slice 2-category

The **slice 2-category** of a 2-category $K$ over an object $X$ is the 2-category whose objects are morphisms $A\to X$ in $K$, whose morphisms are triangles in $K$ that commute up to a specified isomorphism, and whose 2-cells are 2-cells in $K$ forming a commutative 2-diagram with the specified isomorphisms in triangles.

In particular, any 2-cell in $K/X$ must become an isomorphism in $X$. This means that more information is lost when passing to slice 2-categories than for slice 1-categories, and slice 2-categories are not always well-behaved; they often fail to inherit useful properties of $K$. Frequently a better replacement is the fibrational slice.

If $K$ has pullbacks, then for any $f:X\to Y$ there is a pullback functor ${f}^{*}:K/Y\to K/X$. However, this does *not* make the assignation $X\mapsto K/X$ into a functor ${K}^{\mathrm{op}}\to \mathrm{Cat}$ or ${K}^{\mathrm{coop}}\to \mathrm{Cat}$, since there is no way to define it on 2-cells. This is one reason to use fibrational slices instead.

Just as in the 1-categorical case, the pullback functor ${f}^{*}:K/Y\to K/X$ always has a left adjoint ${\Sigma}_{f}:K/X\to K/Y$ given by composition with $f$. However, ${f}^{*}$ cannot be expected to have a right adjoint ${\Pi}_{f}$ for *all* maps $f$, since this fails even in $\mathrm{Cat}$. It is true in $\mathrm{Cat}$ when $f$ is a fibration or opfibration, however; see exponentials in a 2-category.

Revised on February 20, 2009 19:30:35
by Mike Shulman
(75.3.140.11)