REFERENCE SHEET

Universal properties, products & adjunctions

Specify an object by the maps it uniquely permits rather than by one construction.

Before this Functors, naturality & Yoneda

HomD(FX,Y) ≅ HomC(X,UY), naturally in X and Y

Definitions

Product

An object P with projections to A and B such that any pair X→A and X→B factors through one unique map X→P.

Coproduct

The dual construction: injections from A and B into C such that maps A→X and B→X extend uniquely to C→X.

Adjunction

Functors F:C→D and U:D→C with the displayed bijections natural in both variables. F is left adjoint to U.

Key theorems

Uniqueness of universal objects

Two products of the same objects are uniquely isomorphic by an isomorphism preserving their projections; analogous statements hold for other universal properties.

Free–forgetful adjunction

The free-group functor is left adjoint to the underlying-set functor: a set map X→U(G) extends uniquely to a group homomorphism F(X)→G.

Preservation

Right adjoints preserve limits and left adjoints preserve colimits, whenever the relevant limits or colimits exist.

Notable applications

Construction transfer

Products, quotients, and free objects can be recognized by the same mapping patterns across different categories.

Gluing

Pushouts formalize compatible gluing; van Kampen expresses a fundamental group as a group pushout.

Work it out

Try before revealing

01 / Compare

What are the coproducts of two sets in Set and of two groups in Grp?

Hint

Ask which construction accepts arbitrary pairs of outgoing maps.

Solution

In Set it is disjoint union. In Grp it is the free product, not generally the direct product, because maps from the two groups need not have commuting images.

02 / Prove

Why are two products P and Q of A,B isomorphic?