## Functors as shadows

The last example in the previous post said that the collection of all algebraic gadgets of a given kind and structure-preserving maps between them forms a category. The example given was the category of rings. It’s also true that *a category itself* is an algebraic gadget with structure (the ability to compose morphisms); a structure-preserving map between categories is called a *functor*. A functor between categories and maps

- objects of to objects of
- morphisms of to morphisms of

such that identities and composition are preserved.

One way of thinking about a functor from the category to the category is as a “copy” or a “shadow” of in For example, recall that graphs and manifolds both give rise to categories. If we let be the graph of a cube, where

and be the plane, where

- objects are points of the plane and
- morphisms are paths in the plane,

then a functor is a two dimensional picture of a cube, a shadow cast by the cube. The functor takes

- each vertex to a point in the plane, and
- each path on the cube to a path in the plane

such that

- composing paths is associative and
- identity paths on the cube map to constant paths in the plane.

These last two requirements imply that the paths in the shadow of the cube are generated by the shadows of each edge.

Illustrated are the images of four functors from the cube to the plane . Figure (a) maps each vertex to a distinct point and each edge to a distinct path. Figure (b) maps the front face of the cube and the back face of the cube to the same square, and maps the edges running from the front to the back to the constant paths at the corners. This is a *degenerate* functor, i.e. a functor that maps multiple vertices to the same point in Figure (c) maps all the points of the cube to the same point and all the edges to the constant path at that point. This functor is *totally degenerate*, since there’s no way to map to a smaller subset of points in

Exercise: define a functor such that figure d) is the image of .

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