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InfinityCosmos.ForMathlib.AlgebraicTopology.SimplicialCategory.Limits

A limit cone c in a simplicial category A is a simplicially enriched limit if for every X : C, the cone obtained by applying the simplicial coyoneda functor (X ⟶[A] -) to c is a limit cone in SSet.

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    Conical simplicial limits are also limits in the unenriched sense.

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      Characterization in terms of the comparison map. #

      There is a canonical comparison map with the limit in SSet, the following proves that a limit cone in A is a simplicially enriched limit if and only if the comparison map is an isomorphism for every X : A.

      ConicalLimitCone F contains a cone over F together with the information that it is a conical limit.

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        HasConicalLimit F represents the mere existence of a limit for F.

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          The projection from the conical limit object to a value of the functor.

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            C has conical limits of shape J if there exists a conical limit for every functor F : J ⥤ C.

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              If a E ⋙ F has a limit, and E is an equivalence, we can construct a limit of F.

              C has all conical limits of size v₁ u₁ (HasLimitsOfSize.{v₁ u₁} C) if it has conical limits of every shape J : Type u₁ with [Category.{v₁} J].

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                @[reducible, inline]

                C has all (small) conical limits if it has limits of every shape that is as big as its hom-sets.

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                  @[reducible, inline]

                  An abbreviation for HasSLimit (Discrete.functor f).

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