Klein defined geometry in terms of invariance under groups actions; here we give a discrete (partial) converse of this, interpreting all (finitely presentable) groups in terms of the geometry of hyperbolic 3-manifolds (whose fundamental groups are, appropriately, Kleinian groups). For G∗ a Kleinian group of isometries of hyperbolic 3-space H, with MG∗ ∼= H3/G∗ a non-compact N -cusped orientable...

In a locally λ-presentable category, with λ a regular cardinal, classes of objects that are injective with respect to a family of morphisms whose domains and codomains are λ-presentable, are known to be characterized by their closure under products, λ-directed colimits and λ-pure subobjects. Replacing the strict commutativity of diagrams by “commutativity up to ε”, this paper provides an “appro...

The category HopfC of Hopf monoids in a symmetric monoidal category C, assumed to be locally finitely presentable as a category, is analyzed with respect to its categorical properties. Assuming that the functors “tensor squaring” and “tensor cubing” on C preserve directed colimits one has the following results: (1) If, in C, extremal epimorphisms are stable under tensor squaring, then HopfC is ...

The systematic study of quadratic forms over an arbitrary field of characteristic = 2 was initiated by Witt [W]. Two milestone papers in the area are a paper of Pfister [P0] relating quadratic forms and orderings and the paper of Milnor [M] pointing out a possible relationship between the Witt ring of quadratic forms and the Galois cohomology of the field (a relationship eventually verified by ...

This paper draws close connections between the ease of presenting a given complexity class C and the position of the index sets IC = f i : L(Mi) 2 C g and JC = f i : Mi is total ^ L(Mi) = 2 C g in the arithmetical hierarchy. For virtually all classes C studied in the literature, the lowest levels attainable are IC 2 P03 and JC 2 Q02; the rst holds i C is 02-presentable, and the second i C is re...

We define filter quotients of $(\infty,1)$-categories and prove that preserve the structure an elementary $(\infty,1)$-topos in particular lift quotient underlying topos. then specialize to case products a characterization theorem for equivalences product. Then we use construct large class $(\infty,1)$-toposes are not Grothendieck $(\infty,1)$-toposes. Moreover, give one detailed example intere...