Editing Symmetric algebra (section)

==Categorical properties==
Given a [[module (mathematics)|module]] {{mvar|V}} over a [[commutative ring]] {{mvar|K}}, the symmetric algebra {{math|''S''(''V'')}} can be defined by the following [[universal property]]:

::For every {{mvar|K}}-[[linear map]] {{mvar|f}} from {{mvar|V}} to a commutative {{mvar|K}}-algebra {{mvar|A}}, there is a unique {{mvar|K}}-[[algebra homomorphism]] <math>g:S(V)\to A</math> such that <math>f=g\circ i,</math> where {{mvar|i}} is the inclusion of {{mvar|V}} in {{math|''S''(''V'')}}.

As for every universal property, as soon as a solution exists, this defines uniquely the symmetric algebra, [[up to]] a [[canonical isomorphism]]. It follows that all properties of the symmetric algebra can be deduced from the universal property. This section is devoted to the main properties that belong to [[category theory]].

The symmetric algebra is a [[functor]] from the [[category (mathematics)|category]] of {{mvar|K}}-modules to the category of {{mvar|K}}-commutative algebra, since the universal property implies that every [[module homomorphism]] <math>f:V\to W</math> can be uniquely extended to an [[algebra homomorphism]] <math>S(f):S(V)\to S(W).</math>

The universal property can be reformulated by saying that the symmetric algebra is a [[left adjoint]] to the [[forgetful functor]] that sends a commutative algebra to its underlying module.