Lemma [efr-D7EH]

Let \mathcal {T} be a 2-limit sketch. Then the functor \mathsf {Mod}(\mathcal {T},\mathbb {C}) \to [\mathbb {C}^\mathrm {op},\mathsf {Mod}(\mathcal {T})] given by A \in \mathsf {Mod}(\mathcal {T},\mathbb {C}) \mapsto (C \mapsto (S \in \mathcal {T} \mapsto \mathbb {C}(C,A(S)))) is fully faithful, and its essential image consists of those functors F so that each F(-)(S) : \mathbb {C}^\mathrm {op} \to \mathsf {Cat}, S \in \mathcal {T} is representable,

Proof

First note that the codomain can be identified with the subcategory of [\mathbb {C}^\mathrm {op} \times \mathcal {T}, \mathsf {Cat}] spanned by those F where each F(C,-) is a model. Since the Yoneda embedding preserves limits, the functor A \mapsto \operatorname {\mathrm {Hom}}(-, A(=)) from \mathsf {Mod}(\mathcal {T}) clearly lands inside here. Since \mathsf {Mod}(\mathcal {T},\mathbb {C}) is itself a full subcategory of the functor category [\mathcal {T},\mathbb {C}], this is fully faithful.

Clearly for each model A and for each S \in \mathcal {T}, the functor \mathbb {C}(-,A(S)) is representable, by A(S). Conversely, if F(C,S) is such that each F(-,S) is representable, then the currying of F \mathcal {T} \to [\mathbb {C}^\mathrm {op},\mathsf {Cat}] factors over \mathbb {C}, and since the Yoneda embedding preserves those limits that exist, this factorization must be a model as well.

Context

2-Limit sketches [efr-9I1M]

Definition Weighted limit [efr-L9JB]

Let \mathbb {C} be a 2-category, let D: \mathbb {I} \to \mathbb {C} be a 2-diagram in it, and let W: \mathbb {I} \to \mathsf {Cat} be another 2-functor. A limit of D weighted by {W} is an object \lim ^W D \in \mathbb {C} equipped with a natural isomorphism of categories \mathbb {C}(X,\lim ^W D) \cong [ {\mathbb {I} }, {\mathbb {C}} ] (W, \mathbb {C}(X,D(-))) (natural in X \in \mathbb {C}).

Example Particular weighted limits [efr-K3YI]

If W(-) = * is constant at the point, and \mathbb {I} is an 1-category, then a weighted limit is simply a L \to D(i) in the underlying category \mathbb {C}_0 so that there is additionally an isomorphism of categories \mathbb {C}(X,L) \xrightarrow {\sim } \lim _i \mathbb {C}(X,D(I)). Note that this is stronger than just a limit in \mathbb {C}_0. We will refer to limits of this form by their ordinary names---speaking for example of pullbacks, products, and so on.

If I = *, D(*) = D \in \mathbb {C}, and W(*) = C \in \mathsf {Cat}, the universal property of the weighted limit is that \mathbb {C}(X, \lim ^W D) = \mathsf {Cat}(C, \mathbb {C}(X,D)). In this case we write C \pitchfork D for this limit if it exists, and call it a power of D by C.

If \mathbb {I} = \{A \to B \leftarrow C\} and the weighting is given by W(B) = \{1 \to 2\}, W(A) = \{1\}, W(B) = \{2\} (with the obvious inclusions), then the weighted limit is called the comma object and will be denoted D(A) \downarrow _{D(B)} D(C). Note that if \mathbb {C} = \mathsf {Cat} this is precisely the ordinary comma category.

The analogue for a cone on an object C in the setting of weighted limits is called a cylinder: it is a natural transformation W(-) \to \mathbb {C}(C, D(-)). A cylinder on C induces a natural transformation \mathbb {C}(X,C) \to [ {\mathbb {I} }, {\mathbb {C}} ] (W(-),\mathbb {C}(X,D(-))), and we say it's a limit cylinder if this is an isomorphism.

Definition 2-limit sketch [efr-40Q5]

A 2-limit sketch is a small 2-category \mathcal {T} equipped with a (small) set of cylinders \Theta . A model of the sketch in \mathbb {C} is a 2-functor \mathcal {T} \to \mathbb {C} which carries each cylinder in \Theta to a limit cylinder. We write the category of models and natural transformations \mathsf {Mod}(\mathcal {T},\mathbb {C}) (leaving the set of cylinders implicit). If \mathbb {C} = \mathsf {Cat}, we write simply \mathsf {Mod}(\mathcal {T}).

Lemma [efr-D7EH]

Let \mathcal {T} be a 2-limit sketch. Then the functor \mathsf {Mod}(\mathcal {T},\mathbb {C}) \to [\mathbb {C}^\mathrm {op},\mathsf {Mod}(\mathcal {T})] given by A \in \mathsf {Mod}(\mathcal {T},\mathbb {C}) \mapsto (C \mapsto (S \in \mathcal {T} \mapsto \mathbb {C}(C,A(S)))) is fully faithful, and its essential image consists of those functors F so that each F(-)(S) : \mathbb {C}^\mathrm {op} \to \mathsf {Cat}, S \in \mathcal {T} is representable,

Proof

First note that the codomain can be identified with the subcategory of [\mathbb {C}^\mathrm {op} \times \mathcal {T}, \mathsf {Cat}] spanned by those F where each F(C,-) is a model. Since the Yoneda embedding preserves limits, the functor A \mapsto \operatorname {\mathrm {Hom}}(-, A(=)) from \mathsf {Mod}(\mathcal {T}) clearly lands inside here. Since \mathsf {Mod}(\mathcal {T},\mathbb {C}) is itself a full subcategory of the functor category [\mathcal {T},\mathbb {C}], this is fully faithful.

Clearly for each model A and for each S \in \mathcal {T}, the functor \mathbb {C}(-,A(S)) is representable, by A(S). Conversely, if F(C,S) is such that each F(-,S) is representable, then the currying of F \mathcal {T} \to [\mathbb {C}^\mathrm {op},\mathsf {Cat}] factors over \mathbb {C}, and since the Yoneda embedding preserves those limits that exist, this factorization must be a model as well.

Proposition [efr-9O7K]

Let \mathcal {T},\mathcal {T}' be limit sketches and suppose given a functor F: \mathsf {Mod}(\mathcal {T}) \to \mathsf {Mod}(\mathcal {T}') which preserves limits and is accessible, that is it preserved \kappa -filtered colimits for some \kappa . Then F admits a left adjoint L.

In particular, F(A)(S) = \mathsf {Mod}(\mathcal {T})(L(y(S)),A)