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(i) ‘A topos is a category of sheaves on a site’
(ii) ‘A topos is a category with finite limits and power-objects’
(iii) ‘A topos is (the embodiment of) an intuitionistic higher-order theory’
(iv) ‘A topos is (the extensional essence of) a first-order (infinitary) geometric theory’
(v) ‘A topos is a totally cocomplete object in the meta-2-category CART of cartesian (i.e. , finitely complete) categories’
(vi) ‘A topos is a generalized space’
(vii) ‘A topos is a semantics for intuitionistic formal systems’
(viii) ‘A topos is a Morita equivalence class of continuous groupoids’
(ix) ‘A topos is the category of maps of a power allegory’
(x) ‘A topos is a category whose canonical indexing over itself is complete and well-powered’
(xi) ‘A topos is the spatial manifestation of a giraud frame’
(xii) ‘A topos is a setting for synthetic differential geometry’
(xiii) ‘A topos is a setting for synthetic domain theory’