### Representation Theory for Simplicial Sets

see wiki.

The Yoneda Embedding (see page here)

Assume a category C where the objects are sets of points and the morphisms are total functions. |

allows us to represent categories in set.

The opposite category to this This corresponds to face maps and degeneracy maps in simplicial sets. |

Adding an additional point (origin) allows each point to be represented as a vector. Then the morphism is represented by a matrix. |

More about representation theory for simplicial sets on page here.

Can we use a category theory representable functor, that is an object with a homset to every other object? Would that give a vector space as above? |

### Cayley's theorem

A simpler situation is representations of finite groups.

Cayley's theorem - every group G is isomorphic to a permutation group of the permutations of the underlying set.

As an example here is a simple cyclic group defined by a Cayley graph. What permutation group is it isomorphic to? |

See page about Homotopy Groups for algorithm to convert Cayley graph to group.

### Groupoids

If permutations are conditional, for example quantifier in predicate logic, see groupoid page.