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# AplK # The minimal Sullivan model of a simplicial complex
The rational homotopy type of a topological space is a simplified version of the homotopy type, where all homotopy groups are tensored by $\mathbb{Q}$. Despite the lost information, rational homotopy has the advantage of being constructive. Due to Sullivan, for a particular topological space $X$ that satisfies some conditions, it is known theoretically how to obtain its rational homotopy type via the construction of a commutative differential graded algebra, called the Sullivan model of $X$. This algebra is quasi-isomorphic to the normalized singular cochain algebra of $X$, $C^*(X)$, and it allows us to establish a categorical equivalence between homotopy types of spaces and isomorphism classes of Sullivan models: The rational homotopy type of a topological space is a simplified version of the homotopy type, where all homotopy groups are tensored by $\mathbb{Q}$. Despite the lost information, rational homotopy has the advantage of being constructive. Due to Sullivan, for a particular topological space $X$ that satisfies some conditions, it is known theoretically how to obtain its rational homotopy type via the construction of a commutative differential graded algebra, called the Sullivan model of $X$. This algebra is quasi-isomorphic to the normalized singular cochain algebra of $X$, $C^*(X)$, and it allows us to establish a categorical equivalence between homotopy types of spaces and isomorphism classes of Sullivan models: