Types of Subcategories
A subcategory S of C is said to be isomorphism-closed or replete if every isomorphism k : X → Y in C such that Y is in S also belongs to S. A isomorphism-closed full subcategory is said to be strictly full.
A subcategory of C is wide or lluf (a term first posed by P. Freyd) if it contains all the objects of C. A lluf subcategory is typically not full: the only full lluf subcategory of a category is that category itself.
A Serre subcategory is a non-empty full subcategory S of an abelian category C such that for all short exact sequences
in C, M belongs to S if and only if both and do. This notion arises from Serre's C-theory.
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“Our children evaluate themselves based on the opinions we have of them. When we use harsh words, biting comments, and a sarcastic tone of voice, we plant the seeds of self-doubt in their developing minds.... Children who receive a steady diet of these types of messages end up feeling powerless, inadequate, and unimportant. They start to believe that they are bad, and that they can never do enough.”
—Stephanie Martson (20th century)