Enriched and straightened categories

Definition: A category $\mathcal C$ is monoidal if it is accompanied by

• a functor $\otimes \colon \mathcal C\times \mathcal C\to \mathcal C$,
• an object $\mathbf{1}\in \Obj(\mathcal C)$, and
• isomorphisms
• $\alpha_{X,Y,Z}\in \Hom_{\mathcal C}((X\otimes Y)\otimes Z,X\otimes(Y\otimes Z))$,
• $\lambda_X\in \Hom_{\mathcal C}(\mathbf{1}\otimes X,X)$, and
• $\rho_X\in \Hom_{\mathcal C}(X\otimes \mathbf{1},X)$,

for all $X,Y,Z,W\in \Obj(\mathcal C)$, such that $\otimes$ is unital and $\alpha$ is associative over $\otimes$. That is, the diagrams below commute.

Definition: Let $\mathcal C$ be monoidal as above. A category $\mathcal D$ is enriched over $\mathcal C$ if it is accompanied by

• an object $\mathcal D(P,Q)\in \Obj(\mathcal C)$ for every $P,Q\in \Obj(\mathcal D)$, and
• morphisms
• $\gamma_{P,Q,R}\in \Hom_{\mathcal C}(\mathcal D(Q,R)\otimes \mathcal D(P,Q),\mathcal D(P,R))$, and
• $i_P\in \Hom_{\mathcal C}(\mathbf{1},\mathcal D(P,P))$,

for all $P,Q,R,S\in \Obj(\mathcal D)$, such that $\gamma$ is unital and associative over $\otimes$. The category $\mathcal D$ is weakly enriched over $\mathcal C$ if $\gamma$ is unital and associative over $\otimes$ up to homotopy. That is, the diagrams below commute for $\mathcal D$ enriched, and commute up to homotopy for $\mathcal D$ weakly enriched.

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 $\mathcal{CG}$ 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 $\mathcal {CG}$.
• A bicategory, or weak 2-category, is a category weakly enriched over $\mathcal Cat$, the category of small categories.

Definition: Let $\mathcal C,\mathcal D$ be bicategories. An assignment $F\colon \mathcal C\to \mathcal D$ is a pseudofunctor when it has

• an object $F(X)\in \Obj(\mathcal D)$,
• a functor $F(X,Y)\colon \mathcal C(X,Y)\to \mathcal D(F(X),F(Y))$, and
• invertible 2-morphisms
• $F(\id_X)\colon \id_X \Rightarrow F(X,X)(\id_X)$, and
• $F(X,Y,Z)(f,g) \colon F(Y,Z)(g)\circ F(X,Y)(f)\Rightarrow F(X,Z)(g\circ f)$,

for all $X,Y,Z\in \Obj(C)$, such that $F(X,Y)$ is unital and associative over composition. The assignment $F$ is a lax functor when the last two morphisms are not necessarily invertible.

Definition: Let $\mathcal C,\mathcal D$ be categories and $F\colon\mathcal C\to \mathcal D$ a functor. A morphism $f\in \Hom_{\mathcal C}(A,B)$ is $F$-cartesian if

commutes for some unique $g\in \Hom_{\mathcal C}(A,Y)$ (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 $Ff=d_1\Delta^2$ for some $\Delta^2\in \mathcal D_2$, then every $\Lambda^2\in \mathcal C$ with $\Lambda^2_1 = f$ and $F\Lambda^2_0 = d_0\Delta^2$ can be filled in by $g$ with $Fg=d_2\Delta^2$.

Definition: Let $f\colon \mathcal C\to \mathcal D$ be a functor.

• The category $\mathcal C$ is $F$-fibered over $\mathcal D$ if for every morphism $h\in \Hom_{\mathcal D}(U,V)$ and every $B\in \Obj(\mathcal C)$ with $F(B)=V$, there is some $F$-cartesian $f\in \Hom_{\mathcal C}(-,B)$ with $Ff=h$.
• A cleavage of an $F$-fibered category $\mathcal C$ is a class of cartesian morphisms $K$ in $\mathcal C$ such that for every morphism $h\in \Hom_{\mathcal D}(U,V)$ and every $B\in \Obj(\mathcal C)$ with $F(B)=V$, there is a unique $F$-cartesian $f\in K$ with $Ff=h$.
• A cleavage of $\mathcal C$ is a splitting if it contains all the the identity morphisms and is closed under composition.

If $\mathcal C$ is $F$-fibered over $\mathcal D$ and $\mathcal C'$ is $F'$-fibered over $\mathcal D$, then a functor $\mathcal F\colon \mathcal C\to \mathcal C'$ is a \emph{morphism of fibered categories} if $F = F'\circ \mathcal F$ and $\mathcal Ff$ is $F'$-cartesian whenever $f$ is $F$-cartesian.

Theorem: Let $\mathcal C$ be $F$-fibered over $\mathcal D$.

• Every cleavage of $\mathcal C$ defines a pseudofunctor $\mathcal D\to \mathcal Cat$.
• Every pseudofunctor $\mathcal D\to \mathcal Cat$ defines an $F'$-fibered category $\mathcal C'$ with a cleavage over $\mathcal D$.

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 $S\in \sSet$ and the category of functors $\text{sCat}\to \sSet$. 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)