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Modos
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In mathematics, specifically in category theory, modos is the modification of the default-colimiting which says that « each functor is the sum/colimit of its elements » . Modos shows the interdependence of computational-logic ( soundness lemma , cut-elimination , confluence lemma , completeness lemma ... ) along geometry ( completion , topos ... ) . Any secondary-school person is able to know such mathematics , which has been programmed onto the Coq computer. Default (common) colimiting Remember that given two functions and , oneself may form the pairing into the product such that the two projections of this product and cancel this pairing : and . Also remember that in presheaves , the default-colimiting says that « each functor is the sum/colimit of its elements ; which is that the (outer) indexings/cocones of the elements of some target functor over all the elements of some source functor correspond with the (inner) transformations from this source functor into this target functor » . In other words : any outer indexing corresponds with some inner transformation . Modos modifies such common things , but by holding some more-motivated more-grammatical more-complete context . Modified colimiting The ends is to start with some given viewing-data ( "coverings" ) on some generator-morphology ( "category" ) and then to modify the above default-colimiting . The modified-colimiting presents this above summing/copairing correspondence when any such indexing ( « real polymorph-cocones » ) is over only some viewing-elements of this source « viewing-functor » ( "local epimorphism" ) , as long as the corresponding transformation is into the (tautologically extended) « viewed-functor » ( "sheafification") of this target functor . Memo that when the target functor is already viewed-functor ( "sheaf" ) then this modified-colimiting becomes the default-colimiting . But where are those modified-colimits ? Oneself could get them as metafunctors over this generator-morphology , as long as oneself grammatically-distinguishes whatever-is-interesting . Memo that in contrast to the more-common mutually-inductive types , this above grammatical presentation of limit objects shall somehow-depend on the morphisms , but this dependence need-not be grammatical because this dependence is via the sense-decoding ( Yoneda lemma ) of the morphisms . 
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