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In [2], classes of objects injective with respect to a set M of morphisms of a locally presentable category K were characterized: they are precisely the classes closed under products,
The reason all coherent 2-groups with the same underlying weak 2-group are isomorphic is that we have defined a homomorphism of coherent 2-groups to be a weak monoidal functor,
In order to give an example of a protomodular locally finitely presentable category C with a zero object which does not have a semi-abelian generator, we will present it as
To obtain the classical examples K is CAT the 2-category of categories, T is the symmetric monoidal category monad, to say that A has a monoidal pseudo-T -algebra structure is to
We call them the left double , right double and double of the monoidal.. V
We then prove embedding theorems: any locally small pre-Hilbert category whose monoidal unit is a simple generator embeds (weakly) monoidally into the category of pre-Hilbert spaces
riety isa locally finitely presentable category; in [1] Ad´amek and Porst characterized (the dual of) those essentially algebraic theories whose models form a quasivariety as being
A Gray-monoid [DS97, Section 1] may be considered to be a monoidal2-category, and in fact every monoidal 2-category is biequivalent to a Gray-monoid [GPS95, Section 8.1]; indeed this
