A limit of a (small) diagram $d : J \to E$ in a complete category $E$ can be thought of as specifying a set of equations involving the objects of $E$. To motivate this intuitively, one can think of each object $d(j)$ as a "variable" and each morphism in $J$ as a "constraint" connecting these variables. If $E$ has an initial object, a natural question arises: does our set of equations have any solution at all? Equivalently, we can ask: is the limit of $d$ initial? In this paper we consider the computational problem that, given finite diagram $d$ in a finitely complete category $E$, asks whether its limit is empty. We construct a fast algorithm (in the sense of parameterized complexity theory) that solves this problem when $E$ is of the form $\mathbf{FinSet}^{J}$ for a finite category $J$ and $d$ is a structured co-decomposition, i.e. a diagram arising from the opposite of the Grothendieck construction of a simple graph.

翻译：在完备范畴$E$中，（小）图表$d : J \to E$的极限可视为指定一组涉及$E$中对象的方程。直观理解，可将每个对象$d(j)$看作"变量"，而$J$中的每个态射看作连接这些变量的"约束"。若$E$存在始对象，自然产生一个问题：这组方程是否有解？等价地，我们可问：$d$的极限是否为始对象？本文考虑以下计算问题：给定有限完备范畴$E$中的有限图表$d$，判断其极限是否为空。当$E$为$\mathbf{FinSet}^{J}$形式（其中$J$为有限范畴），且$d$为结构化余分解（即由简单图之Grothendieck构造的对偶所导出的图表）时，我们构建了一个快速算法（基于参数化复杂度理论）。