Let $\mathcal{C}$ be the category of $\mathbb{Z}$-graded rings. I have a sequence $A_i$ of $\mathbb{Z}$-graded rings, and split monomorphisms $\varphi_i:A_i\to A_{i+1}$ i.e. $$A_0\rightarrowtail A_1\rightarrowtail\cdots.$$ Let $A$ be the colimit/direct limit of this sequence, with morphisms $\psi_i:A_i\to A$. Are the $\psi_i$ necessarily split monos?
Does this work more generally in concrete categories?
This works in any category. To say $\varphi_i$ is split mono is to say there is a retraction $\rho_i\colon A_{i+1}\rightarrow A_i$. Then, you obtain morphisms $\rho_i\circ\dotsc\circ\rho_{j-1}\colon A_j\rightarrow A_i$ for all $j\ge i$. Then, since $\rho_i\circ\dotsc\circ\rho_{j-1}\circ\varphi_{j-1}=\rho_i\circ\dotsc\circ\rho_{j-2}$, it follows that these are compatible, so they induce a map $\chi_i\colon A\rightarrow A_i$ by the universal property of the colimit such that $\chi_i\circ\psi_j=\rho_i\circ\dotsc\circ\rho_{j-1}$ for all $j\ge i$. In particular, $\chi_i\circ\psi_i=\mathrm{id}_{A_i}$. Thus, $\psi_i$ is a split mono, as desired. This argument can be generalized to general filtered colimits.