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  "body": "\n## Summary\n\nA fibred category is a functor $p \\colon \\mathcal{E} \\to \\mathcal{B}$ along which objects can be pulled back.\nFor every object $y$ of $\\mathcal{E}$ and every morphism $f$ into $p(y)$, there is a universal, or cartesian, morphism over $f$ ending at $y$.\nFibred categories describe families of categories indexed by the objects of $\\mathcal{B}$, such as sheaves or vector bundles over varying spaces.\nThey are the setting of Grothendieck's descent theory and of stacks.\n\n## Explanations\n\n### L1 human\n\nThink of $\\mathcal{B}$ as a category of spaces and of the objects of $\\mathcal{E}$ over a space $b$ as structures on $b$, for example vector bundles on $b$.\nGiven a map of spaces $f \\colon b' \\to b$ and a bundle on $b$, one can pull the bundle back to $b'$.\nA fibred category is the abstract form of this pullback operation.\nThe pulled-back object is determined up to unique isomorphism by a universal property, not chosen once and for all.\n",
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