Exempel på pullback

• Furthermore, since the bottom, front and top squares are pullbacks, the back square is also a pullback.

• This category has weak pullbacks and setindexed products, but, in general, no pullbacks.

• The interior operation is not assumed to be natural; that is, it is not assumed to commute with pullbacks.

• First it can be easily checked that the following diagram is a pullback.

• For example, collections can be zipped if and only if the monad's endofunctor preserves pullbacks.

• We will also have use for inverse images, which are pullbacks of two morphisms at least one of which is a monomorphism.

• The role of pullbacks in categorical interpretations of logic is to perform substitution of terms into predicates and formulas.

• Let be a weak inclusive category that has pullbacks.

• The gist of affine equivalence techniques is to use these estimates in combination with the pullback formulas (2.16)-(2.19) for vector proxies.

• Techniques based on the pullback of differential forms are instrumental in getting interpolation estimates.

• Let us note a general 2-categorical fact, that pullback functors between slices preserve existing exponentials.

• For the rest, we add as before the pullback tiles that provide the coordination mechanism about local instantiations.

• Then (1) is a pullback if and only if it satisfies the reduced chain-condition and f, g is monomorphic.

• Indeed, to interpret substitution, we need to find a canonical choice of pullbacks that is functorial and is strictly preserved by the categorical structure.

• Here we require that the category must be at least lex, since we want to interpret substitution via pullback.