2
$\begingroup$

In Exercise $4D$ of "The Joy of Cats", the authors ask for a proof of the following facts concerning the category of sets:

  1. ${\bf Set}$ has precisely three full, isomorphism-closed, reflective subcategories.

  2. ${\bf Set}$ 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?

$\endgroup$
0

2 Answers 2

3
$\begingroup$

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 $\mathbf{D}$.

$\endgroup$
3
  • $\begingroup$ Why the category of cardinal sets is not isomorphism-closed? $\endgroup$ Commented Jun 18, 2022 at 13:19
  • $\begingroup$ Because any set which is not a cardinal is isomorphic to a cardinal, but is not in the subcategory. $\endgroup$ Commented Jun 18, 2022 at 14:07
  • $\begingroup$ Which is the reflection in the case of cardinal sets? And secondly, is such a subcategory closed under products and equalizers? $\endgroup$ Commented Jun 18, 2022 at 17:02
3
$\begingroup$

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.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .