Definition: A category C is monoidal if it is accompanied by
Definition: Let C be monoidal as above. A category D is enriched over C if it is accompanied by
Definition: A topological space X is compactly generated if its basis of topology of closed sets is given by continuous images of compact Hausdorff spaces K whose preimages are closed in K. A topological space is weakly Hausdorff if the continous image of every compact Hausdorff space is closed in X.
We write CG for the category of compactly generated and weakly Hausdorff spaces. This is a monoidal category with the usual product of topological spaces.
Example: Here are some examples of enriched categories.
Definition: Let C,D be bicategories. An assignment F:C→D is a pseudofunctor when it has
Definition: Let C,D be categories and F:C→D a functor. A morphism f∈HomC(A,B) is F-cartesian if
commutes for some unique g∈HomC(A,Y) (all the vertical arrows are F).
This definition can be rephrased in the language of simplicial sets: the morphism f is F-cartesian if whenever Ff=d1Δ2 for some Δ2∈D2, then every Λ2∈C with Λ21=f and FΛ20=d0Δ2 can be filled in by g with Fg=d2Δ2.
Definition: Let f:C→D be a functor.
If C is F-fibered over D and C′ is F′-fibered over D, then a functor F:C→C′ is a \emph{morphism of fibered categories} if F=F′∘F and Ff is F′-cartesian whenever f is F-cartesian.
Theorem: Let C be F-fibered over D.
The above result follows from sections 3.1.2 and 3.1.3 of Vistoli. Theorem 2.2.1.2 of Lurie generalizes this and provides an equivalence between the category of fibered simplicial sets over S∈sSet and the category of functors sCat→sSet. The forward direction is called straightening and he backward direction is called unstraightening.
References: nLab (articles "Monoidal category," "enriched category," and "pseudofunctor."), Strickland (The category of CGWH spaces), Vistoli (Notes on Grothendieck topologies, Chapter 3), Noohi (A quick introduction), Lurie (Higher Topos Theory, Section 2.2)
- a functor ⊗:C×C→C,
- an object 1∈Obj(C), and
- isomorphisms
- αX,Y,Z∈HomC((X⊗Y)⊗Z,X⊗(Y⊗Z)),
- λX∈HomC(1⊗X,X), and
- ρX∈HomC(X⊗1,X),
Definition: Let C be monoidal as above. A category D is enriched over C if it is accompanied by
- an object D(P,Q)∈Obj(C) for every P,Q∈Obj(D), and
- morphisms
- γP,Q,R∈HomC(D(Q,R)⊗D(P,Q),D(P,R)), and
- iP∈HomC(1,D(P,P)),
Definition: A topological space X is compactly generated if its basis of topology of closed sets is given by continuous images of compact Hausdorff spaces K whose preimages are closed in K. A topological space is weakly Hausdorff if the continous image of every compact Hausdorff space is closed in X.
We write CG for the category of compactly generated and weakly Hausdorff spaces. This is a monoidal category with the usual product of topological spaces.
Example: Here are some examples of enriched categories.
- A topological category is a category enriched over CG.
- A bicategory, or weak 2-category, is a category weakly enriched over Cat, the category of small categories.
Definition: Let C,D be bicategories. An assignment F:C→D is a pseudofunctor when it has
- an object F(X)∈Obj(D),
- a functor F(X,Y):C(X,Y)→D(F(X),F(Y)), and
- invertible 2-morphisms
- F(idX):idX⇒F(X,X)(idX), and
- F(X,Y,Z)(f,g):F(Y,Z)(g)∘F(X,Y)(f)⇒F(X,Z)(g∘f),
Definition: Let C,D be categories and F:C→D a functor. A morphism f∈HomC(A,B) is F-cartesian if
commutes for some unique g∈HomC(A,Y) (all the vertical arrows are F).
This definition can be rephrased in the language of simplicial sets: the morphism f is F-cartesian if whenever Ff=d1Δ2 for some Δ2∈D2, then every Λ2∈C with Λ21=f and FΛ20=d0Δ2 can be filled in by g with Fg=d2Δ2.
Definition: Let f:C→D be a functor.
- The category C is F-fibered over D if for every morphism h∈HomD(U,V) and every B∈Obj(C) with F(B)=V, there is some F-cartesian f∈HomC(−,B) with Ff=h.
- A cleavage of an F-fibered category C is a class of cartesian morphisms K in C such that for every morphism h∈HomD(U,V) and every B∈Obj(C) with F(B)=V, there is a unique F-cartesian f∈K with Ff=h.
- A cleavage of C is a splitting if it contains all the the identity morphisms and is closed under composition.
If C is F-fibered over D and C′ is F′-fibered over D, then a functor F:C→C′ is a \emph{morphism of fibered categories} if F=F′∘F and Ff is F′-cartesian whenever f is F-cartesian.
Theorem: Let C be F-fibered over D.
- Every cleavage of C defines a pseudofunctor D→Cat.
- Every pseudofunctor D→Cat defines an F′-fibered category C′ with a cleavage over D.
The above result follows from sections 3.1.2 and 3.1.3 of Vistoli. Theorem 2.2.1.2 of Lurie generalizes this and provides an equivalence between the category of fibered simplicial sets over S∈sSet and the category of functors sCat→sSet. The forward direction is called straightening and he backward direction is called unstraightening.
References: nLab (articles "Monoidal category," "enriched category," and "pseudofunctor."), Strickland (The category of CGWH spaces), Vistoli (Notes on Grothendieck topologies, Chapter 3), Noohi (A quick introduction), Lurie (Higher Topos Theory, Section 2.2)