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Monday, December 7, 2020

**Abstract:** Day convolution is a very useful tool in many places, and at its most basic level endows a hom between two monoidal objects in some 2-categorical gadgets with a monoidal structure of its own subject to some conditions. We review Day convolution in 1-category theory, make some tiny observations, and then observe what happens in a formal category theoretic setting. Afterwards, we move on to the (infinity, 1)-categorical case and take a quick peek at what happens for a very certain class of double infinity-categories. We'll see some applications including operads and enriched categories in the Gepner-Haugseng/Hinich sense. Afterwards if there is time we will look at some more applications and end on a few questions. Please email vb8 at illinois dot edu for the zoom details.