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A fundamental construction in higher algebra is that of Day convolution. This construction, if it exists, defines an internal hom-object in the oo-category of oo-operads. In this talk, I will describe an explicit condition for the existence of such internal hom-objects for other operad-like structures. Even in the case of oo-operads, our proof is very different from previous constructions. The existence condition we obtain is very similar to the Conduché criterion for detecting exponentiable functors between oo-categories, and in many examples it is both a necessary and sufficient condition. If time permits, I will describe how this construction can be used to prove a new universal property of span categories.

This is joint work with Félix Loubaton and Jaco Ruit.