Related papers: Internal languages of locally cartesian closed $(\…
Reasoning about weak higher categorical structures constitutes a challenging task, even to the experts. One principal reason is that the language of set theory is not invariant under the weaker notions of equivalence at play, such as…
We establish a Dwyer-Kan equivalence of relative categories of combinatorial model categories, presentable quasicategories, and other models for locally presentable (infinity,1)-categories. This implies that the underlying quasicategories…
We introduce $L^2_{K,P}$, a monadic second-order language for reasoning about trees which characterizes the strongly Context-Free Languages in the sense that a set of finite trees is definable in $L^2_{K,P}$ iff it is (modulo a projection)…
We introduce fibred type-theoretic fibration categories which are fibred categories between categorical models of Martin-L\"{o}f type theory. Fibred type-theoretic fibration categories give a categorical description of logical predicates…
We investigate the properties of relative analogues of admissible Ind, Pro, and elementary Tate objects for pairs of exact categories, and give criteria for those categories to be abelian. A relative index map is introduced, and as an…
Here are considered some categorical aspects of "Differential calculus" archetype of local approximation of arbitrary morphisms by "linear" ones.
We extend the model structure on the category $\mathbf{Cat}(\mathcal{E})$ of internal categories studied by Everaert, Kieboom and Van der Linden to an algebraic model structure. Moreover, we show that it restricts to the category of…
We present a type theory dealing with non-linear, "ordinary" dependent types (which we will call cartesian) and linear types, where both constructs may depend on terms of the former. In the interplay between these, we find new type formers…
In this paper, we define indexed type theories which are related to indexed ($\infty$-)categories in the same way as (homotopy) type theories are related to ($\infty$-)categories. We define several standard constructions for such theories…
The local intertwining relation is an identity that gives precise information about the action of normalized intertwining operators on parabolically induced representations. We prove several instances of the local intertwining relation for…
A locally compact contraction group is a pair (G,f) where G is a locally compact group and f an automorphism of G which is contractive in the sense that the forward orbit under f of each g in G converges to the neutral element e, as n tends…
We attach to each weak model category $\mathcal{M}$ a class of first order formulas about the fibrant objects of $\mathcal{M}$ whose validity is invariant under homotopies and weak equivalences. This is a generalization of the classical…
The cartesian structure possessed by relations, spans, profunctors, and other such morphisms is elegantly expressed by universal properties in double categories. Though cartesian double categories were inspired in part by the older program…
A type theory is presented that combines (intuitionistic) linear types with type dependency, thus properly generalising both intuitionistic dependent type theory and full linear logic. A syntax and complete categorical semantics are…
We prove a compactness result with respect to $\Gamma$-convergence for a class of integral functionals which are expressed as a sum of a local and a non-local term. The main feature is that, under our hypotheses, the local part of the…
For certain theories of existentially closed topological differential fields, we show that there is a strong relationship between $\mathcal L\cup\{D\}$-definable sets and their $\mathcal L$-reducts, where $\mathcal L$ is a relational…
We describe a non-extensional variant of Martin-L\"of type theory which we call two-dimensional type theory, and equip it with a sound and complete semantics valued in 2-categories.
We set up a formalism of Maurer-Cartan moduli sets for L-infinity algebras and associated twistings based on the closed model category structure on formal differential graded algebras (a.k.a. differential graded coalgebras). Among other…
Eilenberg's variety theorem, a centerpiece of algebraic automata theory, establishes a bijective correspondence between varieties of languages and pseudovarieties of monoids. In the present paper this result is generalized to an abstract…
We present a new coherence theorem for comprehension categories, providing strict models of dependent type theory with all standard constructors, including dependent products, dependent sums, identity types, and other inductive types.…