Burritos for the Hungry Mathematician

The advent of fast-casual Mexican-style dining establishments, such as Chipotle and Qdoba, has greatly improved the productivity of research mathematicians and theoretical computer scientists in recent years. Still, many experience confusion upon encountering burritos for the first time. Numerous burrito tutorials (of varying quality) are to be found on the Internet. Some describe a burrito as the image of a crêpe under the action of the new-world functor. But such characterizations merely serve to reindex the confusion contravariantly. Others insist that the only way to really understand burritos is to eat many different kinds of burrito, until the common underlying concept becomes apparent. It has been recently remarked by Yorgey [9] that a burrito can be regarded as an instance of a universally-understood concept, namely, that of monad. It is this characterization that we intend to explicate here. To wit, a burrito is just a strong monad in the symmetric monoidal category of food, what’s the problem? 1 The Category of Food The category of food, Füd, is a full subcategory of the category of stuff, Stf, whose objects are the stuff you can eat. A morphism of this category is known as a recipe. This category inherits the ambient monoidal product: if you can eat a carrot and you can eat a potato, then you can eat a carrot and a potato. The monoidal unit in the category of food is nothing, ε, which is to be found in many graduate students’ refrigerators, typically, tensored with beer. The monoidal category Füd is symmetric: if you can eat a carrot and a potato then you can eat a potato and a carrot. However, it fails to be cartesian. We lack the projections: if we have made the mistake of ordering a whiskey sour, there is generally no way to recover just the whiskey. Nor do we have a diagonal natural transformation, ∆ ∶ id ⟶ −⊗−, giving us two whiskeys for the price of one – alas. For further details on the theory of categorical cookery, see [1].