Michael Shulman
slice 2-category

The slice 2-category of a 2-category K over an object X is the 2-category whose objects are morphisms AX 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.

Pullbacks and adjoints

If K has pullbacks, then for any f:XY there is a pullback functor f *:K/YK/X. However, this does not make the assignation XK/X into a functor K opCat or K coopCat, 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/X always has a left adjoint Σ f:K/XK/Y given by composition with f. However, f * cannot be expected to have a right adjoint Π f for all maps f, since this fails even in Cat. It is true in 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)