This paper presents a tensor network formulation of symmetry-protected topological (SPT) phases in mixed states. The authors define and classify SPT phases in mixed states using the tensor network formulation of the density matrix. In one dimension, they introduce strong injective matrix product density operators (MPDO), which describe a broad class of short-range correlated mixed states, including locally decohered SPT states. They map strong injective MPDO to a pure state in the doubled Hilbert space and define SPT phases according to the cohomology class of the symmetry group in the doubled state. The SPT phases are constrained by the Hermiticity and semi-positivity of the density matrix. The authors obtain a complete classification of SPT phases with a direct product of strong G and weak K unitary symmetry given by the cohomology group $ \mathcal{H}^{2}(G,\mathrm{U}(1))\oplus\mathcal{H}^{1}(K,\mathcal{H}^{1}(G,\mathrm{U}(1))) $. They also extend their results to two-dimensional mixed states described by strong semi-injective tensor network density operators and classify the possible SPT phases. The paper discusses the resilience of SPT order against local decoherence and the methodologies for defining and classifying mixed-state SPT phases. The authors show that the SPT phases in their definition are preserved under symmetric local circuits consisting of non-degenerate channels. They also prove that an alternative definition of SPT phases according to the equivalence class of mixed states under a "one-way" connection using symmetric non-degenerate channels reproduces the cohomology classification. The paper concludes with a discussion of the implications of these results for the classification of SPT phases in mixed states.This paper presents a tensor network formulation of symmetry-protected topological (SPT) phases in mixed states. The authors define and classify SPT phases in mixed states using the tensor network formulation of the density matrix. In one dimension, they introduce strong injective matrix product density operators (MPDO), which describe a broad class of short-range correlated mixed states, including locally decohered SPT states. They map strong injective MPDO to a pure state in the doubled Hilbert space and define SPT phases according to the cohomology class of the symmetry group in the doubled state. The SPT phases are constrained by the Hermiticity and semi-positivity of the density matrix. The authors obtain a complete classification of SPT phases with a direct product of strong G and weak K unitary symmetry given by the cohomology group $ \mathcal{H}^{2}(G,\mathrm{U}(1))\oplus\mathcal{H}^{1}(K,\mathcal{H}^{1}(G,\mathrm{U}(1))) $. They also extend their results to two-dimensional mixed states described by strong semi-injective tensor network density operators and classify the possible SPT phases. The paper discusses the resilience of SPT order against local decoherence and the methodologies for defining and classifying mixed-state SPT phases. The authors show that the SPT phases in their definition are preserved under symmetric local circuits consisting of non-degenerate channels. They also prove that an alternative definition of SPT phases according to the equivalence class of mixed states under a "one-way" connection using symmetric non-degenerate channels reproduces the cohomology classification. The paper concludes with a discussion of the implications of these results for the classification of SPT phases in mixed states.