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  "body": "\n## Definition\n\nLet $\\mathcal{E}$ and $\\mathcal{F}$ be [[toposes|6837RS7N82]].\nA *geometric morphism* $f \\colon \\mathcal{E} \\to \\mathcal{F}$ is a pair of functors $f_* \\colon \\mathcal{E} \\to \\mathcal{F}$ and $f^* \\colon \\mathcal{F} \\to \\mathcal{E}$ such that:\n\n1. $f^*$ is [[left adjoint|TZHF6ZM92S]] to $f_*$;\n2. $f^*$ [[preserves finite limits|H4MY8JN8W4]].\n\nThe functor $f_*$ is the *direct image* and $f^*$ is the *inverse image* of $f$.\n\n## Remarks\n\nExample: let $g \\colon X \\to Y$ be a continuous map of topological spaces.\nThe direct image $g_* F(V) = F(g^{-1}(V))$ and the inverse image $g^*$ of sheaves form a geometric morphism $\\mathrm{Sh}(X) \\to \\mathrm{Sh}(Y)$.\nThe inverse image preserves finite limits, because stalks of $g^*G$ are stalks of $G$ and finite limits of sheaves are computed stalkwise.\nNon-example: on $\\mathbf{Set}$, take $f^* = - \\times A$ and $f_* = (-)^A$ for a set $A$ with two elements.\nThen $f^* \\dashv f_*$, but $f^*$ sends the terminal set $1$ to $A$, which is not terminal, so clause 2 fails.\nFor a topos $\\mathcal{E}$, a *point* of $\\mathcal{E}$ is a geometric morphism $\\mathbf{Set} \\to \\mathcal{E}$.\nEvery Grothendieck topos has, up to isomorphism, exactly one geometric morphism to $\\mathbf{Set}$.\n",
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