If S is an order-adjoint monad, that is, a monad on Set that factors through the category of ordered sets with left adjoint maps, then any monad morphism τ : S → T makes T orderadjoint, and the Eilenberg-Moore category of T is monadic over the category of monoids in the Kleisli category of S.

We introduce the concept of firm classes of morphisms as basis for the axiomatic study of completions of objects in arbitrary categories. Results on objects injective with respect to given morphism classes are included. In a finitely well-complete category, firm classes are precisely the coessential first factors of morphism factorization structures.

We characterize the (effective) E-descent morphisms in the category Cat of small categories, when E is the class of discrete fibrations or the one of discrete cofibrations, and prove that every effective global-descent morphism is an effective E-descent morphism while its converse fails.

̄ As an example, the negation map that send P ∈ E(k) to its additive inverse is an isogeny from E to itself; as noted in Lecture 23, it is an automorphism, hence a surjective morphism, and it clearly fixes the identity element (the distinguished rational point O). Recall that a morphism of projective curves is either constant or surjective, so any nonconstant morphism that maps O1 to O2 is autom...

Problem 1. Let C be a category with the zero object OC (that is, an object which is both universal and co-universal). In particular, for any pair of objects M and N , we have a distinguished morphism 0M,N ∈ HomC (M,N) , called the zero morphism, which is the composition of the unique morphisms M → OC and OC → N . Let M and N be objects in C and f ∈ HomC (M,N) . Recall that a kernel of f is a pa...

We study cocompleteness, co-wellpoweredness, and generators in the centralizer category of an object or morphism a monoidal category, center weak category. explicitly give some answers for when colimits, these categories can be inherited from their base monidal categories. Most importantly, we investigate cofree objects comonoids

Abstract. We introduce the concept of a morphism between coloured nets. Our definition generalizes Petris definition for ordinary nets. A morphism of coloured nets maps the topological space of the underlying undirected net as well as the kernel and cokernel of the incidence map. The kernel are flows along the transitionbordered fibres of the morphism, the cokernel are classes of markings of th...

In [6] Higson showed that the formal properties of the Kasparov KK -theory groups are best understood if one regards KK (A, B) for separable C∗-algebras A, B as the morphism set of a category KK . In category language the composition and exterior KK product give KK the structure of a symmetric monoidal category which is enriched over abelian groups. We show that the enrichment of KK can be lift...

A poset can be regarded as a category in which there is at most one morphism between objects, and such that at most one of Hom(c, c′) and Hom(c′, c) is nonempty for c 6= c′. If we keep in place the latter axiom but allow for more than one morphism between objects, we have a sort of generalized poset in which there are multiplicities attached to the covering relations, and possibly nontrivial au...