… an exactness sequent with β a constructible subsketch inclusion and if F is an X -structure in
a category C , the functor β C : B C → A C restricts to a full and faithful functor β … surjective on …

… -functor and prove some basic properties which will be used in Sect. 3, where we introduce
free \(\Phi \)-exact completions in … on why we need to consider the notion of \(\Phi \)-exactness …

… functors on the category of groups. Namely, we give several equivalent descriptions of right
exact functors on the … The exactness of the vertical left-hand sequence follows from the snake …

… whose homotopy category has nice exactness properties. … Then we can construct a path
object PY on Y and a fibration … So we can define a functor ρ: Hex(C)/i(X) → Hex(C(X)) by …

… on closure-type operators on the frame C. In the latter case we learn, in particular, that a
distributive complete … Thus we have a contravariant functor Ω : Top → Frm , where Top is the …

… a coaction functor with the full-crossed-product functor on … In this lemma, we were interested
in characterizing exactness of the … certainly has a unique commuting completion, and Ä is a .ı …

… properties of the category of simplicial sheaves on a topological space. Like categories of
fibrant … Thus, our work builds on the recently discovered interpretation of Martin–Löf's theory of …

… finite limit preserving functors from C to Set. Many so-called exactness properties have been
… , Mal’tsev, coherent with finite products and so on. In some sense, the authors’ approach to …

… Lastly, in Section 7, we outline how one can introduce completed Ext-functors and complete
… Because Mk is projective, it follows from the exactness of the above and from the definition …

… the notion of exactness for (n,1)-categories. We show that the exact completion provides a
… We denote by Catreg the subcategory of Cat on the regular categories and regular functors. …