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 $a\colon A\to X$ in $C$;

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

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

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 $(f,\phi)$ such that $\phi$ is an identity, and all 2-morphisms between these.

If, on the other hand, we do not require $\phi : a \to b f$ to be invertible, then we obtain the lax-slice 2-category $C\sslash X$. This can be regarded as an F-category whose tight morphisms are those where $\phi$ is invertible (or an identity).

Dually, the morphisms in the colax-slice 2-category involve a $\phi : b f \to a$. It is not clear whether there is a universally accepted convention as to which is the lax-slice and which is the colax-slice; the one adopted here is that used by Johnson-Yau and is such that for the lax-slice $C\sslash X$, the canonical transformation is a lax natural transformation.

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

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.

Related concepts

• fibrational slice

• slice category

• slice (∞,1)-category

References

• Niles Johnson, Donald Yau, 2-Dimensional Categories (arXiv:2002.06055)

On 2-colimits in slice 2-categories:

• Luca Mesiti, Colimits in 2-dimensional slices [arXiv:2305.01494]