Isomorphism of Filtered Objects?

Question

Let $\mathscr{C}$ be a category and let $Filt(\mathscr{C})$ denote the category of filtered objects in $\mathscr{C}$. So an object $X$ in $Filt(\mathscr{C})$ look something like $X_0\subset X_1\subset ... \subset X_k=X$. What are isomorphisms in $Filt(\mathscr{C})$? Are they level-wise isomorphisms between the objects?

Answer 1

It depends on what notion of morphism you want for this category. My gut reaction would be to define morphisms between the filtered objects to be a collection of arrows $X_i \to Y_i$ in $\mathcal{C}$ which commute with the inclusions in the filtration. Indeed, this is the notion of morphism adopted in this text. Then isomorphisms would be level-wise isomorphisms, which additionally must commute with the inclusions.