## A general notion of sketch

### John Power – 29 June 2022

This is research from about 25 years ago, joint with Yoshiki Kinoshita and Makoto Takeyama. There has long been a fine body of research about sketches, providing a graphical notion of presentation, contrasting with logical or algebraic such notions. It has traditionally been based on limits and colimits, e.g., finite product sketches, finite limit sketches and geometric sketches. But what about monoidal examples? or binders? I wanted to include the following: given any finitary monad T, equivalently algebraic structure or Lawvere Cat-theory, on Cat or variants, define and develop a notion of T-sketch. So I shall discuss a general setting. In particular, we can define T-sketch and T-model for a finitary monad T on an lfp category C, then prove that every T-sketch has a generic T-model.

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