Papers

We study the tableaux monoid from the perspective of rewriting theory. We first construct a finite quadratic convergent presentation based on column generators and prove that it yields a biautomatic structure. We then extend this presentation to a finite coherent presentation in the sense of polygraphic rewriting. These results provide canonical normal forms, yield homological and algorithmic consequences, and produce a higher-dimensional presentation that can be used to describe actions of the tableaux monoid on categories.

In this note, for every topological colored nonsymmetric operad with CW spaces in each arity we construct a homotopy isomorphism with a CW homotopy operad (i.e. all higher homotopies are cellular with respect to the product structure and CW structure on cubes) not changing its objects. By using the two-sided bar construction, we rectify this to a homotopy isomorphism with a strict CW operad. This applies more generally to topological algebras over colored operads in sets. In the case of diagrams of finite CW complexes over finite posets, the resulting diagram consists of finite CW spaces enhancing the Quillen adjunction between simplicial sets and topological spaces. As a byproduct, we produce an embedding of topological homotopy algebras over operads into the presheaf category over the branch resolution for the category of $P$-trees generalizing the known case of topological operads.

In this work we investigate a relative notion of protomodularity. We say that a finitely complete category $\mathsf{C}$ satisfies the \defn{$\mathscr{M}$-version of protomodularity} when the change-of-base functors (concerning the fibration of points) are conservative with respect to a class of morphisms $\mathscr{M}$ of $\mathsf{C}$. If $\mathscr{M}$ is the class of all morphisms or the class of all monomorphisms of $\mathsf{C}$, then this notion is precisely that of an “ordinary” protomodular category. We analyse what conditions the class $\mathscr{M}$ should have to obtain the relative $\mathscr{M}$-versions of some well-known properties of protomodular categories, such as the Split Short Five Lemma (when $\mathsf{C}$ is also pointed) and their relation with Mal'tsev categories. We also give diverse examples of categories satisfying the $\mathscr{M}$-version of protomodularity for different choices of $\mathscr{M}$.

We give a negative answer to the open problem posed by Adámek and Rosický in 2001, asking whether the cocompletion of any small category under sifted colimits is finitely accessible. We will construct an infinite family of counterexamples.

We prove a two-dimensional version of Hermida's equivalence result on representable multicategories and classify its stricter versions. The obtained three-dimensional equivalence of the respective totality categories is of categories enriched over the multicategory $\mathsf{Bicat}$ of Verity. There is a basic enriched equivalence with respect to inherently lax monoidal pseudofunctors, and a one with respect to pseudo-monoidal pseudofunctors, related to the latter via the forgetful enriched functors. We extend Slattery's result on a bi-multicategory structure of the Kleisli bicategory of a 2-multicategory encompassing bi-representability.

We study the truncation $(-)^{\leq0}$ of dg categories. We first show that given a dg category $\mathscr{C}$ with shifts, for example a pretriangulated dg category, the canonical functor $\mathscr{C}^{\leq0}\to\mathscr{C}$ is a localization whose kernel is compactly generated by the $0$-th cohomology. Next we demonstrate that the derived category of the truncation serves as a triangulated analogue of the Auslander's category of coherent functors over abelian categories. We give a description of the Auslander-Reiten-Serre duality in terms of the inverse dualizing bimodule of the truncation. Building on truncations, we introduce the notion of connective resolutions of dg categories, defined as a localization functor from a connective dg category. This notion encompasses non-commutative resolutions of algebras in module categories, cluster tilting objects in triangulated categories, and also the truncation as the universal connective resolution. We give a sufficient condition for an object in a triangulated category to give a connective resolution of its enhancement in terms of finiteness of resolution dimension. Furthermore, we show that every proper dg module over a connective dg algebra is a direct summand of a dg module giving a connective resolution. Consequently, for every proper connective dg algebra, its bounded dg derived category has a connective resolution. Finally, we give some explicit description of the truncation for some categories including the derived and cluster categories of a Dynkin quiver, and the Yoneda category of a finite dimensional algebra.

We investigate how to systematically construct continuous dualities from zero-dimensional dualities, employing well-known methods from algebra, topology, category theory, and domain theory. While our method is general, this paper focusses on the move from Stone spaces to compact Hausdorff spaces and the move from Priestley spaces to compact ordered Hausdorff spaces. The engine of our approach is Stone duality for relations: on the space side quotienting by a preorder turns zero-dimensional spaces into continuous ones, while distributive lattices with a proximity relation are their algebraic duals. Our duality for relations is inherently order-enriched. Double categories organise both functional and relational morphism in the same structure. The move from zero-dimensional to continuous dualities is then a three-step construction: extend a duality from functional to relational morphism, split idempotents, restrict to maps.

We prove that the categories of strictly unital (resp.\ non unital) A$_{\infty}$-algebras and A$_{\infty}$-categories, with strictly unital (resp.\ non unital) A$_{\infty}$-morphisms and A$_{\infty}$-functors have no coequalizers.

We show that, under some mild assumptions, every quasi-pointed category C carries a canonical near-torsion theory (T,F), where T consists of the 0-supported objects, that is, those admitting a morphism to the initial object 0, and F consists of the subterminal objects. The coreflection and the reflection of an object A are then given by the projection $A \times 0 \to A$ and by the strong-epimorphic part of $A \to 1$, respectively. Using the general construction of the "largest" non-pointed torsion theory associated with a near-torsion theory, we prove that (T,F) is a (non-pointed) torsion theory in the full subcategory of those objects A for which the canonical sequence $t(A) \to A \to f(A)$ is short exact, in the sense that it is both a kernel and a cokernel diagram. This subcategory is the whole of C whenever C is sequentiable, that is, a regular protomodular and quasi-pointed category. We describe the resulting torsion theories in the category of Lie-Rinehart algebras and in the categories of crossed modules over a fixed group. We also show that (T,F) is a torsion theory in the (non-protomodular) category of monoids over a group G all of whose elements have finite order.

We prove that every finite symmetric tensor category over an algebraically closed field of positive characteristic admits a symmetric tensor functor to a higher Verlinde category. This was conjectured for arbitrary symmetric tensor categories of moderate growth by Benson, Etingof and Ostrik. The proof uses higher Frobenius functors, introduced by the authors in [CF1]. The main effort is directed towards a proof of a conjecture from [CF1] about thick tensor ideals in the representation category of elementary abelian $p$-groups, which implies that the higher Frobenius functors are symmetric monoidal. In fact, we prove a refinement of the latter conjecture that relates tensoring with restricted Steinberg modules to an iterated generalised cohomology of modules of elementary abelian $p$-groups.

Graphoid automata evaluate graphs by composing relations between words of states. We examine the equations that make this evaluation independent of the chosen graph expression, and what changes when relations are replaced by matrices over a commutative semiring. We prove that the classical graphoid equations are equivalent to the special commutative Frobenius equations together with the required symmetry laws. The known cospan and hypergraph presentations then describe equality of structural and labelled expressions, with the bone law deleting precisely the isolated internal vertices. We use established partition representations to compare this equality with equality of evaluated matrices. For any nonempty finite state set $Q$ and commutative semiring $K$, we give a uniform criterion for two structural expressions to have the same matrix. The criterion determines which isolated-vertex counts are distinguished and when a vanishing scalar makes matrices from different boundary partitions equal. For the canonical structure, the bone law holds precisely when $|Q|\cdot1_K=1_K$. Preserving all relational compositions is a separate requirement, equivalent to additive idempotency for $|Q|\geq2$. We then examine the characteristic matrices of the known relational structures on $Q$ given by disjoint unions of Abelian groups. Their Frobenius equations hold over every commutative semiring, whereas specialness depends on the order of each group and the bone law depends on the number of groups. These conditions specify which equations remain valid when weights are assigned to a graphoid automaton.

We develop a higher-geometric framework for magnetic higher-form symmetries of pre-quantum field theories. The magnetic (topological) charges of a sigma-model with stacky target are encoded by the shape of the target, equivalently of its stack of fields, which is op-dual to the flat automorphism group controlling electric symmetries. Magnetic symmetry backgrounds are obtained by dualizing these charges against a chosen moduli stack of topological field theories, generalizing Pontryagin duality, and act on the theory by stacking. Cohesive modalities allow for targets with non-finite and non-discrete data. When the dualizing stack is a grouplike $E_\infty$-object, we prove a fibration theorem expressing the magnetic symmetries of an electrically gauged theory in terms of those of the ungauged theory and of the gauge group, computed by a homotopy-fixed-point spectral sequence with an explicit first page. For moduli stacks of higher $U(1)$-gerbes with connection, encoding bulk topological local Lagrangians, we prove a factorization theorem: the magnetic symmetry backgrounds of any connected target split, non-canonically, into Pontryagin duals of its integral homology groups, and are necessarily flat, as a consequence of dualizing discrete charges against a target with connections. We illustrate both theorems with pure $\mathrm{Pin}^\pm(2)$ gauge theory: $O(2)$ carries invertible magnetic $\mathbb{Z}_2$ symmetries in form degrees $d-3$, $d-4$ and $d-5$ that are absent for $\mathrm{Pin}^-(2)$, the first two distinguished by a single differential of the spectral sequence. Gauging and physical applications are developed in upcoming work.

We study the construction of ternary semi-direct products in semi-abelian categories, following a definition of higher semi-direct product previously introduced by Carrasco and Cegarra for groups and Lie algebras. We show that these objects are determined by actions and equivariant morphisms satisfying a certain compatibility condition in the ambient category, and that actions by semi-direct products are a special case of this construction. We also show how their structure can be further simplified in algebraically coherent or locally algebraically cartesian closed categories. We also give a concrete description of the structure of ternary semi-direct products in concrete categories, and how our categorical interpretation of the necessary structure relates to the group and Lie-theoretic versions given by Carrasco and Cegarra.

Let $(\mathcal{C},\mathbb{E},\mathfrak s)$ be an extriangulated category equipped with a proper class $\xi$ of $\mathbb E$-triangles.We establish stability under iteration for the quasi-$\xi$-Gorenstein projective and injective subcategories and extend the relative Ext-vanishing criteria for their finite homological dimensions to larger coefficient subcategories. An example in a bounded homotopy category shows that $\mathcal{P}(\xi)\subsetneq\mathcal{QGP}(\xi)\subsetneq\mathcal{GP}(\xi)$. For module categories, every quasi-Gorenstein projective module over a left perfect ring or a commutative Noetherian ring of finite Krull dimension is projective. When a commutative ring admits a finite chain of trace and nilpotent quotients ending in an Artinian ring and the endomorphism rings at the trace steps have finite right global dimension, we construct finite families of quasi-projective test modules that detect prescribed upper bounds on projective dimension for modules of bounded cardinality through Ext vanishing in finitely many consecutive degrees.

Following Miyamoto, we consider Q-charts, a class of subductions $\phi\colon X\to Y$ of diffeological spaces along which Blohmann's elasticity descends. We show that the Riemannian apparatus of elastic diffeology—vertical, horizontal and full connections, Riemannian metrics, Levi-Civita connections and geodesics—also descends along Q-charts. More precisely, structures on $Y$ correspond bijectively to structures on $X$ that are invariant under the pseudogroup $\Psi(\phi)$ of fibre-preserving transitions, and torsion-freeness, effectivity, flatness, fullness, the Levi-Civita property and the geodesic equation are preserved in both directions. The key input is that the iterated tangent bundles $T^nX$ and their fibre products $T_kX$ are the pullbacks along $\phi$ of the corresponding bundles over $Y$. For quotients by principal actions of diffeologically discrete groups, we lift geodesics globally, prove that the class of elastic spaces on which $\R$ is a curve object is closed under such quotients, and show that complete geodesic flows descend. As an application, we determine the Riemannian geometry of the irrational tori $\Tor_\alpha=\R/(\Z+\alpha\Z)$ completely. Let $\vartheta$ be the one-form on $\Tor_\alpha$ induced by $dt$ on $\R$. Every Riemannian metric is a positive constant multiple of $\vartheta^2$, and the vertical connections form a one-parameter family of flat effective connections, a single member of which is the unique Levi-Civita connection of every metric. This connection has a complete geodesic flow, yet the induced pseudo-distance vanishes identically.

A dagger on a category assigns to every morphism $f \colon X \to Y$ a morphism $f^\dagger \colon Y \to X$, contravariantly and involutively, in the way the transpose reverses a real matrix. We show that on the categories of linear algebra and differential geometry, daggers encode metric and Riemannian structures. As is known, on a single vector space, an anti-involution of the endomorphism ring is the adjoint of a form that is determined up to a scalar. But more generally, on the category of all finite-dimensional vector spaces, a dagger is the adjoint for an involution of the scalars together with a unique unimodular Hermitian form on every space. The same holds for finitely generated projective modules over a commutative ring, and hence, through the Serre-Swan correspondence, for vector bundles, where a dagger is a choice of bundle metrics. The dagger structure expresses notions of orthogonality categorically: orthogonal complements and normal bundles acquire universal properties, and Hermitian modules form a dagger compact category. Moreover, daggers on Lie groups, Lie groupoids and their infinitesimal counterparts capture unitary representations and metric connections: a connection is metric precisely when its parallel transport is a dagger functor, or, infinitesimally, when its splitting of the Atiyah sequence commutes with the daggers. Finally, we classify the daggers on the locally graded category of differential operators of a Lie-Rinehart algebra. They correspond to metric-divergence structures, consisting of a real structure, a divergence, and a unimodular Hermitian form on every finitely generated projective module, and they act on operators as formal adjoints; Green's formula expresses the difference between an operator and its adjoint as a divergence.

Børve, Horiatakis, and Kalck defined the magnitude of the module category of a representation-finite algebra as the sum of the entries of the inverse of the matrix of Hom dimensions between indecomposable modules. They showed it is an Euler characteristic, the alternating count of vertices, arrows, and meshes of the Auslander–Reiten quiver, and conjectured that it is at least the number of simple modules, with equality precisely for special biserial algebras. We prove this conjecture over an algebraically closed field. For a representation-directed algebra, we relate the magnitudes of the module category, the subcategory of modules without a given simple composition factor, and the ideal quotient by it. By the theory of hammocks of Ringel and Vossieck, this quotient is equivalent to a category of representations of a finite poset, whose magnitude we compute. The general case reduces to the representation-directed case through algebras obtained by grading a standard form.

We describe hearts of strongly deflation-exact categories with kernels as localizations of categories of three-term complexes up to homotopy, without assuming admissibility of kernels. We construct a calculus of fractions and give explicit formulas for kernels and cokernels. For complete locally convex spaces, whose failure to be quasiabelian was established by Prosmans (2000), our description and its dual provide derived equivalent abelian models through two choices of a notion of exactness. For quasi-complete, sequentially complete and locally (a.k.a. Mackey) complete spaces, we show that the topologically exact sequences form the maximal deflation-exact structure, although these categories are not extension-closed in the category of locally convex Hausdorff spaces. Our results make them nevertheless accessible to homological algebra with respect to the induced notion of exactness. Further applications concern the quasiabelian category of bornological modules equipped with the non-maximal linear split exact structure.

This version contains the introduction and conclusion of my PhD thesis, "Freely Generated Categorical Structures and Automatic Differentiation", together with its English and Dutch summaries. The full thesis consists of an introductory chapter, six joint research papers, and a concluding chapter, developed during my PhD studies at Utrecht University under the supervision of Gabriele Keller and Matthijs Vákár. The research papers are available separately and are not reproduced here. The introduction presents the scope of the thesis, explains the contributions of the six papers and their connections, and introduces the categorical foundations of our approach. The guiding idea is that programming languages, viewed as freely generated categorical structures, provide a principled setting for constructing structure-preserving program transformations and proving their correctness. Automatic differentiation supplies the central application: we study forward- and reverse-mode differentiation for expressive typed languages, including higher-order functions, recursive types, iteration and partiality. The semantic requirements of these transformations also motivate independent mathematical results on free distributive and extensive categories, cartesian closedness, and Grothendieck constructions. The conclusion brings these contributions together, discusses their limitations, and outlines further directions. Throughout, the thesis develops a dialogue between theory and practice: categorical semantics guides the construction of reliable and practically useful program transformations, while the demands of computation lead to new categorical structures and results.

Let $B$ and $D$ be both algebras and coalgebras in a braided monoidal category $\mathcal{C}$. In the literature, there are equivalent conditions for $(B,D)$ to form a generalized smash biproduct (or cross product) denoted by $B\times D$, which were mainly given by Bespalov and Drabant in 1999 as well as by Bulacu, Caenepeel and Torrecillas in 2013. This paper is devoted to constructing another biproduct $D\times B^\ast$ when $B$ has a left dual object $B^\ast$ in $\mathcal{C}$. As results, we show that $B\times D$ is left smash over $D\times B^\ast$, and establish a specific isomorphism ${}_{B\times D}\mathfrak{YD}(\mathcal{C})^{B\times D}\cong{}_{D\times B^\ast}\mathfrak{YD}(\mathcal{C})^{D\times B^\ast}$ between the categories of the (left-right) Yetter-Drinfeld modules in $\mathcal{C}$. The constructions and results are provided in three levels: (1) $B\times D$ is an algebra and a coalgebra; (2) $B\times D$ is a bialgebra; (3) $B\times D$ is a Hopf algebra.

This paper studies orthogonal model structures, i.e., model structures such that a lifting in the Lifting axiom is unique. Cofibrant (fibrant) objects are defined without initial (terminal) objects. For an orthogonal model structure on a category with enough cofibrant objects and fibrant objects, it is proved that the homotopy category is equivalent to the full subcategory of cofibrant-fibrant objects. TTF model structures, bi-reflective model structures, and torsion model structures, are introduced. They are all orthogonal. One to one correspondences between TTF model structures and TTF triples in an abelian category, bi-reflective model structures and bi-reflective pairs in any category, and torsion model structures and twin torsion pairs in an abelian category, are established in a constructive way. Torsion model structures on a poset, on the category of $G$-sets, and on the category of topological groups, are also constructed. The homotopy categories of all these model structures are computed. In particular, the homotopy category of a TTF model structure is an abelian category.

The nerve theorem provides a combinatorial model for the homotopy type of a topological space from a suitable open cover. We extend this local-to-global approach by replacing the intersections of the cover with compatible categorical models. These local models are assembled through the Grothendieck construction, yielding a small category whose classifying space has the weak homotopy type of the original space. Under suitable hypotheses, the resulting category is finite and acyclic. We develop two versions of this construction. The first is indexed by the usual poset of nonempty finite intersections of the cover. The second uses the membership poset, inspired by the finite-space construction of Sancho de Salas, which retains only the membership patterns realized by points of the space and is defined for point-finite covers. The classical nerve theorem and its componentwise variant are recovered as particular cases. The construction is recursive: categorical models of local pieces, together with functors representing their inclusions, can be combined to produce categorical models of more complex spaces.

Concurrent games and strategies based on event structures are an established and expressive model of interactive computation. In this paper we revisit the problem of equipping this model with symmetry. Symmetry is essential for many applications, but it is also a source of mathematical complexity, not least because strategies with symmetry can be compared up to several distinct notions of equivalence. Our first contribution is a complete characterization of two classes of strategies for which composition admits an identity. For these two classes the identity laws hold, respectively, up to a "weak" and a "strong" notion of equivalence. The main result is that the weak class of strategies corresponds precisely to those inducing Street fibrations over the game. We also show how the established "thin" approach to symmetry also fits in this general framework. A second contribution is a formal connection between games and setoid-valued profunctors. This follows a long tradition of relating games and relational models, but here additionally includes a very general model of symmetries. Our characterization of strategies as Street fibrations makes the connection to profunctors significantly easier to establish.

We extend a six-functor formalism $D\colon\mathrm{Span}(C,E)\to\mathrm{Cat}$ to a lax symmetric monoidal functor of $(\infty,2)$-categories $\mathbf{Span}^2(C,E)^P_I\to\mathbf{Cat}$, where $P$ and $I$ are the classes of $D$-proper and $D$-étale morphisms, respectively. This proves a conjecture of Mann and generalizes a special case of a theorem of Cnossen, Lenz, and Linskens. To prove this result, we develop a theory of internal $\mathsf{E}$-monoidal categories and $\mathsf{E}$-operads, where $\mathsf{E}$ is a local class of morphisms in an $\infty$-topos. These notions generalize the internal symmetric monoidal categories and operads developed by Martini and Wolf.

In this article, we introduce a new rewriting method for computing hom-bases of diagrammatic categories. Our approach is based on stratified presentations, obtained by partitioning an equational presentation into strata according to the rewriting behavior of its relations. We establish modularity principles for termination and confluence in stratified rewriting systems with coefficients in a ring. We then introduce a completion procedure that transforms a stratified presentation into a normalized one, in which mixed interactions between successive strata are absorbed into the normalized rewriting rules, thereby inducing modularity of confluence. This leads to a stratified hom-basis theorem for diagrammatic categories. We illustrate the method by applying it to the $q$-Schur category, for which we construct a new hom-basis.

Lax extensions (also called relators or relation liftings) are a categorical notion to reason about functors acting on functions and relations in a compatible way. They play a central role to develop sound proof principles for behavioral equivalence of state-based systems and are also important for establishing contextual equivalence for effectful programs. In this paper, we develop the theory of lax extensions for parametrized functors and monads and consider notions of behavioral preorders, equivalence relations or metrics which can now be modulated by additional parameters. From an operational viewpoint, we replace standard contextual equivalence where we quantify over all possible contexts by a refined notion of equivalence where the user can regulate the allowed contexts via chosen parameters.

We show that under mild assumptions a functor is a prefibration if and only if it has terminal objects in its fibers and admits precartesian liftings of monomorphisms. We use this to show that the following conditions on a pointed category $\mathbb{C}$ are equivalent: (a) The category of morphisms of $\mathbb{C}$ admits generic split extensions; (b) Each functor category of $\mathbb{C}$ with finite domain category admits generic split extensions; (c) $\mathbb{C}$ admits normalizers and generic split extensions; (d) The kernel functor from the category split extensions in $\mathbb{C}$ to $\mathbb{C}$ is a prefibration. In addition we show that the category of morphisms of a pointed protomodular $\mathbb{C}$ admits generic split extensions if and only if for each morphism $f:X\to Z$ the functor sending each object $B$ to the isomorphism class of split extensions in $\mathbb{C}^{\mathbf{2}}$ of $(B,B,1_B)$ with kernel $(X,Z,f)$, is representable.

We prove that the sheaf $\infty$-topos associated to a geometric site in the sense of Lurie can be written as a partially lax limit of smaller sheaf $\infty$-topoi. This is thus a formalization of Lurie's vision for fractured $\infty$-topoi to axiomatize the relation between gros and petit topoi. As a consequence, we realize the $\infty$-topos of $\kappa$-small condensed anima as a partially lax limit of sheaf $\infty$-topoi over extremally disconnected spaces, marked at the open embeddings. Moreover, we deduce a gros version of the étale exodromy theorem.

Given a monoid that can be embedded in a group and that acts partially on a set, we show that under certain conditions the category associated with the partial action has a groupoid of fractions. By imposing some topological conditions on the action, we show that the groupoid is actually étale. This étale groupoid arises naturally in $C^{\ast}$-algebra theory.

Canonical ideal quotients of extriangulated categories admit natural one-sided triangulated or pretriangulated structures. A fundamental question is whether these quotient structures still retain enough information to recover higher orthogonality properties of the subcategory being factored out. We show that, for a strongly functorially finite $n$-rigid subcategory, such information is encoded by the nilpotency of the canonical suspension and, equivalently, of the canonical loop. More precisely, this nilpotency characterizes two-sided maximal $n$-orthogonality. We further establish an objectwise recognition criterion that combines one-sided higher orthogonality with the vanishing of the $n$th suspension or loop. In the triangulated setting, our results give a converse to the known truncation construction and show that the quotient is $n$-truncated if and only if the subcategory is $(n+1)$-cluster tilting. Moreover, in the nonsplit case, the nilpotency index is exactly $n$. These results turn truncation from a consequence of higher orthogonality into a sharp criterion for recognizing it.