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Mathlib.CategoryTheory.Limits.Preserves.Shapes.Terminal

Preserving terminal object #

Constructions to relate the notions of preserving terminal objects and reflecting terminal objects to concrete objects.

In particular, we show that terminalComparison G is an isomorphism iff G preserves terminal objects.

The map of an empty cone is a limit iff the mapped object is terminal.

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The property of reflecting terminal objects expressed in terms of IsTerminal.

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A functor that preserves and reflects terminal objects induces an equivalence on IsTerminal.

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Preserving the terminal object implies preserving all limits of the empty diagram.

If C has a terminal object and G preserves terminal objects, then D has a terminal object also. Note this property is somewhat unique to (co)limits of the empty diagram: for general J, if C has limits of shape J and G preserves them, then D does not necessarily have limits of shape J.

If the terminal comparison map for G is an isomorphism, then G preserves terminal objects.

If there is any isomorphism G.obj ⊤ ⟶ ⊤, then G preserves terminal objects.

If there is any isomorphism G.obj ⊤ ≅ ⊤, then G preserves terminal objects.

If G preserves terminal objects, then the terminal comparison map for G is an isomorphism.

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The map of an empty cocone is a colimit iff the mapped object is initial.

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A functor that preserves and reflects initial objects induces an equivalence on IsInitial.

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Preserving the initial object implies preserving all colimits of the empty diagram.

If C has an initial object and G preserves initial objects, then D has an initial object also. Note this property is somewhat unique to colimits of the empty diagram: for general J, if C has colimits of shape J and G preserves them, then D does not necessarily have colimits of shape J.

If the initial comparison map for G is an isomorphism, then G preserves initial objects.

If there is any isomorphism ⊥ ⟶ G.obj ⊥, then G preserves initial objects.

If there is any isomorphism ⊥ ≅ G.obj ⊥, then G preserves initial objects.

If G preserves initial objects, then the initial comparison map for G is an isomorphism.

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