical perspective, cleaner than previous proofs of the same results. Moreover, as expected, they are applicable more widely. We believe that, in particular in the case of PCSPs, category theory brings insights that can help solve some of the current challenges of the field.

翻译：自21世纪初以来，所谓的约束满足问题（CSP）代数方法一直是研究这类问题复杂性的主流方法。该方法的核心是多态性概念，它决定了问题的复杂性（至多对数空间归约）。近年来，出现了一种更通用的CSP新版本——承诺约束满足问题（PCSP），并且多态性概念以及代数方法的大多数核心论点都被推广到了承诺设置中。然而，最近的研究也表明，包括代数拓扑在内的其他领域的见解在研究PCSP中极为有用。在本文中，我们为范畴论研究者提供了一个进入CSP和PCSP复杂性研究的切入点。我们证明许多标准的CSP概念都有清晰且众所周知的范畴论对应物。例如，多态性的代数结构可以被描述为定义为右Kan扩展的集合函子。我们提供了代数方法核心结果的纯粹范畴论证明，包括一个证明复杂性仅取决于多态性的证明。与先前的相同结果证明相比，我们的新证明明显更简短，并且从范畴论的角度来看更简洁。此外，正如预期的那样，它们适用范围更广。我们相信，特别是在PCSP的情况下，范畴论带来的见解可能有助于解决该领域当前的一些挑战。