Concrete Category - Further Examples

Further Examples

1. Any group G may be regarded as an "abstract" category with one object, and one morphism for each element of the group. This would not be counted as concrete according to the intuitive notion described at the top of this article. But every faithful G-set (equivalently, every representation of G as a group of permutations) determines a faithful functor G → Set. Since every group acts faithfully on itself, G can be made into a concrete category in at least one way.
2. Similarly, any poset P may be regarded as an abstract category with a unique arrow x → y whenever x ≤ y. This can be made concrete by defining a functor D : P → Set which maps each object x to and each arrow x → y to the inclusion map .
3. The category Rel whose objects are sets and whose morphisms are relations can be made concrete by taking U to map each set X to its power set and each relation to the function defined by . Noting that power sets are complete lattices under inclusion, those functions between them arising from some relation R in this way are exactly the supremum-preserving maps. Hence Rel is equivalent to a full subcategory of the category Sup of complete lattices and their sup-preserving maps. Conversely, starting from this equivalence we can recover U as the composite Rel → Sup → Set of the forgetful functor for Sup with this embedding of Rel in Sup.
4. The category Setop can be embedded into Rel by representing each set as itself and each function f: X → Y as the relation from Y to X formed as the set of pairs (f(x),x) for all x ∈ X; hence Setop is concretizable. The forgetful functor which arises in this way is the contravariant powerset functor Setop → Set.
5. It follows from the previous example that the opposite of any concretizable category C is again concretizable, since if U is a faithful functor C → Set then Cop may be equipped with the composite Cop → Setop → Set.
6. If C is any small category, then there exists a faithful functor P : SetCop → Set which maps a presheaf X to the coproduct . By composing this with the Yoneda embedding Y:C → SetCop one obtains a faithful functor C → Set.
7. For technical reasons, the category Ban₁ of Banach spaces and linear contractions is often equipped not with the "obvious" forgetful functor but the functor U₁ : Ban₁ → Set which maps a Banach space to its (closed) unit ball.