We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored *-categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called ``probabilistic relations''. The compact closed structure associated with the category of relations does not generalize directly, instead one obtains nuclear ideals. We introduce the notion of nuclear ideal to analyze these classes of morphisms. In compact closed categories, we see that all morphisms are nuclear, and in the category of Hilbert spaces, the nuclear morphisms are the Hilbert-Schmidt maps.
We also introduce two new examples of tensored *-categories, in which integration plays the role of composition. In the first, morphisms are a special class of distributions, which we call tame distributions. We also introduce a category of probabilistic relations which was the original motivating example.
Finally, we extend the recent work of Joyal, Street and Verity on traced monoidal categories to this setting by introducing the notion of a trace ideal. For a given symmetric monoidal category, it is not generally the case that arbitrary endomorphisms can be assigned a trace. However, we can find ideals in the category on which a trace can be defined satisfying equations analogous to those of Joyal, Street and Verity. We establish a close correspondence between nuclear ideals and trace ideals in a tensored *-category, suggested by the correspondence between Hilbert-Schmidt operators and trace operators on a Hilbert space.
Our subject is that of categories, functors and distributors enriched in a base quantaloid Q. We show how cocomplete Q-categories are precisely those which are tensored and conically cocomplete, or alternatively, those which are tensored, cotensored and order-cocomplete. Bearing this in mind, we analyze how Sup-valued homomorphisms on Q are related to Q-categories. With an appendix on action, representation and variation.
Monoidal categories enriched in a braided monoidal category $\mathcal{V}$ are classified by braided oplax monoidal functors from $\mathcal{V}$ to the Drinfeld centers of ordinary monoidal categories. In this article, we prove that this classifying functor is strongly monoidal if and only if the original $\mathcal{V}$-monoidal category is tensored over $\mathcal{V}$. We then define a completion operation which produces a tensored $\mathcal{V}$-monoidal category $\overline{\mathcal{C}}$ from an arbitrary $\mathcal{V}$-monoidal category $\mathcal{C}$, and we determine many equivalent conditions w…
This paper shows how internal models for polymorphic lambda calculi arise in any 2-category with a notion of discreteness. We generalise to a 2-categorical setting the famous theorem of Peter Freyd saying that there are no sufficiently (co)complete non-degenerate categories. As a simple corollary, we obtain a variant of Freyd theorem for categories internal to any tensored category. Also, with help of introduced concept of an associated category, we prove a representation theorem relating our internal models with well-studied fibrational models for polymorphism.
This paper introduces a skew variant of the notion of enriched category, suitable for enrichment over a skew-monoidal category, the main novelty of which is that the elements of the enriched hom-objects need not be in bijection with the morphisms of the underlying category. This is the natural setting in which to introduce the notion of locally weak comonad, which is fundamental to the theory of enriched algebraic weak factorisation systems. The equivalence, for a monoidal closed category $\mathcal{V}$, between tensored $\mathcal{V}$-categories and hommed $\mathcal{V}$-actegories is extended t…
We prove that given $\mathcal{C}$ a presentably symmetric monoidal $\infty$-category, and any essentially small $\infty$-operad $\mathcal{O}$, the $\infty$-category of $\mathcal{O}$-algebras in $\mathcal{C}$ is enriched, tensored and cotensored over the presentably symmetric monoidal $\infty$-category of $\mathcal{O}$-coalgebras in $\mathcal{C}$. We provide a higher categorical analogue of the universal measuring coalgebra. For categories in the usual sense, the result was proved by Hyland, López Franco, and Vasilakopoulou.
For a monoidal $\infty$-category $\mathcal{M}$ with colimits, we study colimits of $\mathcal{M}$-functors $\mathcal{A}\to\mathcal{B}$ where $\mathcal{B}$ is left-tensored over $\mathcal{M}$ and $\mathcal{A}$ is an $\mathcal{M}$-enriched category. We prove that the enriched Yoneda embedding $Y:\mathcal{A}\to P_{\mathcal{M}}(\mathcal{A})$ yields a universal $\mathcal{M}$-functor and, in the case when $\mathcal{A}$ has a certain monoidal structure, the category of enriched presheaves $P_{\mathcal{M}}(\mathcal{A})$ inherits the same monoidal structure.
We continue the study of enriched infinity categories, using a definition equivalent to that of Gepner and Haugseng. In our approach enriched infinity categories are associative monoids in an especially designed monoidal category of enriched quivers. We prove that, in case the monoidal structure in the basic category M comes from direct product, our definition is essentially equivalent to the approach via Segal objects. Furthermore, we compare our notion with the notion of category left-tensored over M, and prove a version of Yoneda lemma in this context. Version 2: An error in 2.6.2 corrected…
This paper deals with questions relating to Haghverdi and Scott's notion of partially traced categories. The main result is a representation theorem for such categories: we prove that every partially traced category can be faithfully embedded in a totally traced category. Also conversely, every symmetric monoidal subcategory of a totally traced category is partially traced, so this characterizes the partially traced categories completely. The main technique we use is based on Freyd's paracategories, along with a partial version of Joyal, Street, and Verity's Int-construction.
We define a traced pseudomonoid as a pseudomonoid in a monoidal bicategory equipped with extra structure, giving a new characterisation of Cauchy complete traced monoidal categories as algebraic structures in Prof, the monoidal bicategory of profunctors. This enables reasoning about the trace using the graphical calculus for monoidal bicategories, which we illustrate in detail. We apply our techniques to study traced ∗-autonomous categories, proving a new equivalence result between the left ⊗-trace and the right ⅋-trace, and describing a new condition under which traced ͨ…
In [BaSc2] the authors introduced a much weaker homotopical structure than a model category, called a "weak cofibration category". We further showed that a small weak cofibration category induces in a natural way a model category structure on its ind-category, provided the ind-category satisfies a certain two out of three property. The purpose of this paper is to serve as a companion to the papers above, proving results which say that if a certain property or structure exists in the weak cofibration category, then the same property or structure also holds in the induced model structure on the …
We give an alternate conception of string diagrams as labeled 1-dimensional oriented cobordisms, the operad of which we denote by Cob/O, where O is the set of string labels. The axioms of traced (symmetric monoidal) categories are fully encoded by Cob/O in the sense that there is an equivalence between (Cob/O)-algebras, for varying O, and traced categories with varying object set. The same holds for compact (closed) categories, the difference being in terms of variance in O. As a consequence of our main theorem, we give a characterization of the 2-category of traced categories solely in terms …
This dissertation has two main parts. The first part deals with questions relating to Haghverdi and Scott's notion of partially traced categories. The main result is a representation theorem for such categories: we prove that every partially traced category can be faithfully embedded in a totally traced category. Also conversely, every monoidal subcategory of a totally traced category is partially traced, so this characterizes the partially traced categories completely. The main technique we use is based on Freyd's paracategories, along with a partial version of Joyal, Street, and Verity's Int…
In this note we provide an example of an endomorphism of a short exact sequence of perfect complexes, with the trace of the middle map not equal to the sum of the traces of the two other ones. The point is that the squares involved are commutative only up to homotopy. In view of this example I have found in 1968, Deligne immediately created his "categories spectrales", and soon afterwards Illusie introduced the "filtered derived categories" where a satisfactory kind of additivity is restored for the trace. This paper, written in French, ends up with a brief chronological comment.
The purpose of this expository note is to describe duality and trace in a symmetric monoidal category, along with important properties (including naturality and functoriality), and to give as many examples as possible. Among other things, this note is intended as background for the generalizations to the context of bicategories and indexed monoidal categories.
By the Lefschetz fixed point theorem, if an endomorphism of a topological space is fixed-point-free, then its Lefschetz number vanishes. This necessary condition is not usually sufficient, however; for that we need a refinement of the Lefschetz number called the Reidemeister trace. Abstractly, the Lefschetz number is a trace in a symmetric monoidal category, while the Reidemeister trace is a trace in a bicategory; in this paper we relate these contexts using indexed symmetric monoidal categories. In particular, we will show that for any symmetric monoidal category with an associated indexed sy…
This paper is about skew monoidal tensored V-categories (= skew monoidal hommed V-actegories) and their categories of modules. A module over <M,*,R> is an algebra for the monad T = R * _ on M. We study in detail the skew monoidal structure of M^T and construct a skew monoidal forgetful functor from M^T to the category of E-objects in M where E=M(R,R) is the endomorphism monoid of the unit object R. Then we give conditions for the forgetful functor to be strong monoidal and for the category M^T of modules to be monoidal. In formulating these conditions a notion of `self-cocomplete' subcategorie…
The additivity of traces in certain tensor triangulated categories for endomorphisms of finite order of distinguished triangles is investigated. For the identity endomorphism this has been fully established by J. P. May ("The additivity of traces in triangulated categories", Adv. Math., 2001, 163, 34-73). By imposing extra conditions on the coefficients we show how May's result implies a stronger additivity.
The main result of this paper is the construction of a trace and a trace pairing for endomorphisms satisfying suitable conditions in a monoidal category. This construction is a common generalization of the trace for endomorphisms of dualizable ob jects in a balanced monoidal category and the trace of nuclear operators on a locally convex topological vector space with the approximation property.
While not obvious from its initial motivation in linear algebra, there are many context where iterated traces can be defined. In this paper we prove a very general theorem about iterated 2-categorical traces. We show that many Lefschetz-type theorems in the literature are consequences of this result and the new perspective we provide allows for immediate spectral generalizations.
The category-valued trace assigns to a bimodule category over a linear monoidal category a linear category. It generalizes Drinfeld centers of monoidal categories and the relative Deligne product of bimodule categories. In this article, we study bimodule categories that are given as categories of bicomodules over a Hopf algebra. Our main result is a representation-theoretic realization of the category-valued trace as a category of generalized Hopf bimodules.
We show that in any symmetric monoidal category, if a weight for colimits is absolute, then the resulting colimit of any diagram of dualizable objects is again dualizable. Moreover, in this case, if an endomorphism of the colimit is induced by an endomorphism of the diagram, then its trace can be calculated as a linear combination of traces on the objects in the diagram. The formal nature of this result makes it easy to generalize to traces in homotopical contexts (using derivators) and traces in bicategories. These generalizations include the familiar additivity of the Euler characteristic an…
We generalize the notion of a modified trace (or m-trace) to the setting of non-unimodular categories. M-traces are known to play an important role in low-dimensional topology and representation theory, as well as in studying the category itself. Under mild conditions we give existence and uniqueness results for m-traces in pivotal categories.
We consider traces on module categories over pivotal fusion categories which are compatible with the module structure. It is shown that such module traces characterise the Morita classes of special haploid symmetric Frobenius algebras. Moreover, they are unique up to a scale factor and they equip the dual category with a pivotal structure. This implies that for each pivotal structure on a fusion category over the complex numbers there exists a conjugate pivotal structure defined by the canonical module trace.
We prove a version of J.P. May's theorem on the additivity of traces, in symmetric monoidal stable $\infty$-categories. Our proof proceeds via a categorification, namely we use the additivity of topological Hochschild homology as an invariant of stable $\infty$-categories and construct a morphism of spectra $\mathrm{THH}(\mathbf C)\to \mathrm{End}(\mathbf 1_\mathbf C)$ for $\mathbf C$ a stably symmetric monoidal rigid $\infty$-category. We also explain how to get a more general statement involving traces of finite (homotopy) colimits.
We present an $\ell$-adic trace formula for saturated and admissible dg-categories over a base monoidal dg-category. Moreover, we prove Künneth formulas for dg-category of singularities, and for inertia-invariant vanishing cycles. As an application, we prove a version of Bloch's Conductor Conjecture (stated by Spencer Bloch in 1985), under the additional hypothesis that the monodromy action of the inertia group is unipotent.
We prove that topological Hochschild homology (THH) arises from a presheaf of circles on a certain combinatorial category, which gives a universal construction of THH for any enriched infinity category. Our results rely crucially on an elementary, model-independent framework for enriched higher category theory, which may be of independent interest.
We study a 2-functor that assigns to a bimodule category over a finite k-linear tensor category a k-linear abelian category. This 2-functor can be regarded as a category-valued trace for 1-morphisms in the tricategory of finite tensor categories. It is defined by a universal property that is a categorification of Hochschild homology of bimodules over an algebra. We present several equivalent realizations of this 2-functor and show that it has a coherent cyclic invariance. Our results have applications to categories associated to circles in three-dimensional topological field theories with defe…
Man-in-the-Middle (MM) is not only a ubiquitous attack pattern in security, but also an important paradigm of network computation and economics. Recognizing ongoing MM-attacks is an important security task; modeling MM-interactions is an interesting task for semantics of computation. Traced monoidal categories are a natural framework for MM-modelling, as the trace structure provides a tool to hide what happens *in the middle*. An effective analysis of what has been traced out seems to require an additional property of traces, called *normality*. We describe a modest model of network computatio…
We give a proof of a formula for the trace of self-braidings (in an arbitrary channel) in UMTCs which first appeared in the context of rational conformal field theories (CFTs). The trace is another invariant for UMTCs which depends only on modular data, and contains the expression of the Frobenius-Schur indicator as a special case. Furthermore, we discuss some applications of the trace formula to the realizability problem of modular data and to the classification of UMTCs.
We study the question when a $*$-autonomous Mix-category has a representation as a $*$-autonomous Mix-subcategory of a compact one. We define certain partial trace-like operation on morphisms of a Mix-category, which we call a mixed trace, and show that any structure preserving embedding of a Mix-category into a compact one induces a mixed trace on the former. We also show that, conversely, if a Mix-category ${\bf K}$ has a mixed trace, then we can construct a compact category and structure preserving embedding of ${\bf K}$ into it, which induces the same mixed trace. Finally, we find a specif…
We define a tracelike transformation to be a natural family of conjugation invariant maps $T_{x,C}: hom_C(x,x) \to hom_C(1,1)$ for all dualisable objects $x$ in any symmetric monoidal infinity-category $C$. This generalises the trace from linear algebra that assigns a scalar $Tr(f) \in k$ to any endomorphism $f:V \to V$ of a finite-dimensional $k$-vector space. Our main theorem computes the moduli space of tracelike transformations using the one-dimensional cobordism hypothesis with singularities. As a consequence we show that the trace $Tr$ can be uniquely extended to a tracelike transformati…
We study the additivity of various geometric invariants involved in Reimann-Roch type formulas and defined via the trace map. To do so in a general context we prove that given any Grothendieck category A, the derived category D(A) has a compatible triangulation in the sense of [May, J.P. :The Additivity of Traces in Triangulated Categories, Advances in Mathematics 163, (2001), 34-73], but not resorting to model categories. The result is proved just using the structural properties inherent to D(A). In the second part of the paper we apply compatibility to prove additivity of traces firstly and …
We give an explicit point-set construction of the Dennis trace map from the $K$-theory of endomorphisms $K\mathrm{End}(\mathcal{C})$ to topological Hochschild homology $\mathrm{THH}(\mathcal{C})$ for any spectral Waldhausen category $\mathcal{C}$. We describe the necessary technical foundations, most notably a well-behaved model for the spectral category of diagrams in $\mathcal{C}$ indexed by an ordinary category via the Moore end. This is applied to define a version of Waldhausen's $S_{\bullet}$-construction for spectral Waldhausen categories, which is central to this account of the Dennis t…
The primary contribution of this paper is to give a formal, categorical treatment to Penrose's abstract tensor notation, in the context of traced symmetric monoidal categories. To do so, we introduce a typed, sum-free version of an abstract tensor system and demonstrate the construction of its associated category. We then show that the associated category of the free abstract tensor system is in fact the free traced symmetric monoidal category on a monoidal signature. A notable consequence of this result is a simple proof for the soundness and completeness of the diagrammatic language for trac…
Every small monoidal category with universal finite joins of central idempotents is monoidally equivalent to the category of global sections of a sheaf of local monoidal categories on a topological space. Every small stiff monoidal category monoidally embeds into such a category of global sections. An infinitary version of these theorems also holds in the spatial case. These representation results are functorial and subsume the Lambek-Moerdijk-Awodey sheaf representation for toposes, the Stone representation of Boolean algebras, and the Takahashi representation of Hilbert modules as continuous…
We study the biclosedness of the monoidal categories of modules and comodules over a (left or right) Hopf algebroid, along with the bimodule category centres of the respective opposite categories and a corresponding categorical equivalence to anti Yetter-Drinfel'd contramodules and anti Yetter-Drinfel'd modules, respectively. This is directly connected to the existence of a trace functor on the monoidal categories of modules and comodules in question, which in turn allows to recover (or define) cyclic operators enabling cyclic cohomology.
In this paper we show that to a unital associative algebra object (resp. co-unital co-associative co-algebra object) of any abelian monoidal category $\mathcal{C}$ endowed with a symmetric $2$-trace, one can attach a cyclic (resp. cocyclic) module, and therefore speak of the cyclic (co)homology of the (co)algebra "with coefficients in $F$". We observe that if $\mathcal{M}$ is a $\mathcal{C}$-bimodule category equipped with a stable central pair then $\mathcal{C}$ acquires a symmetric 2-trace. The dual notions of symmetric $2$-contratraces and stable central contrapairs are derived as well. As …
For C a factorisable and pivotal finite tensor category over an algebraically closed field of characteristic zero we show: 1) C always contains a simple projective object; 2) if C is in addition ribbon, the internal characters of projective modules span a submodule for the projective SL(2,Z)-action; 3) the action of the Grothendieck ring of C on the span of internal characters of projective objects can be diagonalised; 4) the linearised Grothendieck ring of C is semisimple iff C is semisimple. Results 1-3 remain true in positive characteristic under an extra assumption. Result 1 implies that t…
Differential categories provide an axiomatization of the basics of differentiation and categorical models of differential linear logic. As differentiation is an important tool throughout quantum mechanics and quantum information, it makes sense to study applications of the theory of differential categories to categorical quantum foundations. In categorical quantum foundations, compact closed categories (and therefore traced symmetric monoidal categories) are one of the main objects of study, in particular the category of finite-dimensional Hilbert spaces FHilb. In this paper, we will explain w…
We replace a ring with a small $\mathbb C$-linear category $\mathcal{C}$, seen as a ring with several objects in the sense of Mitchell. We introduce Fredholm modules over this category and construct a Chern character taking values in the cyclic cohomology of $\mathcal C$. We show that this categorified Chern character is homotopy invariant and is well-behaved with respect to the periodicity operator in cyclic cohomology. For this, we also obtain a description of cocycles and coboundaries in the cyclic cohomology of $\mathcal C$ (and more generally, in the Hopf-cyclic cohomology of a Hopf modul…
A variant of the trace in a monoidal category is given in the setting of closed monoidal derivators, which is applicable to endomorphisms of fiberwise dualizable objects. Functoriality of this trace is established. As an application, an explicit formula for the trace of the homotopy colimit of endomorphisms over finite EI-categories is deduced. This result can be seen as a generalization of the additivity of traces in monoidal categories with a compatible triangulation.
We generalize the definition of satellites with respect to presheaves (and copresheaves) with trace in the sense of Inassaridze; 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 the construction of satellites is given.
We study structures which have arisen in recent work by the present author and Bob Coecke on a categorical axiomatics for Quantum Mechanics; in particular, the notion of strongly compact closed category. We explain how these structures support a notion of scalar which allows quantitative aspects of physical theory to be expressed, and how the notion of strong compact closure emerges as a significant refinement of the more classical notion of compact closed category. We then proceed to an extended discussion of free constructions for a sequence of progressively more complex kinds of structured …
We define the affinization of an arbitrary monoidal category $\mathcal{C}$, corresponding to the category of $\mathcal{C}$-diagrams on the cylinder. We also give an alternative characterization in terms of adjoining dot generators to $\mathcal{C}$. The affinization formalizes and unifies many constructions appearing in the literature. In particular, we describe a large number of examples coming from Hecke-type algebras, braids, tangles, and knot invariants. When $\mathcal{C}$ is rigid, its affinization is isomorphic to its horizontal trace, although the two definitions look quite different. In…
We consider algebras in a modular tensor category C. If the trace pairing of an algebra A in C is non-degenerate we associate to A a commutative algebra Z(A), called the full centre, in a doubled version of the category C. We prove that two simple algebras with non-degenerate trace pairing are Morita-equivalent if and only if their full centres are isomorphic as algebras. This result has an interesting interpretation in two-dimensional rational conformal field theory; it implies that there cannot be several incompatible sets of boundary conditions for a given bulk theory.
The purpose of this work is to provide details about the construction of the Chern character for categorical sheaves mentioned in our previous work "Chern character, loop spaces and derived algebraic geometry". For this, we introduce and study the notion of rigid symmetric monoidal \infty-category. We show how trace maps can be constructed in this higher categorical setting, and using a recent work of Hopkins-Lurie we prove the existence of a "cyclic trace", which is the main ingredient in the construction of the Chern character. Our Chern character is then constructed for any pair (T,A), cons…
We continue the investigation of tabular algebras with trace (a certain class of associative ${\Bbb Z}[v, v^{-1}]$-algebras equipped with distinguished bases) by determining the extent to which the tabular structure may be recovered from a knowledge of the structure constants. This problem is equivalent to understanding a certain category (the category of table data associated to a tabular algebra) which we introduce. The main result is that this category is equivalent to another category (the category of based posets associated to a tabular algebra) whose structure we describe explicitly.
We define higher Frobenius-Schur indicators for objects in linear pivotal monoidal categories. We prove that they are category invariants, and take values in the cyclotomic integers. We also define a family of natural endomorphisms of the identity endofunctor on a $k$-linear semisimple rigid monoidal category, which we call the Frobenius-Schur endomorphisms. For a $k$-linear semisimple pivotal monoidal category -- where both notions are defined --, the Frobenius-Schur indicators can be computed as traces of the Frobenius-Schur endomorphisms.
