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  "path": "/abs/2605.24240v1",
  "publishedAt": "2026-05-26T00:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Ernst Althaus",
    "Benjamin Merlin Bumpus",
    "James Fairbanks",
    "Emilio Minichiello",
    "Daniel Rosiak"
  ],
  "textContent": "**Authors:** Ernst Althaus, Benjamin Merlin Bumpus, James Fairbanks, Emilio Minichiello, Daniel Rosiak\n\nA 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.",
  "title": "A Parameterized Algorithm for Testing whether the Limit of a Diagram is Empty"
}