Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.
Sign up to join this community
In Exercise $4D$ of "The Joy of Cats", the authors ask for a proof of the following facts concerning the category of sets:
${\bf Set}$ has precisely three full, isomorphism-closed, reflective subcategories.
${\bf Set}$ has infinitely many reflective subcategories (non-necessarily full or isomorphism-closed or both).
Now, my question is: does there exist a full not isomorphism-closed reflective subcategory?
Sure, for instance take $\mathbf{C}\subset \mathbf{Set}$ the full subcategory consisting of the cardinal sets. In general, if $\mathbf{C}\subset \mathbf{D}$ is a reflexive subcategory, then any skeleton of $\mathbf{C}$ is also reflexive in $\mathbf{D}$.
Any full subcategory of a category consisting of terminal objects is reflective, but the only isomorphism-closed such subcategory is the one consisting of all terminal objects. In particular, if a category has more than one terminal object (i.e. a non-"strict" terminal object), then it has full reflective subcategories that are not closed under isomorphism (those consisting of some but not all terminal objects). Moreover, these subcategories have all (even large) limits and colimits (since they are equivalent to the category with one object).
Indeed, any assignment of an object in a full subcategory of terminal objects to each object of a category realizes the the condition of being reflective since each object in the category would have a unique morphism to its assigned terminal object through which any other morphism to a terminal object (in the subcategory) would factor via a unique morphism from the assigned terminal object to the other terminal object.
Interestingly a subcategory consisting of a single object and its identity morphism is reflective if and only if the object is terminal, in which case the subcategory is full.
