nLab
slice 2-category

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Idea

The generalization of a slice category to 2-categories.

Definition

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

  • as objects, the 1-morphisms a:AXa\colon A\to X in CC;

  • as 1-morphisms from a:AXa\colon A\to X to b:BXb\colon B\to X, the pairs (f,ϕ)(f,\phi) where f:ABf\colon A\to B is a 1-morphism in CC and ϕ:abf\phi\colon a \cong b f is a 2-isomorphism in CC.

  • as 2-morphisms from (f,ϕ)(f,\phi) to (g,ψ)(g,\psi), the 2-morphisms ξ:fg\xi\colon f \to g such that (bξ).ϕ=ψ(b \xi) . \phi = \psi.

If CC is a strict 2-category, then so is C/XC/X. Moreover, in this case, we can also define the strict-slice 2-category to be the subcategory C/ sXC/_s X of C/XC/X containing all the objects, only those morphisms (f,ϕ)(f,\phi) such that ϕ\phi is an identity, and all 2-morphisms between these.

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

Dually, the morphisms in the colax-slice 2-category involve a ϕ:bfa\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 CXC\sslash X, the canonical transformation

is a lax natural transformation.

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

Remarks

When CC 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 CC 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.

References

Last revised on January 11, 2021 at 11:29:04. See the history of this page for a list of all contributions to it.