Kan Extension Research Articles

Using Kan Extensions to Motivate the Design of a Surprisingly Effective Unsupervised Linear SVM on the Occupancy Dataset

Recent research has suggested that category theory can provide useful insights into the field of machine learning (ML). One example is improving the connection between an ML problem and the design of a corresponding ML algorithm. A tool from category theory called a Kan extension is used to derive the design of an unsupervised anomaly detection algorithm for a commonly used benchmark, the Occupancy dataset. Achieving an accuracy of 93.5% and an ROCAUC of 0.98, the performance of this algorithm is compared to state-of-the-art anomaly detection algorithms tested on the Occupancy dataset. These initial results demonstrate that category theory can offer new perspectives with which to attack problems, particularly in making more direct connections between the solutions and the problem’s structure.

Homotopy Sheaves on Generalised Spaces

We study the homotopy right Kan extension of homotopy sheaves on a category to its free cocompletion, i.e. to its category of presheaves. Any pretopology on the original category induces a canonical pretopology of generalised coverings on the free cocompletion. We show that with respect to these pretopologies the homotopy right Kan extension along the Yoneda embedding preserves homotopy sheaves valued in (sufficiently nice) simplicial model categories. Moreover, we show that this induces an equivalence between sheaves of spaces on the original category and colimit-preserving sheaves of spaces on its free cocompletion. We present three applications in geometry and topology: first, we prove that diffeological vector bundles descend along subductions of diffeological spaces. Second, we deduce that various flavours of bundle gerbes with connection satisfy (∞,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\infty ,2)$$\\end{document}-categorical descent. Finally, we investigate smooth diffeomorphism actions in smooth bordism-type field theories on a manifold. We show how these smooth actions allow us to extract the values of a field theory on any object coherently from its values on generating objects of the bordism category.

Cellular automata and Kan extensions

In this paper, we formalize precisely the sense in which the application of a cellular automaton to partial configurations is a natural extension of its local transition function through the categorical notion of Kan extension. In fact, the two possible ways to do such an extension and the ingredients involved in their definition are related through Kan extensions in many ways. These relations provide additional links between computer science and category theory, and also give a new point of view on the famous Curtis–Hedlund theorem of cellular automata from the extended topological point of view provided by category theory. These links also allow to relatively easily generalize concepts pioneered by cellular automata to arbitrary kinds of possibly evolving spaces. No prior knowledge of category theory is assumed for the most part.

Fast Left Kan Extensions Using the Chase

We show how computation of left Kan extensions can be reduced to computation of free models of cartesian (finite-limit) theories. We discuss how the standard and parallel chase compute weakly free models of regular theories and free models of cartesian theories and compare the concept of “free model” with a similar concept from database theory known as “universal model”. We prove that, as algorithms for computing finite-free models of cartesian theories, the standard and parallel chase are complete under fairness assumptions. Finally, we describe an optimized implementation of the parallel chase specialized to left Kan extensions that achieves an order of magnitude improvement in our performance benchmarks compared to the next fastest left Kan extension algorithm we are aware of.

Kan Extensions are Partial Colimits

One way of interpreting a left Kan extension is as taking a kind of “partial colimit”, whereby one replaces parts of a diagram by their colimits. We make this intuition precise by means of the partial evaluations sitting in the so-called bar construction of monads. The (pseudo)monads of interest for forming colimits are the monad of diagrams and the monad of small presheaves, both on the (huge) category CAT of locally small categories. Throughout, particular care is taken to handle size issues, which are notoriously delicate in the context of free cocompletion. We spell out, with all 2-dimensional details, the structure maps of these pseudomonads. Then, based on a detailed general proof of how the restriction-of-scalars construction of monads extends to the case of pseudoalgebras over pseudomonads, we consider a morphism of monads between them, which we call image. This morphism allows in particular to generalize the idea of confinal functors, i.e. of functors which leave colimits invariant in an absolute way. This generalization includes the concept of absolute colimit as a special case. The main result of this paper spells out how a pointwise left Kan extension of a diagram corresponds precisely to a partial evaluation of its colimit. This categorical result is analogous to what happens in the case of probability monads, where a conditional expectation of a random variable corresponds to a partial evaluation of its center of mass.

A class of exactness properties characterized via left Kan extensions

We consider a general class of exactness properties on a finitely complete category, all of which can be expressed as the condition that a certain morphism in a diagram is a strong epimorphism . For each such exactness property, we characterize finitely bicomplete categories having the property by restricting the condition to those diagrams built from only one object in the category via a left Kan extension. In the regular context, this generalizes the theory of approximate co-operations introduced by D. Bourn and Z. Janelidze. As an application, we deduce from this a characterization of (essentially) algebraic categories satisfying such a given exactness property. The pointed version of these exactness properties is also studied.

Internal coalgebras in cocomplete categories: Generalizing the Eilenberg–Watts theorem

The category of internal coalgebras in a cocomplete category [Formula: see text] with respect to a variety [Formula: see text] is equivalent to the category of left adjoint functors from [Formula: see text] to [Formula: see text]. This can be seen best when considering such coalgebras as finite coproduct preserving functors from [Formula: see text], the dual of the Lawvere theory of [Formula: see text], into [Formula: see text]: coalgebras are restrictions of left adjoints and any such left adjoint is the left Kan extension of a coalgebra along the embedding of [Formula: see text] into [Formula: see text]. Since [Formula: see text]-coalgebras in the variety [Formula: see text] for rings [Formula: see text] and [Formula: see text] are nothing but left [Formula: see text]-, right [Formula: see text]-bimodules, the equivalence above generalizes the Eilenberg–Watts theorem and all its previous generalizations. By generalizing and strengthening Bergman’s completeness result for categories of internal coalgebras in varieties, we also prove that the category of coalgebras in a locally presentable category [Formula: see text] is locally presentable and comonadic over [Formula: see text] and, hence, complete in particular. We show, moreover, that Freyd’s canonical constructions of internal coalgebras in a variety define left adjoint functors. Special instances of the respective right adjoints appear in various algebraic contexts and, in the case where [Formula: see text] is a commutative variety, are coreflectors from the category [Formula: see text] into [Formula: see text].

Polynomial functors in manifold calculus

Let M be a smooth manifold, and let O(M) be the poset of open subsets of M. Manifold calculus, due to Goodwillie and Weiss, is a calculus of functors suitable for studying contravariant functors (cofunctors) F:O(M)⟶Spaces from O(M) to the category of spaces. Weiss showed that polynomial cofunctors of degree ≤k are determined by their values on Ok(M), where Ok(M) is the full subposet of O(M) whose objects are open subsets diffeomorphic to the disjoint union of at most k balls. Afterwards Pryor showed that one can replace Ok(M) by more general subposets and still recover the same notion of polynomial cofunctor. In this paper, we generalize these results to cofunctors from O(M) to any simplicial model category M. If Fk(M) stands for the unordered configuration space of k points in M, we also show that the category of homogeneous cofunctors O(M)⟶M of degree k is weakly equivalent to the category of linear cofunctors O(Fk(M))⟶M provided that M has a zero object. Using a new approach, we also show that if M is a general model category and F:Ok(M)⟶M is an isotopy cofunctor, then the homotopy right Kan extension of F along the inclusion Ok(M)↪O(M) is also an isotopy cofunctor.

Decomposition spaces, incidence algebras and Möbius inversion II: Completeness, length filtration, and finiteness

This is the second in a trilogy of papers introducing and studying the notion of decomposition space as a general framework for incidence algebras and Möbius inversion, with coefficients in ∞-groupoids. A decomposition space is a simplicial ∞-groupoid satisfying an exactness condition weaker than the Segal condition. Just as the Segal condition expresses composition, the new condition expresses decomposition.In this paper, we introduce various technical conditions on decomposition spaces. The first is a completeness condition (weaker than Rezk completeness), needed to control simplicial nondegeneracy. For complete decomposition spaces we establish a general Möbius inversion principle, expressed as an explicit equivalence of ∞-groupoids. Next we analyse two finiteness conditions on decomposition spaces. The first, that of locally finite length, guarantees the existence of the important length filtration for the associated incidence coalgebra. We show that a decomposition space of locally finite length is actually the left Kan extension of a semi-simplicial space. The second finiteness condition, local finiteness, ensures we can take homotopy cardinality to pass from the level of ∞-groupoids to the level of Q-vector spaces.These three conditions — completeness, locally finite length, and local finiteness — together define our notion of Möbius decomposition space, which extends Leroux's notion of Möbius category (in turn a common generalisation of the locally finite posets of Rota et al. and of the finite decomposition monoids of Cartier–Foata), but which also covers many coalgebra constructions which do not arise from Möbius categories, such as the Faà di Bruno and Connes–Kreimer bialgebras.Note: The notion of decomposition space was arrived at independently by Dyckerhoff and Kapranov [6] who call them unital 2-Segal spaces.

Effect Algebras as Presheaves on Finite Boolean Algebras

For an effect algebra $A$, we examine the category of all morphisms from finite Boolean algebras into $A$. This category can be described as a category of elements of a presheaf $R(A)$ on the category of finite Boolean algebras. We prove that some properties (being an orthoalgebra, the Riesz decomposition property, being a Boolean algebra) of an effect algebra $A$ can be characterized by properties of the category of elements of the presheaf $R(A)$. We prove that the tensor product of of effect algebras arises as a left Kan extension of the free product of finite Boolean algebras along the inclusion functor. As a consequence, the tensor product of effect algebras can be expressed by means of the Day convolution of presheaves on finite Boolean algebras.

The Category of Matroids

The structure of the category of matroids and strong maps is investigated: it has coproducts and equalizers, but not products or coequalizers; there are functors from the categories of graphs and vector spaces, the latter being faithful and having a nearly full Kan extension; there is a functor to the category of geometric lattices, that is nearly full; there are various adjunctions and free constructions on subcategories, inducing a simplification monad; there are two orthogonal factorization systems; some, but not many, combinatorial constructions from matroid theory are functorial. Finally, a characterization of matroids in terms of optimality of the greedy algorithm can be rephrased in terms of limits.

Differential forms in positive characteristic avoiding resolution of singularities

This paper studies several notions of sheaves of differential forms that are better behaved on singular varieties than Kahler differentials. Our main focus lies on varieties that are defined over fields of positive characteristic. We identify two promising notions: the sheafification with respect to the cdhtopology, and right Kan extension from the subcategory of smooth varieties to the category of all varieties. Our main results are that both are cdh-sheaves and agree with Kahler differentials on smooth varieties. They agree on all varieties under weak resolution of singularities. A number of examples highlight the difficulties that arise with torsion forms and with alternative candiates.

Algebraic Kan extensions along morphisms of internal algebra classifiers

An “algebraic left Kan extension” is a left Kan extension which interacts well with the algebraic structure present in the given situation, and these appear in various subjects such as the homotopy theory of operads and in the study of conformal field theories. In the most interesting examples, the functor along which we left Kan extend goes between categories that enjoy universal properties which express the meaning of the calculation we are trying to understand. These universal properties say that the categories in question are universal examples of some categorical structure possessing some kind of internal structure, and so fall within the theory of “internal algebra classifiers” described in earlier work of the author. In this article conditions of a monad-theoretic nature are identified which give rise to morphisms between such universal objects, which satisfy the key condition of Guitart-exactness, which guarantees the algebraicness of left Kan extending along them. The resulting setting explains the algebraicness of the left Kan extensions arising in operad theory, for instance from the theory of “Feynman categories” of Kaufmann and Ward, generalisations thereof, and also includes the situations considered by Batanin and Berger in their work on the homotopy theory of algebras of polynomial monads.

Re-Gauging Groupoid, Symmetries and Degeneracies for Graph Hamiltonians and Applications to the Gyroid Wire Network

We study a class of graph Hamiltonians given by a type of quiver representation to which we can associate (non)-commutative geometries. By selecting gauging data, these geometries are realized by matrices through an explicit construction or a Kan extension. We describe the changes in gauge via the action of a re-gauging groupoid. It acts via matrices that give rise to a noncommutative 2-cocycle and hence to a groupoid extension (gerbe). We furthermore show that automorphisms of the underlying graph of the quiver can be lifted to extended symmetry groups of re-gaugings. In the commutative case, we deduce that the extended symmetries act via a projective representation. This yields isotypical decompositions and super-selection rules. We apply these results to the primitive cubic, diamond, gyroid and honeycomb wire networks using representation theory for projective groups and show that all the degeneracies in the spectra are consequences of these enhanced symmetries. This includes the Dirac points of the G(yroid) and the honeycomb systems.

Positive fragments of coalgebraic logics

Positive modal logic was introduced in an influential 1995 paper of Dunn as the positive fragment of standard modal logic. His completeness result consists of an axiomatization that derives all modal formulas that are valid on all Kripke frames and are built only from atomic propositions, conjunction, disjunction, box and diamond. In this paper, we provide a coalgebraic analysis of this theorem, which not only gives a conceptual proof based on duality theory, but also generalizes Dunn's result from Kripke frames to coalgebras for weak-pullback preserving functors. To facilitate this analysis we prove a number of category theoretic results on functors on the categories $\mathsf{Set}$ of sets and $\mathsf{Pos}$ of posets: Every functor $\mathsf{Set} \to \mathsf{Pos}$ has a $\mathsf{Pos}$-enriched left Kan extension $\mathsf{Pos} \to \mathsf{Pos}$. Functors arising in this way are said to have a presentation in discrete arities. In the case that $\mathsf{Set} \to \mathsf{Pos}$ is actually $\mathsf{Set}$-valued, we call the corresponding left Kan extension $\mathsf{Pos} \to \mathsf{Pos}$ its posetification. A $\mathsf{Set}$-functor preserves weak pullbacks if and only if its posetification preserves exact squares. A $\mathsf{Pos}$-functor with a presentation in discrete arities preserves surjections. The inclusion $\mathsf{Set} \to \mathsf{Pos}$ is dense. A functor $\mathsf{Pos} \to \mathsf{Pos}$ has a presentation in discrete arities if and only if it preserves coinserters of `truncated nerves of posets'. A functor $\mathsf{Pos} \to \mathsf{Pos}$ is a posetification if and only if it preserves coinserters of truncated nerves of posets and discrete posets. A locally monotone endofunctor of an ordered variety has a presentation by monotone operations and equations if and only if it preserves $\mathsf{Pos}$-enriched sifted colimits.

Topological K-theory of complex noncommutative spaces

The purpose of this work is to give a definition of a topological K-theory for dg-categories over$\mathbb{C}$and to prove that the Chern character map from algebraic K-theory to periodic cyclic homology descends naturally to this new invariant. This topological Chern map provides a natural candidate for the existence of a rational structure on the periodic cyclic homology of a smooth proper dg-algebra, within the theory of noncommutative Hodge structures. The definition of topological K-theory consists in two steps: taking the topological realization of algebraic K-theory and inverting the Bott element. The topological realization is the left Kan extension of the functor ‘space of complex points’ to all simplicial presheaves over complex algebraic varieties. Our first main result states that the topological K-theory of the unit dg-category is the spectrum$\mathbf{BU}$. For this we are led to prove a homotopical generalization of Deligne’s cohomological proper descent, using Lurie’s proper descent. The fact that the Chern character descends to topological K-theory is established by using Kassel’s Künneth formula for periodic cyclic homology and the proper descent. In the case of a dg-category of perfect complexes on a separated scheme of finite type, we show that we recover the usual topological K-theory of complex points. We show as well that the Chern map tensorized with$\mathbb{C}$is an equivalence in the case of a finite-dimensional associative algebra – providing a formula for the periodic homology groups in terms of the stack of finite-dimensional modules.

Freer monads, more extensible effects

We present a rational reconstruction of extensible effects, the recently proposed alternative to monad transformers, as the confluence of efforts to make effectful computations compose. Free monads and then extensible effects emerge from the straightforward term representation of an effectful computation, as more and more boilerplate is abstracted away. The generalization process further leads to freer monads, constructed without the Functor constraint. The continuation exposed in freer monads can then be represented as an efficient type-aligned data structure. The end result is the algorithmically efficient extensible effects library, which is not only more comprehensible but also faster than earlier implementations. As an illustration of the new library, we show three surprisingly simple applications: non-determinism with committed choice (LogicT), catching IO exceptions in the presence of other effects, and the semi-automatic management of file handles and other resources through monadic regions. We extensively use and promote the new sort of `laziness', which underlies the left Kan extension: instead of performing an operation, keep its operands and pretend it is done.

Derived Kan Extension for Strict Polynomial Functors

We investigate fundamental properties of adjoint functors to the precomposition functor in the category of strict polynomial functors.

Left Kan extensions preserving finite products

We give conditions that ensure the preservation of finite products by left Kan extensions into cocomplete subcanonical sites. The conditions involve a suitable notion of flatness (interpreted in the internal logic of the site) of the functor that is extended and a good behavior of colimits used in the calculation of the left Kan extension. When the recipient category is a Grothendieck topos the good behavior of the colimits is granted and the flatness conditions turn out to be necessary as well as sufficient. We also show that the category of compactly generated Hausdorff spaces and that of small categories are equipped with suitable topologies that account for the well-known phenomenon of preservation of finite products by the geometric and categorical realization, respectively, of simplicial sets.

Homotopy nilpotent groups

We study the connection between the Goodwillie tower of the identity and the lower central series of the loop group on connected spaces. We define the simplicial theory of homotopy n-nilpotent groups. This notion interpolates between infinite loop spaces and loop spaces. We prove that the set-valued algebraic theory obtained by applying $\pi_0$ is the theory of ordinary n-nilpotent groups and that the Goodwillie tower of a connected space is determined by a certain homotopy left Kan extension. We prove that n-excisive functors of the form $\Omega F$ have values in homotopy n-nilpotent groups.