We discuss a new type of weak factorization system. Although these systems provide (up to isomorphism) uniquely determined decompositions of morphisms, in general they do not constitute orthogonal factorizations and are not even functorial. Nevertheless, they arise naturally, as injective hulls or projective covers in comma categories. Surprisingly, often injective hulls and projective covers c...

For small involutive and integral quantaloids Q it is shown that covariant presheaves on symmetric Q-categories are equivalent to certain subalgebras of a speci c monad on the category of symmetric Q-categories. This construction is related to a weakening of the subobject classi er axiom which does not require the classi cation of all subalgebras, but only guarantees that classi able subalgebra...

It is well known that we may represent (strict) ω-categories as simplicial sets, via Street’s ω-categorical nerve construction [2]. What may be less well known, is that we may extend Street’s nerve functor to one which has been shown to be fully-faithful (Verity [3]). This is achieved by augmenting each simplicial set in the codomain of this functor with a specified subset of thin simplices and...

We propose a notion of weak (n + k, n)-category, which we call (n + k, n)Θ-spaces. The (n + k, n)-Θ-spaces are precisely the fibrant objects of a certain model category structure on the category of presheaves of simplicial sets on Joyal’s category Θn. This notion is a generalization of that of complete Segal spaces (which are precisely the (∞, 1)-Θ-spaces). Our main result is that the above mod...