MATH.16:4.1 - Express the needed use as maps and equations

Describe what a user of the mathematical object will put in and obtain. Introduce symbols after those meanings are clear. A function f:X -> A takes an input in X and returns one element of A. Its direction matters: receiving an A and returning an X is another operation.

Ask what information completely determines the proposed result. If later use needs an extra choice, include that choice in the input. If the supplied data should suffice, require the resulting map to be unique.

The following contrasts help select a construction; they are examples of different mapping requirements:

Needed use	Maps and question to formulate
Carry two components together and recover each one.	Seek maps from the new object to both component objects. What combined map is determined by supplying the two components?
Carry either kind of input, then process it according to that kind.	Seek maps from each input object into the new object. Do the two processing rules determine one rule on the combined object?
Combine components that must agree about a common quantity.	Express both accounts of that quantity in one codomain and require equality.
Identify inputs while retaining selected answers.	Which maps give the same answer on identified inputs, and therefore can operate on classes? MATH.2 supplies that quotient construction.
Interpret everything built from generating operations.	Which assignment to generators extends to a map preserving the operations? MATH.5 supplies the extension and its uniqueness.

Choose the mathematical setting along with the maps. In the main route, the objects are sets and the permitted maps are all functions between them. A category specifies a setting through its objects, maps, identity maps and associative composition.

Additional structure – groups and topology. For groups, choose homomorphisms preserving the group operation; for topological spaces, choose continuous functions. These choices change what must be constructed and proved.