# Puppe sequence

The construction can then be iterated to obtain the exact Puppe sequence

*(the) various constructions (of the coexact sequence) involve quotient spaces instead of subspaces, and so all maps and homotopies require more scrutiny to ensure that they are well-defined and continuous.*

From this, the Puppe sequence gives the **homotopy sequence of a fibration**:

This bijection can be used in the relative homotopy sequence above, to obtain the **homotopy sequence of a weak fibration**, having the same form as the fibration sequence, although with a different connecting map.

It is a simple exercise in topology to see that every three elements of a Puppe sequence are, up to a homotopy, of the form:

By "up to a homotopy", we mean here that every 3 elements in a Puppe sequence are of the above form if regarded as objects and morphisms in the homotopy category.