Category theory mereologically considered

Suppose consider arrows $f :B \rightarrow A$ in a category as expressing generalized parthood and read them as $A$ is an $f$-part of $B$.   But what if we also consider a $g :  C \rightarrow B$ ? Then by composition $C$ is a $ f \circ g$-part of $A$.  What is the concept of sieve mereologically speaking ?

A sieve on $x$ is a set (class) of parts $y$ of $x$, $Pyz$, such that if $z$ is a part of $y$ then $z$ also belongs to this set.

\[Siev(A,x) \equiv \forall y. ((y \in A \rightarrow Pyx) \& \forall z. Pzy \rightarrow Pzx) \]

A sieve on $x$ is a set of parts of $x$ plus all the parts of those parts. It is a $P$-filter on $z$.

A covering sieve is a generalization of fusion: $x$ is to be seen as a generalized fusion of any one of its covering sieves.  The axioms for a Grothendieck topology make sense for fusions. If $z$ is a fusion of the $\phi(x)$ and $Pyz$ then we can consider $\phi_y(x)$ (the $y$-restriction of $\phi(x)$) expressing elements which are overlaps of elements which satisfy $\phi(x)$ with $y$. Then we should have that $y$ is the fusion of the $\phi_y(x)$.

Suppose $z$ is a fusion of the $\phi(x)$. And consider a $P$-filter on $z$ given by $\psi(y)$. If for every element $x$ which satisfies $\phi(x)$ we have that restriction $x$ is a fusion of the $\psi_x(y)$ then $z$ is also a fusion of the $\psi(y)$.

Finally each $z$ is the fusion of all its parts.

The Aristotelic concept of the topos of $x$ involves considering a $z$ with $Pxz$ such that for all $y$ such that $Pxy$ we have $Pzy$ (a minimal cover). A modal version would be more appropriate.  Then the topos itself should be a kind of completement $t$ such that $t$ and $x$ do not overlap and $t + x = z$.