# Change of Base for Commutative Algebras - Mathematics > Category Theory

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Abstract: In this paper we examine on a pair of adjoint functors $\phi^{\ast},\phi {\ast}$ for a subcategory of the category of crossed modules overcommutative algebras where $\phi ^{\ast}:\mathbf{XMod}$\textbf{-}$%Q ightarrow $ $\mathbf{XMod-}P$, pullback, which enables us to move fromcrossed $Q$-modules to crossed $P$-modules by an algebra morphism $\phi:P ightarrow Q$ and $\phi {\ast}:\mathbf{XMod}$\textbf{-}$P ightarrow $$\mathbf{XMod-}Q$, induced. We note that this adjoint functor pair $\phi^{\ast},\phi {\ast}$ makes $p:\mathbf{XMod} ightarrow $ $k$\textbf{-Alg}intoa bifibred category over $k$\textbf{-Alg}, the category of commutativealgebras, where $p$ is given by $pC,R,\partial=R.$Also, some examples and results on induced crossed modules are given.

Author: ** U. Ege Arslan, Ö. Gürmen**

Source: https://arxiv.org/