Editing Power set (section)

== Functors and quantifiers ==
There is both a covariant and contravariant power set [[functor]], {{math|{{itco|{{mathcal|P}}}}: Set → Set}} and {{math|{{overbar|{{itco|{{mathcal|P}}}}}}: Set {{sup|op}} → Set}}. The covariant functor is defined more simply as the functor which sends a set {{math|''S''}} to {{math|{{mathcal|P}}(''S'')}} and a morphism {{math|''f'': ''S'' → ''T''}} (here, a function between sets) to the image morphism. That is, for <math>A = \{x_1, x_2, ...\} \in \mathsf{P}(S), \mathsf{P}f(A) = \{f(x_1), f(x_2), ...\} \in \mathsf{P}(T)</math>. Elsewhere in this article, the power set was defined as the set of functions of {{Math|''S''}} into the set with 2 elements. Formally, this defines a natural isomorphism <math>\overline{\mathsf{P}} \cong \text{Set}(-,2)</math>. The contravariant power set functor is different from the covariant version in that it sends {{math|''f''}} to the ''pre''image morphism, so that if <math>f(A) = B \subseteq T, \overline\mathsf{P}f(B) = A</math>. This is because a general functor <math>\text{C}(-,c)</math> takes a morphism <math>h:a \rightarrow b</math> to precomposition by ''h'', so a function <math>h^*: C(b,c) \rightarrow C(a,c)</math>, which takes morphisms from ''b'' to ''c'' and takes them to morphisms from ''a'' to ''c'', through ''b'' via ''h''. <ref>{{Cite book |last=Riehl |first=Emily |title=Category Theory in Context |date=16 November 2016 |publisher=Courier Dover Publications |isbn=978-0486809038}}</ref>

In [[category theory]] and the theory of [[elementary topos|elementary topoi]], the [[universal quantifier]] can be understood as the [[right adjoint]] of a [[functor]] between power sets, the [[inverse image]] functor of a function between sets; likewise, the [[existential quantifier]] is the [[left adjoint]].{{sfn|ps=|Mac Lane|Moerdijk|1992|p=58}}