(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’