The Yoneda Lemma without category theory: algebra and applications

However it can just as well be considered a fundamental representation theorem of universal algebra, via the connection between the homfunctor of a category and free algebras for the theory represented by that category. This connection is not generally appreciated outside category theoretic circles, which this section endeavors to correct by presenting the relevant concepts from an algebraic perspective. The algebraically motivated notations T , s, A, h, h∗, and a∗ below correspond to the respective notations C, c, F , τ , α(τ), and α−1(a) (for a ∈ F (c)) above, with the free algebra Ts corresponding to the representable functor C(c,−) : C → Set, fu = Tuf to C(c, f), and f∗ = Tf to C(f,−). The remaining sections of the paper apply the Yoneda Lemma to a proposed simplification and generalization of algebra, illustrated with applications. Algebras and their homomorphisms have in effect been defined as respectively functors and natural transformations long before category theory came into existence. That is, we do not see any essential differences between the definitions, and category theorists are just as entitled to characterize algebra as merely another language for category theory as algebraists are to the converse claim. Sections 2 and 3 on respectively dense and didense extensions of categories exploit the Yoneda Lemma to replace these conventional definitions of algebra/functor and homomorphism/natural transformation with definitions that