The paper introduces a tensor network approach called the locally purified density operator (LPDO) to investigate symmetry-protected topological (SPT) phases in open quantum systems. The authors extend the concept of injectivity, originally associated with matrix product states (MPS) and projected entangled pair states (PEPS), to LPDOs in $(1+1)$D and $(2+1)$D systems. They identify two distinct types of injectivity conditions for short-range entangled density matrices. Within the LPDO framework, they outline a classification scheme for decohered average symmetry-protected topological (ASPT) phases, consistent with earlier results obtained through spectrum sequence techniques. The approach offers an intuitive and explicit construction of ASPT states, particularly in the context of weak global symmetry and strong fermion parity symmetry. The authors derive both the classification data and the explicit forms of the obstruction functions using the LPDO formalism, especially in cases where nontrivial group extensions between strong and weak symmetries exist, leading to intrinsic ASPT phases. They demonstrate constructions of fixed-point LPDOs for ASPT phases in $(1+1)$D and $(2+1)$D and discuss their physical realization in decohered or disordered systems.The paper introduces a tensor network approach called the locally purified density operator (LPDO) to investigate symmetry-protected topological (SPT) phases in open quantum systems. The authors extend the concept of injectivity, originally associated with matrix product states (MPS) and projected entangled pair states (PEPS), to LPDOs in $(1+1)$D and $(2+1)$D systems. They identify two distinct types of injectivity conditions for short-range entangled density matrices. Within the LPDO framework, they outline a classification scheme for decohered average symmetry-protected topological (ASPT) phases, consistent with earlier results obtained through spectrum sequence techniques. The approach offers an intuitive and explicit construction of ASPT states, particularly in the context of weak global symmetry and strong fermion parity symmetry. The authors derive both the classification data and the explicit forms of the obstruction functions using the LPDO formalism, especially in cases where nontrivial group extensions between strong and weak symmetries exist, leading to intrinsic ASPT phases. They demonstrate constructions of fixed-point LPDOs for ASPT phases in $(1+1)$D and $(2+1)$D and discuss their physical realization in decohered or disordered systems.