MATH.16:4 - Solution

Local mantra: start from the use; choose maps and laws; construct the object; derive its uses; revise the changed requirement.

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:

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.

MATH.16:4.2 - State the universal property before choosing an encoding

Turn the use into a requirement by comparing the proposed object with arbitrary permitted ways of supplying or processing its data:

1. Fix the objects already given by the question, together with their maps and laws. They remain the same throughout this comparison.
2. Introduce a variable object X for a possible source of data, or Z for a possible destination. Describe the maps and equations that make it an allowed instance of the needed use. Let it vary over every permitted instance, not just one sample. For example, a source of two components supplies functions from X to each fixed component set.
3. Seek an object P with the maps through which it will be used. Ask how an allowed instance should relate to P: should its data assemble into P, or should processing extend from P to a destination? Draw that comparison map in the corresponding direction.
4. Combine the comparison map with P’s use maps in the order their domains require, and equate the resulting routes with the original maps. Require a comparison map for every allowed instance, and uniqueness when the supplied data should determine it completely.

This specifies the behavior a construction must realize. If the intended use leaves the maps or agreements undecided, return to that particular choice before asking for a universal object. The prepared-function case in :5.4 follows these steps for a use beyond pairing or tagging.

Suppose the question requires two recoverable components in A and B, and those components must determine the entire result. A and B are fixed; a trial source X supplies maps to both. Seek an object P with projections pA:P -> A and pB:P -> B.

For every allowed object X and maps f:X -> A, g:X -> B, require a unique map <f,g>:X -> P such that:

pA composed with <f,g> = f;

pB composed with <f,g> = g.

These equations say that forming the combined result and reading either component returns the supplied component. Uniqueness says there is no further choice in that combination. Any map h:X -> P is consequently recovered from its components:

<pA composed with h, pB composed with h> = h.

An object with these projections and property is a product of A and B. Its specification describes both how to construct a result and how to use it.

Now suppose the components must agree through maps s:A -> C and t:B -> C. Seek projections that satisfy:

s composed with pA = t composed with pB.

Require a unique combined map only for f and g satisfying s composed with f = t composed with g. This is a pullback of s and t: it combines precisely the data compatible under that equation.

To choose an object for either kind of input, reverse the mapping question. Seek maps iA:A -> S and iB:B -> S that place the inputs in the new object. For maps f:A -> Z and g:B -> Z, require a unique map [f,g]:S -> Z with [f,g] composed with iA = f and [f,g] composed with iB = g. This is a coproduct. In sets, a tag can preserve which input was supplied, so the processing rule can select the correct branch.

The comparison points into the product or pullback and out of the coproduct. Its direction follows the needed operation: assemble supplied components or extend supplied processing rules.

MATH.16:4.3 - Construct an object and establish the property

In sets, the product is the set of ordered pairs A x B. Use the coordinate projections and define <f,g>(x)=(f(x),g(x)). Both projection equations follow by reading the corresponding coordinate. Any function with those projections must return that same pair at every x, which proves uniqueness.

For the pullback, form the subset:

Q={(a,b) in A x B | s(a)=t(b)}.

The same pairing formula lands in Q exactly when the compatibility equation holds. Reading the coordinates again proves uniqueness. Q may be empty. The empty set is still a valid pullback in sets; it says no pair meets the stated compatibility requirement.

For a coproduct of sets, distinguish the two input branches with tags 0 and 1:

S=({0} x A) union ({1} x B).

The injections are iA(a)=(0,a) and iB(b)=(1,b). Define [f,g](0,a)=f(a) and [f,g](1,b)=g(b). Every element has one of these forms, which establishes existence and forces the rule uniquely. Even when A and B are the same set, the tags keep the two branches distinct. For A=B={0}, handlers f(0)=0 and g(0)=1 therefore extend to a function taking (0,0) to 0 and (1,0) to 1.

Additional structure – groups and topology. Establish that the object and maps have the chosen structure. For groups, define product multiplication componentwise. If f and g preserve multiplication, then their paired map does too, because each coordinate does. For a pullback of group homomorphisms, the compatibility equation is preserved by multiplication and inverses. A construction with continuous maps requires the appropriate topology and continuity arguments.

Use a known applicable construction when available. For quotient and generator questions, MATH.2 and MATH.5 provide the detailed operations. If the required object cannot yet be constructed, identify the unresolved existence or construction question. An existence theorem may supply a result for reasoning while leaving an effective way to obtain its elements for further work.

MATH.16:4.4 - Derive the next map, comparison or operation

Use the universal property to build the map the question needs. For a product, give the two component maps. For a pullback, also establish their agreement in C. For a coproduct, give a processing rule for each input kind.

The property also supplies equality tests. Two maps into a product or pullback are equal when both their projections agree. Two maps out of a coproduct are equal when composing each with iA gives equal maps and composing each with iB gives equal maps. This lets you compare an entire map through its required parts.

It can compare different constructions of the same object. If P and P’ satisfy the same product property for A and B, the projections of each determine a map to the other. Composing these maps preserves both projections. The identity does too, so uniqueness makes the composites identities. Thus the two constructions are isomorphic by the maps that preserve their projections. MATH.7 develops the transport of further structure when a usable bijection has been obtained.

A new operation on the components needs a compatibility test. Here take Q to be the pullback of sets and u:A -> A, v:B -> B to be functions proposing updates. They induce a function (a,b) -> (u(a),v(b)) on Q exactly when:

s(u(a))=t(v(b)) whenever s(a)=t(b).

The original agreement does not settle the changed one. A failed pair identifies which update or agreement condition must change. This gives a way to work on operations themselves while keeping their required uses visible.

Additional structure – groups. When the update must also be a homomorphism, establish preservation of the group operation. Agreement of the updated components alone establishes only a function on the set of compatible pairs.

MATH.16:4.5 - Use the result and return to a changed requirement

Return the construction with the maps and conditions needed by its consumer. A name such as “product” helps recognition; the projections, formation rule and applicable laws let the receiver use it.

For a mathematical question, stop with the required map, equality, usable construction or demonstrated obstruction. For computation, obtain the procedure and resources needed for the chosen representation through C.29.2. A finite pullback can be enumerated by testing pairs, while a large or infinite one needs an appropriate computational method.

For an application to another subject, C.29 supplies the correspondence that gives the mathematical objects and equations their subject meaning. In particular, a compatibility equation must represent the actual agreement needed by the work. A pairing of functions on one input describes a different operation from two executions that modify a shared input. For the latter use, first specify which values each step reads and changes, and which intervening steps are permitted. Use that account to decide whether a function construction represents the work; the pairing alone leaves those interactions unspecified.

When the question changes, return to the affected mapping requirement. Adding agreement can turn a product question into a pullback question. Needing to accept either input can call for a coproduct. Needing only selected answers can call for a quotient. Retain a useful earlier construction while its earlier question remains current.