Suppose $X$ verifies the suitable conditions in which Weil (resp. Cartier) divisors make sense.
The group of Weil divisors $\mathrm{Div}(X)$ on a scheme $X$ is the free abelian group generated by codimention $1$ irreducible subvarieties.
The sheaf of Cartier divisors is $\mathrm{Cart}_X:=\mathcal{K}_X^{\times}/\mathcal{O}_X^{\times}$.
The group of Cartier divisors is $\mathrm{Cart}(X):=\Gamma(X,\mathrm{Cart}_X)$.
The group $Z^p(X)$ of $p$-cycles is the free abelian group generated by irreducible subvarieties of codimension $p$, so in particular $\mathrm{Div}(X)=Z^1(X)$. So the notion of $p$-cycle is a direct generalization of the notion of Weil divisor.
My question:
Is there an analogous notion of group of "Cartier $p$-cycles" $\mathrm{Cart}^p(X)$? If yes, is there a sheaf $\mathrm{Cart}^p_X$ such that (naturally in $X$) we have $\mathrm{Cart}^p(X)=\Gamma(X,\mathrm{Cart}^p_X)$?