# Category Theory (subject)

## Overview and reach

Quite often in maths we have objects or spaces, like groups or topologies and functions between them, like group homomorphisms or continuous maps where given 3 such objects:

• [ilmath]A[/ilmath], [ilmath]B[/ilmath], [ilmath]C[/ilmath] which are say, groups or topologies, and given
• the maps [ilmath]f:A\rightarrow B[/ilmath] and [ilmath]g:B\rightarrow C[/ilmath] which are say, group homomorphisms or continuous maps

Then the map:

• [ilmath]g\circ f:A\rightarrow C[/ilmath] is also a group homomorphism, or continuous map.

Notice also that each space has an "identity map" a map that maps it to iteself. Which is either a group homomorphism, or a continuous map using the terms of this example

Because this structure occurs so frequently it is fit for study and abstraction.

Category theory is the study of objects linked by arrows, where arrows compose. In fact that is the definition of a category, a thing with objects and arrows, where the arrows compose.

The most familiar category to the reader will be [ilmath]\mathrm{SET} [/ilmath]

Before the reader thinks "ah, but are all arrows not just functions? Don't all arrows compose by function composition" they should see Example of a category whose arrows compose not by function composition.

## Functors

A functor comes in two forms, but both are essentially a "morphism between categories" that preserve arrow composition.

### Example

For example, there is a functor between [ilmath]\mathrm{TOP} [/ilmath] and [ilmath]\mathrm{GROUP} [/ilmath], this is given by the map that sends each topology to its fundamental group

Todo (Subject page refactoring): Flesh out page

Any readers reading this with the above "todo" in place, see the pages linked by this page, they should contain enough terms to explore, if not check out the navbox.