Following earlier work, a modal logic for T -coalgebras is a functor L on a suitable variety. Syntax and proof system of the logic are given by presentations of the functor. This paper makes two contributions. First, a previous result characterizing those functors that have presentations is generalized from endofunctors on one-sorted varieties to functors between many-sorted varieties. This yie...

We construct sections of a structure presheaf of rational functions on a differential spectrum using only localization and projective limits. For this purpose we introduce a special form of a multiplicative system generated by one differential polynomial and call it D-localization. This technique allows us to obtain sections of a differential spectrum of a differential ring R without computatio...

The category of all monads over many-sorted sets (and over other " set-like " categories) is proved to have coequalizers and strong coin-tersections. And a general diagram has a colimit whenever all the monads involved preserve monomorphisms and have arbitrarily large joint pre-fixpoints. In contrast, coequalizers fail to exist e.g. for monads over the (presheaf) category of graphs. For more ge...

Introduction In classical C̆ech theory, we “compute” (or better: filter) the cohomology of a sheaf when given an open covering. Namely, if X is a topological space, U = {Ui} is an indexed open covering, and F is an abelian sheaf on X, then we get a C̆ech to derived functor spectral sequence E 2 = H (U,H(F ))⇒ H(X,F ), where H(F ) is the presheaf whose value on an open U is H(U,F |U ) (and we use ...

Abstract We define a motivic conductor for any presheaf with transfers F using the categorical framework developed theory of motives modulus by Kahn, Miyazaki, Saito and Yamazaki. If is reciprocity sheaf, this yields an increasing exhaustive filtration on $F(L)$ , where L henselian discrete valuation field geometric type over perfect ground field. show that if smooth group scheme, then extends ...

For any smooth free action of the unit circle S in a manifold M ; the Gysin sequence of M is a long exact sequence relating the DeRham cohomologies of M and its orbit space M/S. If the action is not free then M/S is not a manifold but a stratified pseudomanifold and there is a Gysin sequence relating the DeRham cohomology of M with the intersection cohomology of M/S. In this work we extend the ...

We study the different guises of the projective objects in Cocont(Q): they are the “completely distributive” cocomplete Q-categories (the left adjoint to the Yoneda embedding admits a further left adjoint); equivalently, they are the “totally continuous” cocomplete Q-categories (every object is the supremum of the presheaf of objects “totally below” it); and also are they the Q-categories of re...

We introduce the notion of state dependent interactive programs for Martin-Löf Type Theory. These programs are elements of coalgebras of an endofunctor on the presheaf category S → Set. We prove the existence of final coalgebras for these functors. This shows as well the consistency of type theory plus rules expressing the existence of weakly final coalgebras for these functors, which represent...

We generalize the definition of satellites with respect to presheaves (and copresheaves) with trace in the sense of [1]; a presheaf with trace is replaced by a graph with a pair of diagrams defined on it. We show that the right satellite functor is left adjoint to the left satellite functor, and that a functor having a right (left) adjoint preserves right (left) satellites. In particular cases ...

Introduction p. 1 1. Concrete presheaf categories p. 20 2. ω-graphs p. 26 3. ω-categories p. 27 4. Adjoining indeterminates p. 32 5. Computads p. 42 6. Multitopes and computopes p. 50 7. Words for computads p. 59 8. Another set of primitive operations for ω-categories p. 66 9. A construction of the one-step extension X[U] p. 75 10. Solution of the word problem p. 88 11. Proof of the existence o...