The monoidal fibered category of Beck modules

Martin Frankland, University of Regina

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In his 1967 thesis, Beck proposed a notion of module over an object in a category C. This provided a natural notion of coefficient module for André-Quillen (co)homology of any algebraic structure, generalizing the original case of commutative rings. Motivated by Quillen homology, I will discuss the tensor product of Beck modules. As one varies the object in C, the categories of Beck modules over different objects assemble into a fibered category over C, sometimes called the tangent category of C. I will describe how this fibered category interacts with the tensor product. Lastly, I will sketch work in progress on the homotopy theory of simplicial Beck modules over simplicial objects, generalizing some work of Quillen on simplicial commutative rings.