… 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 …

… 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 …

… 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. …

… a natural universal property: every regular functor C → E into an … In this paper we look at the
notions of regularity and exactness … good on our claim, we need a preparatory result on the …

… work on preservation of exactness properties by pro-completions, to prove that these exactness
… For any i ≠ j , we define the functor F i : A i → C to be the constant functor mapping each …

… In Section 3, we will first construct the unstable p-completion functor on an arbitrary
presentable ∞-category X, and then show that if X is moreover an ∞-topos, then this …