Knot quandles vs. knot biquandles

As a generalization of quandles, biquandles have given many invariants of classical/surface/virtual links. In this paper, we show that the fundamental quandle [Formula: see text] of any classical/surface link [Formula: see text] detects the fundamental biquandle [Formula: see text]; more precisely, there exists a functor [Formula: see text] from the category of quandles to that of biquandles such that [Formula: see text]. Then, we can expect invariants from biquandles to be reduced to those from quandles. In fact, we introduce a right-adjoint functor [Formula: see text] of [Formula: see text], which implies that the coloring number of a biquandle [Formula: see text] is equal to that of the quandle [Formula: see text].

Categorical definitions and properties via generators

In the present work, we show how the study of categorical constructions does not have to be done with all the objects of the category, but we can restrict ourselves to work with families of generators. Thus, universal properties can be characterized through iterated families of generators, which leads us in particular to an alternative version of the adjoint functor theorem. Similarly, the properties of relations or subobjects algebra can be investigated by this method. We end with a result that relates various forms of compactness through representable functors of generators.

Symmetric Self-adjunctions and Matrices

It is shown that the multiplicative monoids of Brauer's centralizer algebras generated out of the basis are isomorphic to monoids of endomorphisms in categories where an endofunctor is adjoint to itself, and where, moreover, a kind of symmetry involving the self-adjoint functor is satisfied. As in a previous paper, of which this is a companion, it is shown that such a symmetric self-adjunction is found in a category whose arrows are matrices, and the functor adjoint to itself is based on the Kronecker product of matrices.

Hodge realizations of 1-motives and the derived Albanese

We prove that the embedding of the derived category of 1-motives up to isogeny into the triangulated category of effective Voevodsky motives, as well as its left adjoint functor LAlbℚ, commute with the Hodge realization. This result yields a new proof of the rational form of Deligne's conjecture on 1-motives.