# What is algebraic about algebraic effects?

Published as arXiv:1807.05923.

Abstract: This note recapitulates and expands the contents of a tutorial on the mathematical theory of algebraic effects and handlers which I gave at the Dagstuhl seminar 18172 "Algebraic effect handlers go mainstream". It is targeted roughly at the level of a doctoral student with some amount of mathematical training, or at anyone already familiar with algebraic effects and handlers as programming concepts who would like to know what they have to do with algebra. We draw an uninterrupted line of thought between algebra and computational effects. We begin on the mathematical side of things, by reviewing the classic notions of universal algebra: signatures, algebraic theories, and their models. We then generalize and adapt the theory so that it applies to computational effects. In the last step we replace traditional mathematical notation with one that is closer to programming languages.

### Comments

In this paper, you state that "In the opposite category the map turns its direction, and the exponentials become products" - this looks strange to me.

Did you mean exponentials taken in the op category? (in the lucky case they exist) Or did you mean exponentials taken in the original category, and they serve as a product in the op category? I'm sure it's the sum in the original category that serves as a product in the op category (and vice versa).

You are quite right, that bit is misstated. I believe one has to use powers and co-powers instead of exponentials and products, and dualize that accordingly.

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