Michael Shulman
slice 2-category

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

In particular, any 2-cell in K/XK/X must become an isomorphism in XX. 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 KK. Frequently a better replacement is the fibrational slice.

Pullbacks and adjoints

If KK has pullbacks, then for any f:XYf:X\to Y there is a pullback functor f *:K/YK/Xf^*:K/Y\to K/X. However, this does not make the assignation XK/XX\mapsto K/X into a functor K opCatK^{op}\to Cat or K coopCatK^{coop}\to 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/YK/Xf^*:K/Y\to K/X always has a left adjoint Σ f:K/XK/Y\Sigma_f:K/X\to K/Y given by composition with ff. However, f *f^* cannot be expected to have a right adjoint Π f\Pi_f for all maps ff, since this fails even in CatCat. It is true in CatCat when ff 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)