  A category is adhesive if it has all pullbacks, all pushouts along
monomorphisms, and all exactness conditions between pullbacks and pushouts
along monomorphisms which hold in a topos. This condition can be modified by
considering only pushouts along regular monomorphisms, or by asking only for
the exactness conditions which hold in a quasitopos. We prove four
characterization theorems dealing with adhesive categories and their variants.
