Abstract In this work, we conclude our study of fibred $\infty $ -bicategories by providing a Grothendieck construction in this setting. Given a scaled simplicial set S (which need not be fibrant) we construct a 2-categorical version of Lurie’s straightening-unstraightening adjunction, thereby furnishing an equivalence between the $\infty $ -bicategory of 2-Cartesian fibrations over S and the $\infty $ -bicategory of contravariant functors with values in the $\infty $ -bicategory of $\infty $ -bicategories. We provide a relative nerve construction in the case where the base is a 2-category, and use this to prove a comparison to existing bicategorical Grothendieck constructions.

Abstract We construct a monoidal version of Lurie’s un/straightening equivalence. In more detail, for any symmetric monoidal $\infty $ -category $\mathbf {C}$ , we endow the $\infty $ -category of coCartesian fibrations over $\mathbf {C}$ with a (naturally defined) symmetric monoidal structure, and prove that it is equivalent the Day convolution monoidal structure on the $\infty $ -category of functors from $\mathbf {C}$ to $\mathbf {Cat}_\infty $ . In fact, we do this over any $\infty $ -operad by categorifying this statement and thereby proving a stronger statement about the functors that assign to an $\infty $ -category $\mathbf {C}$ its category of coCartesian fibrations on the one hand, and its category of functors to $\mathbf {Cat}_\infty $ on the other hand.

Let C be a small category. In this paper, we mainly study the category of modules M od‐ R on ringed sites ( C , R ) . We firstly reprove the Theorem A of the paper [Wu, M. and Xu,F., Skew category algebras and modules on ringed finite sites. J. A. 631, 2023], then we characterize M od‐ R in terms of the torsion modules on Gr ( R ) , where Gr ( R ) is the linear Grothendieck construction of R . Finally, we investigate the hereditary torsion pairs, TTF triples and Abelian recollements of M od‐ R . When C is finite, the complete classifications of all these are given, respectively.

For a plural signature Σ and with regard to the category NPIAlg(Σ)s, of naturally preordered idempotent Σ-algebras and surjective homomorphisms, we define a contravariant functor LsysΣ from NPIAlg(Σ)s to Cat, the category of categories, that assigns to I in NPIAlg(Σ)s the category I-LAlg(Σ), of I-semi-inductive Lallement systems of Σ-algebras, and a covariant functor (Alg(Σ)↓s·) from NPIAlg(Σ)s to Cat, that assigns to I in NPIAlg(Σ)s the category (Alg(Σ)↓sI), of the coverings of I, i.e., the ordered pairs (A,f) in which A is a Σ-algebra and a surjective homomorphism. Then, by means of the Grothendieck construction, we obtain the categories ∫NPIAlg(Σ)sLsysΣ and ∫NPIAlg(Σ)s(Alg(Σ)↓s·); define a functor LΣ from the first category to the second, which we will refer to as the Lallement functor; and prove that it is a weak right multiadjoint. Finally, we state the relationship between the Płonka functor and the Lallement functor.

Abstract The Grothendieck construction establishes an equivalence between fibrations, a.k.a. fibred categories and indexed categories and is one of the fundamental results of category theory. Cockett and Cruttwell introduced the notion of fibrations into the context of tangent categories and proved that the fibres of a tangent fibration inherit a tangent structure from the total tangent category. The main goal of this paper is to provide a Grothendieck construction for tangent fibrations. Our first attempt will focus on providing a correspondence between tangent fibrations and indexed tangent categories, which are collections of tangent categories and tangent morphisms indexed by the objects and morphisms of a base tangent category. We will show that this construction inverts Cockett and Cruttwell’s result, but it does not provide a full equivalence between these two concepts. In order to understand how to define a genuine Grothendieck equivalence in the context of tangent categories, inspired by Street’s formal approach to monad theory we introduce a new concept: tangent objects. We show that tangent fibrations arise as tangent objects of a suitable $2$ -category and we employ this characterisation to lift the Grothendieck construction between fibrations and indexed categories to a genuine Grothendieck equivalence between tangent fibrations and tangent indexed categories.

We introduce partially lax limits of ∞ \infty -categories, which interpolate between ordinary limits and lax limits. Most naturally occurring examples of lax limits are only partially lax; we give examples arising from enriched ∞ \infty -categories and ∞ \infty -operads. Our main result is a formula for partially lax limits and colimits in terms of the Grothendieck construction. This generalizes a formula of Lurie for ordinary limits and of Gepner, Haugseng, and Nikolaus for fully lax limits.

Correction to: Ramsey Properties of Products and Pullbacks of Categories and the Grothendieck Construction

Abstract We show how to apply forward and reverse mode Combinatory Homomorphic Automatic Differentiation (CHAD) (Vákár (2021). ESOP, 607–634; Vákár and Smeding (2022). ACM Transactions on Programming Languages and Systems44 (3) 20:1–20:49.) to total functional programming languages with expressive type systems featuring the combination of •tuple types;•sum types;•inductive types;•coinductive types;•function types.We achieve this by analyzing the categorical semantics of such types in $\Sigma$ -types (Grothendieck constructions) of suitable categories. Using a novel categorical logical relations technique for such expressive type systems, we give a correctness proof of CHAD in this setting by showing that it computes the usual mathematical derivative of the function that the original program implements. The result is a principled, purely functional and provably correct method for performing forward- and reverse-mode automatic differentiation (AD) on total functional programming languages with expressive type systems.

In this paper we provide purely categorical proofs of two important results of structural Ramsey theory: the result of M. Sokić that the free product of Ramsey classes is a Ramsey class, and the result of M. Bodirsky that adding constants to the language of a Ramsey class preserves the Ramsey property. The proofs that we present here ignore the model-theoretic background of these statements. Instead, they focus on categorical constructions by which the classes can be constructed generalizing the original statements along the way. It turns out that the restriction to classes of relational structures, although fundamental for the original proof strategies, is not relevant for the statements themselves. The categorical proofs we present here remove all restrictions on the signature of first-order structures and provide the information not only about the Ramsey property but also about the Ramsey degrees.

Everyone knows that if you have a bivariant homology theory satisfying a base change formula, you get an representation of a category of correspondences. For theories in which the covariant and contravariant transfer maps are in mutual adjunction, these data are actually equivalent. In other words, a 2-category of correspondences is the universal way to attach to a given 1-category a set of right adjoints that satisfy a base change formula. Through a bivariant version of the Yoneda paradigm, I give a definition of correspondences in higher category theory and prove an extension theorem for bivariant functors. Moreover, conditioned on the existence of a 2-dimensional Grothendieck construction, I provide a proof of the aforementioned universal property. The methods, morally speaking, employ the `internal logic' of higher category theory: they make no explicit use of any particular model.

We provide a categorical semantics for bounded Petri nets, both in the collective- and individual-token philosophy. In both cases, we describe the process of bounding a net internally, by just constructing new categories of executions of a net using comonads, and externally, using lax-monoidal-lax functors. Our external semantics is non-local, meaning that tokens are endowed with properties that say something about the global state of the net. We then prove, in both cases, that the internal and external constructions are equivalent, by using machinery built on top of the Grothendieck construction. The individual-token case is harder, as it requires a more explicit reliance on abstract methods.

In category theory circles it is well-known that the Schreier theory of group extensions can be understood in terms of the Grothendieck construction on indexed categories. However, it is seldom discussed how this relates to extensions of monoids. We provide an introduction to the generalised Grothendieck construction and apply it to recover classifications of certain classes of monoid extensions (including Schreier and weakly Schreier extensions in particular).

We show how a particular variety of hierarchical nets, where the firing of a transition in the parent net must correspond to an execution in some child net, can be modelled utilizing a functorial semantics from a free category -- representing the parent net -- to the category of sets and spans between them. This semantics can be internalized via Grothendieck construction, resulting in the category of executions of a Petri net representing the semantics of the overall hierarchical net. We conclude the paper by giving an engineering-oriented overview of how our model of hierarchical nets can be implemented in a transaction-based smart contract environment.

Diagrams of quantales and Lipschitz norms

We obtain combinatorial model categories of parametrised spectra, together with systems of base change Quillen adjunctions associated to maps of parameter spaces. We work with simplicial objects and use Hovey's sequential and symmetric stabilisation machines. By means of a Grothendieck construction for model categories, we produce combinatorial model categories controlling the totality of parametrised stable homotopy theory. The global model category of parametrised symmetric spectra is equipped with a symmetric monoidal model structure (the external smash product) inducing pairings in twisted cohomology groups. As an application of our results we prove a tangent prolongation of Simpson's theorem, characterising tangent $\infty$-categories of presentable $\infty$-categories as accessible localisations of $\infty$-categories of presheaves of parametrised spectra. Applying these results to the homotopy theory of smooth $\infty$-stacks produces well-behaved (symmetric monoidal) model categories of smooth parametrised spectra. These models provide a concrete foundation for studying twisted differential cohomology, incorporating previous work of Bunke and Nikolaus.

We characterize the category of Sambin’s positive topologies as the result of the Grothendieck construction applied to a doctrine over the category Loc of locales. We then construct an adjunction between the category of positive topologies and that of topological spaces Top, and show that the well-known adjunction between Top and Loc factors through the constructed adjunction.

A (closed) dynamical system is a notion of how things can be, together with a notion of how they may change given how they are. The idea and mathematics of closed dynamical systems has proven incredibly useful in those sciences that can isolate their object of study from its environment. But many changing situations in the world cannot be meaningfully isolated from their environment - a cell will die if it is removed from everything beyond its walls. To study systems that interact with their environment, and to design such systems in a modular way, we need a robust theory of open dynamical systems. In this extended abstract, we put forward a general definition of open dynamical system. We define two general sorts of morphisms between these systems: covariant morphisms which include trajectories, steady states, and periodic orbits; and contravariant morphisms which allow for plugging variables of some systems into parameters of other systems. We define an indexed double category of open dynamical systems indexed by their interface and use a double Grothendieck construction to construct a double category of open dynamical systems. In our main theorem, we construct covariantly representable indexed double functors from the indexed double category of dynamical systems to an indexed double category of spans. This shows that all covariantly representable structures of dynamical systems - including trajectories, steady states, and periodic orbits - compose according to the laws of matrix arithmetic.

In this work, we introduce a 2-categorical variant of Lurie's relative nerve functor. We prove that it defines a right Quillen equivalence which, upon passage to $\infty$-categorical localizations, corresponds to Lurie's scaled unstraightening equivalence. In this $\infty$-bicategorical context, the relative 2-nerve provides a computationally tractable model for the Grothendieck construction which becomes equivalent, via an explicit comparison map, to Lurie's relative nerve when restricted to 1-categories.

The aim of the paper is to introduce an approach to the theory of 2-categories which is based on systematic use of the Grothendieck construction and the Segal Machine and to show how adjunction questions can be investigated by means of this approach and what its connections are with more traditional approaches. As an application, the derived Morita 2-category and the Fourier–Mukai 2-category over a Noetherian ring are constructed and the embedding of the latter in the former is demonstrated. Bibliography: 15 titles.

Enriched Lawvere theories are a generalization of Lawvere theories that allow\nus to describe the operational semantics of formal systems. For example, a\ngraph enriched Lawvere theory describes structures that have a graph of\noperations of each arity, where the vertices are operations and the edges are\nrewrites between operations. Enriched theories can be used to equip systems\nwith operational semantics, and maps between enriching categories can serve to\ntranslate between different forms of operational and denotational semantics.\nThe Grothendieck construction lets us study all models of all enriched theories\nin all contexts in a single category. We illustrate these ideas with the\nSKI-combinator calculus, a variable-free version of the lambda calculus.\n