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  "body": "\n## Definition\n\nLet $\\mathcal{D}$ be a [[triangulated category|T2V7GJJ2EN]] with shift $X \\mapsto X[1]$.\nA *t-structure* on $\\mathcal{D}$ is a pair $(\\mathcal{D}^{\\le 0}, \\mathcal{D}^{\\ge 0})$ of [[full subcategories|V3BFGKBJF7]], each closed under isomorphisms.\nWrite $\\mathcal{D}^{\\le n} = \\mathcal{D}^{\\le 0}[-n]$ and $\\mathcal{D}^{\\ge n} = \\mathcal{D}^{\\ge 0}[-n]$.\nThe pair must satisfy:\n\n1. $\\mathcal{D}^{\\le -1} \\subseteq \\mathcal{D}^{\\le 0}$ and $\\mathcal{D}^{\\ge 1} \\subseteq \\mathcal{D}^{\\ge 0}$;\n2. $\\Hom(X, Y) = 0$ for every $X$ in $\\mathcal{D}^{\\le 0}$ and every $Y$ in $\\mathcal{D}^{\\ge 1}$;\n3. every object $X$ fits into a [[distinguished triangle|ADKK56VZHW]] $A \\to X \\to B \\to A[1]$ with $A$ in $\\mathcal{D}^{\\le 0}$ and $B$ in $\\mathcal{D}^{\\ge 1}$.\n\nThe *heart* of the t-structure is the full subcategory $\\mathcal{D}^{\\le 0} \\cap \\mathcal{D}^{\\ge 0}$.\n\n## Remarks\n\nThe heart is always an abelian category, and the triangle in clause 3 is unique up to a unique isomorphism; this gives truncation functors $\\tau_{\\le 0}$ and $\\tau_{\\ge 1}$.\nExample: let $D(\\mathcal{A})$ be the derived category of an abelian category, written with cochain complexes.\nIts *standard* t-structure has $\\mathcal{D}^{\\le 0}$ the complexes with $H^i = 0$ for $i \u003e 0$ and $\\mathcal{D}^{\\ge 0}$ those with $H^i = 0$ for $i \u003c 0$.\nThe triangle of clause 3 is $\\tau_{\\le 0} X \\to X \\to \\tau_{\\ge 1} X \\to$, built from the truncated complexes, and the heart is equivalent to $\\mathcal{A}$.\nNon-example: if $\\mathcal{D} \\ne 0$, taking $\\mathcal{D}^{\\le 0} = \\mathcal{D}^{\\ge 0} = \\mathcal{D}$ fails clause 2, because for $X \\ne 0$ the identity of $X$ is a nonzero element of $\\Hom(X, X)$, while $X$ lies in both $\\mathcal{D}^{\\le 0}$ and $\\mathcal{D}^{\\ge 1} = \\mathcal{D}$.\n",
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