Abstract: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.
| Comments: | 18 pages, comments welcome! |
| Subjects: | Category Theory (math.CT); Computational Complexity (cs.CC) |
| Cite as: | arXiv:2605.24240 [math.CT] |
| (or arXiv:2605.24240v1 [math.CT] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24240 arXiv-issued DOI via DataCite (pending registration) |
From: Emilio Minichiello [view email]
[v1]
Fri, 22 May 2026 21:32:24 UTC (23 KB)
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