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To make sense of enrichment over a category $V$, you want $V$ to have a monoidal structure. Indeed, you want to be able to compose morphisms so you need a way to go from "something in $\hom(a,b)$ and …

If there is such a pushout, then $B\to D$ is also a monomorphism, i.e. $B$ is a subobject of $D$. Phrased more concretely, you're asking when there is a subobject $B$ of $D$ such that $B+C = D$ and $B …

Here's a useful lemma to check whether a square in an abelian category is a pullback square: Lemma: A square $A_0\to (A_1,A_2)\to A_3$ with $A_0\to A_1, A_2\to A_3$ being monomorphisms is a pullback s …

Yes. In an idempotent complete monoidal category $C$ (abelian implies idempotent complete), if $A$ (e.g. your $X\oplus Y$) has a dual and $B$ (e.g. your $X$) is a retract of $A$, then $B$ also does. N …

If by "rigid" you mean the same thing as dualizable, an example is given by $\mathcal M = Mod_{R[G]}$ for some commutative ring $R$ and group $G$, with monoidal structure given by $\otimes_R$ and the …

Yes, this is true more generally. It follows from the following observation : for a triangulated category $D$, a thick subcategory $C$ and $x \in D$, $x$ is in $C$ if and only if $x=0$ in $D/C$. Combi …