nLab
slice 2-category

Idea

The generalization of a slice category to 2-categories.

Definition

Let $C$ be a (non-strict) 2-category, and $X\in C$ an object. Then the slice 2-category $C/X$ has:

• as objects, the 1-morphisms $a:A\to X$ in $C$;

• as 1-morphisms from $a:A\to X$ to $b:B\to X$, the pairs $(f,\varphi )$ where $f:A\to B$ is a 1-morphism in $C$ and $\varphi :bf\cong a$ is a 2-isomorphism in $C$.

• as 2-morphisms from $(f,\varphi )$ to $(g,\psi )$, the 2-morphisms $\xi :f\to g$ such that $\psi .(b\xi )=\varphi$.

If $C$ is a strict 2-category, then so is $C/X$. Moreover, in this case, we can also define the strict-slice 2-category to be the subcategory $C{/}_{s}X$ of $C/X$ containing all the objects, only those morphisms $(f,\varphi )$ such that $\varphi$ is an identity, and all 2-morphisms between these.

If, on the other hand, we do not require $\varphi$ to be invertible, then we obtain the lax-slice 2-category $C⫽X$ (with evident dual the colax-slice 2-category).

Finally, the subcategory of $C/X$ whose objects are the fibrations and whose morphisms are the maps of fibrations is denoted $\mathrm{Fib}(X)={\mathrm{Fib}}_C(X)={\mathrm{Fib}}_C/X$, and may be called the fibrational-slice 2-category. Similarly we have the opfibrational-slice $\mathrm{Opf}(X)$.

Remarks

When $C$ is a 1-category, the slice, strict-slice, lax- and colax-slice, and fibrational- and opfibrational-slice 2-categories all coincide with the usual slice category. When $C$ is a (2,1)-category, then all of them coincide except the strict one. Thus, when generalizing a concept involving slice categories from categories to 2-categories, it can sometimes be a little subtle to decide on the correct version of slice category to use.