We consider, for each exchange matrix B\documentclass[12pt]{minimal}
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\begin{document}$$B$$\end{document}, a category of geometric cluster algebras over B\documentclass[12pt]{minimal}
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\begin{document}$$B$$\end{document} and coefficient specializations between the cluster algebras. The category also depends on an underlying ring R\documentclass[12pt]{minimal}
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\begin{document}$$R$$\end{document}, usually Z,Q\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {Z},\,\mathbb {Q}$$\end{document}, or R\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}$$\end{document}. We broaden the definition of geometric cluster algebras slightly over the usual definition and adjust the definition of coefficient specializations accordingly. If the broader category admits a universal object, the universal object is called the cluster algebra over B\documentclass[12pt]{minimal}
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\begin{document}$$B$$\end{document} with universal geometric coefficients, or the universal geometric cluster algebra over B\documentclass[12pt]{minimal}
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\begin{document}$$B$$\end{document}. Constructing universal geometric coefficients is equivalent to finding an R\documentclass[12pt]{minimal}
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\begin{document}$$R$$\end{document}-basis for B\documentclass[12pt]{minimal}
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\begin{document}$$B$$\end{document} (a “mutation-linear” analog of the usual linear-algebraic notion of a basis). Polyhedral geometry plays a key role, through the mutation fan FB\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {F}}_B$$\end{document}, which we suspect to be an important object beyond its role in constructing universal geometric coefficients. We make the connection between FB\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {F}}_B$$\end{document} and g\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf{g}$$\end{document}-vectors. We construct universal geometric coefficients in rank 2\documentclass[12pt]{minimal}
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\begin{document}$$2$$\end{document} and in finite type and discuss the construction in affine type.
