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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be an [[(∞,1)-category|YFY1FWA1MY]] with a [[zero object|AQET9YHH6F]] $0$.\nA *triangle* in $\\mathcal{C}$ is a commutative square\n$$\\begin{array}{ccc} X \u0026 \\xrightarrow{f} \u0026 Y \\\\ \\downarrow \u0026 \u0026 \\downarrow g \\\\ 0 \u0026 \\longrightarrow \u0026 Z \\end{array}$$\nwhose lower left corner is $0$.\nIt is a fiber sequence if the square is a [[pullback|Q05K04QJ71]], and a cofiber sequence if the square is a [[pushout|Q699KKMVJD]].\nThe category $\\mathcal{C}$ is *stable* if:\n\n1. $\\mathcal{C}$ has a zero object.\n2. Each morphism $g \\colon Y \\to Z$ has a fiber, that is, a pullback of $Y \\to Z \\leftarrow 0$, and each morphism $f \\colon X \\to Y$ has a cofiber, that is, a pushout of $0 \\leftarrow X \\to Y$.\n3. A triangle in $\\mathcal{C}$ is a fiber sequence if and only if it is a cofiber sequence.\n\nPullbacks and pushouts are meant in the ∞-categorical sense, that is, as homotopy pullbacks and homotopy pushouts.\n\n## Remarks\n\nEquivalently, $\\mathcal{C}$ is stable if it has a zero object and all finite limits and finite colimits, and its pullback squares are exactly its pushout squares.\n\nExample: the ∞-category of [[spectra|1GHF4EKBXS]] is stable.\nThe zero spectrum is a zero object, and spectra have all finite limits and colimits, so clauses 1 and 2 hold.\nClause 3 holds because the suspension functor on spectra is an equivalence.\nIn a pointed ∞-category with finite limits and colimits, this forces fiber and cofiber sequences to agree.\n\nNon-example: the ∞-category of pointed spaces has a zero object, the one-point space, and has fibers and cofibers, but clause 3 fails.\nThe cofiber of $S^1 \\to *$ is $S^2$, while the fiber of $* \\to S^2$ is the loop space $\\Omega S^2$.\nThis is not equivalent to $S^1$, because $\\pi_2(\\Omega S^2) \\cong \\pi_3(S^2) \\cong \\Z$ while $\\pi_2(S^1) = 0$.\n\nThe suspension $\\Sigma X$ is the cofiber of $X \\to 0$, and the loop object $\\Omega X$ is the fiber of $0 \\to X$.\nIn a stable ∞-category, $\\Sigma$ and $\\Omega$ are inverse equivalences.\n",
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