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, 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，所谓约束满足问题（CSP）的代数方法一直是研究此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（在对数空间归约意义下）。过去几年中，出现了一种更广义的CSP变体——承诺约束满足问题（PCSP），多态性概念及代数方法的核心论点大多被推广至承诺设定。然而，近期研究也表明，来自代数拓扑等其他领域的洞见对PCSP研究具有重要价值。本文为范畴理论研究者提供了研究CSP与PCSP复杂性的切入点。我们证明许多标准CSP概念具有明确且广为人知的范畴对应物。例如，多态性的代数结构可描述为通过右Kan延拓定义的集合函子。我们为代数方法的核心结论提供了纯范畴证明，包括复杂性仅取决于多态性的证明。相较于先前对相同结果的证明，我们的新证明在范畴视角下更为简洁清晰，且具有更广泛的应用范围。我们相信，尤其在PCSP研究中，范畴理论带来的洞见将有助于解决该领域当前面临的若干挑战。