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For every distributive law between pseudomonads, the 2-category of pseudoalgebras for the lifted pseudomonad is equivalent to the 2-category of pseudoalgebras for the composite pseudomonad.

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Given a generating family F of subgroups of a group G, closed under conjugation and with partial order compatible with inclusion, a new group S can be constructed, taking into account the multiplication in the subgroups and their mutual actions given by conjugation. The group S is called the active sum of F , has G as a homomorph and is such that S/Z(S) G/Z(G), where Z denotes the center. The basic question we investigate in this paper is: when is the active sum S of the family F isomorphic to the group G? The conditions found to answer this question are often of a homological nature. We show that the following groups are active sums of cyclic subgroups: free groups, semidirect products of cyclic groups, Coxeter groups, Wirtinger approximations, groups of order p 3 with p an odd prime, simple groups with trivial Schur multiplier, and special linear groups SLn(q) with a few exceptions. We show as well that every finite group G such that G/G is not cyclic is the active sum of proper normal subgroups.

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Axiomatic Cohesion proposes that the contrast between cohesion and non-cohesion may be expressed by means of a geometric morphism p : E → S (between toposes) with certain special properties that allow to effectively use the intuition that the objects of E are 'spaces' and those of S are 'sets'. Such geometric morphisms are called (pre-) cohesive. We may also say that E is pre-cohesive (over S). In this case, the topos E determines an S-enriched 'homotopy' category. The purpose of the present paper is to study certain aspects of this homotopy theory. We introduce weakly Kan objects in a pre-cohesive topos, which are analogous to Kan complexes in the topos of simplicial sets. Also, given a geometric morphism g : F → E between pre-cohesive toposes F and E (over the same base), we define what it means for g to preserve pieces. We prove that if g preserves pieces then it induces an adjunction between the homotopy categories determined by F and E, and that the direct image g * : F → E preserves weakly Kan objects. These and other results support the intuition that the inverse image of g is 'geometric realization'. In particular, since Kan complexes are weakly Kan in the pre-cohesive topos of simplicial sets, the result relating g and weakly Kan objects is analogous to the fact that the singular complex of a space is a Kan complex.

We provide a characterization of those morphisms of locales f : L → M for which the direct image functor f * : Sh(L) → Sh(M) is exact.

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