---
tag: K8P15XTH1M
kind: concept
name: Sketch
msc: [18C30]
sources:
  - url: https://en.wikipedia.org/wiki/Sketch_(mathematics)
same_as:
  wikidata: Q7534724
  wikipedia: Sketch (mathematics)
---

## Summary

A sketch is a small category together with chosen cones and cocones in it.
A model of a sketch is a functor that turns the chosen cones into limits and the chosen cocones into colimits.
Sketches give a way to describe mathematical theories, such as the theory of groups or of categories, by diagrams instead of formulas.
The categories of models of sketches in sets are exactly the accessible categories.

## Explanations

### L1 human

A sketch is a blueprint for a kind of structure.
It names some objects and arrows, and it says that certain objects must be products, pullbacks, sums and so on.
A model is an actual structure in sets that follows the blueprint.
For example, a sketch with an object $X$, an object $P$ declared to be $X \times X$, and an arrow $m \colon P \to X$ has as models the sets with a binary operation.