Joyal and Street note in their paper on braided monoidal categories [Braided tensor categories, Advances in Math. 102(1993) 20-78] that the 2-category V-Cat of categories enriched over a braided monoidal category V is not itself braided in any way that is based upon the braiding of V. The exception that they mention is the case in which V is symmetric, which leads to V-Cat being symmetric as well. The symmetry in V-Cat is based upon the symmetry of V. The motivation behind this paper is in part to describe how these facts relating V and V-Cat are in turn related to a categorical analogue of topological delooping. To do so I need to pass to a more general setting than braided and symmetric categories -- in fact the k-fold monoidal categories of Balteanu et al in [Iterated Monoidal Categories, Adv. Math. 176(2003) 277-349]. It seems that the analogy of loop spaces is a good guide for how to define the concept of enrichment over various types of monoidal objects, including k-fold monoidal categories and their higher dimensional counterparts. The main result is that for V a k-fold monoidal category, V-Cat becomes a (k-1)-fold monoidal 2-category in a canonical way. In the next paper I indicate how this process may be iterated by enriching over V-Cat, along the way defining the 3-category of categories enriched over V-Cat. In future work I plan to make precise the n-dimensional case and to show how the group completion of the nerve of V is related to the loop space of the group completion of the nerve of V-Cat.
This paper is an abridged version of `Enrichment as categorical delooping I' math.CT/0304026.
The 2-category V-Cat of categories enriched over a braided monoidal category V is not itself braided in any way that is based upon the braiding of V. The exception is the case in which V is symmetric, which leads to V-Cat being symmetric as well. This paper describes how these facts are related to a categorical analogue of topological delooping. It seems that the analogy of loop spaces is a good guide for how to define the concept of enrichment over various types of monoidal objects, including k-fold monoidal categories and their higher dimensional counterparts. The main result is that for V a…
We develop a notion of iterated monoidal category and show that this notion corresponds in a precise way to the notion of iterated loop space. Specifically the group completion of the nerve of such a category is an iterated loop space and free iterated monoidal categories give rise to finite simplicial operads of the same homotopy type as the classical little cubes operads used to parametrize the higher H-space structure of iterated loop spaces. Iterated monoidal categories encompass, as a special case, the notion of braided tensor categories, as used in the theory of quantum groups.
We propose a recursive definition of V-n-categories and their morphisms. We show that for V k-fold monoidal the structure of a (k-n)-fold monoidal strict (n+1)-category is possessed by V-n-Cat. This article is a completion of the work begun in math.CT/0306086, and the initial sections duplicate the beginning of that paper.
We study the totality of categories weakly enriched in a monoidal bicategory using a notion of enriched icon as 2-cells. We show that when the monoidal bicategory in question is symmetric then this process can be iterated. We show that starting from the symmetric monoidal bicategory Cat and performing the construction twice yields a convenient symmetric monoidal bicategory of partially strict tricategories. We show that restricting to the doubly degenerate ones immediately gives the correct bicategory of "2-tuply monoidal categories" missing from our earlier studies of the Periodic Table. We p…
We define a bar construction endofunctor on the category of commutative augmented monoids $A$ of a symmetric monoidal category $\mathcal{V}$ endowed with a left adjoint monoidal functor $F:s\mathbf{Set}\to \mathcal{V}$. To do this, we need to carefully examine the monoidal properties of the well-known (reduced) simplicial bar construction $B_\bullet(1,A,1)$. We define a geometric realization $|-|$ with respect to the image under $F$ of the canonical cosimplicial simplicial set. This guarantees good monoidal properties of $|-|$: it is monoidal, and given a left adjoint monoidal functor $G:\math…
Operads were originally defined as V-operads, that is, enriched in a symmetric or braided monoidal category V. The symmetry or braiding in V is required in order to describe the associativity axiom the operads must obey, as well as the associativity that must be a property of the action of an operad on any of its algebras. After a review of the role of operads in loop space theory and higher categories we go over definitions of iterated monoidal categories and introduce a large family of simple examples. Then we generalize the definition of operad by defining n-fold operads and their algebras …
Lyubashenko has described enriched 2-categories as categories enriched over V-Cat, the 2-category of categories enriched over a symmetric monoidal V. I have generalized this to the k-fold monoidal V. The symmetric case can easily be recovered. The introduction of this paper proposes a recursive definition of V-n-categories and their morphisms. Then I consider the special case of V-2-categories and give the details of the proof that with their morphisms these form the structure of a 3-category.
We provide a definition of enrichment that applies to a wide variety of categorical structures, generalizing Leinster's theory of enriched $T$-multicategories. As a sample of newly enrichable structures, we describe in detail the examples of enriched monoidal categories and enriched double categories, with a focus on monoidal double categories as broadly convenient bases of enrichment.
We define a notion of category enriched over an oplax monoidal category $V$, extending the usual definition of category enriched over a monoidal category. Even though oplax monoidal structures involve infinitely many functors $V^n\to V$, defining categories enriched over $V$ only requires the lower arity maps $(n \leq 3)$, similarly to the monoidal case. The focal point of the enrichment theory shifts, in the oplax case, from the notion of $V$-category (given by collections of objects and hom-objects together with composition and unit maps) to the one of categories enriched over $V$ (genuine c…
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.
We define the phrase `category enriched in an fc-multicategory' and explore some examples. An fc-multicategory is a very general kind of 2-dimensional structure, special cases of which are double categories, bicategories, monoidal categories and ordinary multicategories. Enrichment in an fc-multicategory extends the (more or less well-known) theories of enrichment in a monoidal category, in a bicategory, and in a multicategory. Moreover, fc-multicategories provide a natural setting for the bimodules construction, traditionally performed on suitably cocomplete bicategories. Although this paper …
I show that the theories of enrichment in a monoidal infinity-category defined by Hinich and by Gepner-Haugseng agree, and that the identification is unique. Among other things, this makes the Yoneda lemma available in the former model.
We study the existence of universal measuring comonoids $P(A,B)$ for a pair of monoids $A$, $B$ in a braided monoidal closed category, and the associated enrichment of a category of monoids over the monoidal category of comonoids. In symmetric categories, we show that if $A$ is a bimonoid and $B$ is a commutative monoid, then $P(A,B)$ is a bimonoid; in addition, if $A$ is a cocommutative Hopf monoid then $P(A,B)$ always is Hopf. If $A$ is a Hopf monoid, not necessarily cocommutative, then $P(A,B)$ is Hopf if the fundamental theorem of comodules holds; to prove this we give an alternative descr…
We introduce a new type of weakly enriched categories over a given symmetric monoidal model category M; these are called Co-Segal categories. Their definition derives from the philosophy of classical (enriched) Segal categories. We study their homotopy theory by giving a model structure on them. One of the motivations of introducing these structure was to have an alternative definition of higher linear categories following Segal-like methods.
In this paper we answer the question: `what kind of a structure can a general multicategory be enriched in?' The answer is, in a sense to be made precise, that a multicategory of one type can be enriched in a multicategory of the type one level up. In the case of ordinary categories this reduces to something surprising: a category may be enriched in an `fc-multicategory', a very general kind of 2-dimensional structure encompassing monoidal categories, plain multicategories, bicategories and double categories. (It turns out that fc-multicategories also provide a natural setting for the bimodule…
We prove a number of results involving categories enriched over \textsc{CMet}, the category of complete metric spaces with possibly infinite distances. The category \textsc{CPMet} of intrinsic complete metric spaces is locally $\aleph_1$-presentable, closed monoidal, and comonadic over \textsc{CMet}. We also prove that the category \textsc{CCMet} of convex complete metric spaces is not closed monoidal and characterize the isometry-$\aleph_0$-generated objects in \textsc{CMet}, \textsc{CPMet} and \textsc{CCMet}, answering questions by Di Liberti and Rosický. Other results include the automatic …
We extend the theory of Sweeder's measuring comonoids to the framework of duoidal categories: categories equipped with two compatible monoidal structures. We use one of the tensor products to endow the category of monoids for the other with an enrichment in the category of comonoids. The enriched homs are provided by the universal measuring comonoids. We study a number of duoidal structures on categories of graded objects and of species and the associated enriched categories, such as an enrichment of graded (twisted) monoids in graded (twisted) comonoids, as well as two enrichments of symmetri…
We introduce the notion of a monoidal category enriched in a braided monoidal category $\mathcal V$. We set up the basic theory, and prove a classification result in terms of braided oplax monoidal functors to the Drinfeld center of some monoidal category $\mathcal T$. Even the basic theory is interesting; it shares many characteristics with the theory of monoidal categories enriched in a symmetric monoidal category, but lacks some features. Of particular note, there is no cartesian product of braided-enriched categories, and the natural transformations do not form a 2-category, but rather sat…
fc-multicategories are a very general kind of two-dimensional structure, encompassing bicategories, monoidal categories, double categories and ordinary multicategories. We define them and explain how they provide a natural setting for two familiar categorical ideas. The first is the bimodules construction, traditionally carried out on suitably cocomplete bicategories but perhaps more naturally carried out on fc-multicategories. The second is enrichment: there is a theory of categories enriched in an fc-multicategory, extending the usual theory of enrichment in a monoidal category. We finish by…
We introduce the theory of enrichment over an internal monoidal category as a common generalization of both the standard theories of enriched and internal categories. The aim of the paper is to justify and contextualize the new notion by comparing it to other known generalizations of enrichment: namely, those for indexed categories and for generalized multicategories. It turns out that both of these notions are closely related to internal enrichment and, as a corollary, to each other.
This paper emerged as a result of tackling the following three issues. Firstly, we would like the well known embedding of bicategories into pseudo double categories to be monoidal, which it is not if one uses the usual notion of a monoidal pseudo double category. Secondly, in \cite{Gabi} the question was raised: which would be an alternative notion to intercategories of Grandis and Paré, so that monoids in Böhm's monoidal category $(Dbl,\ot)$ of strict double categories and strict double functors with a Gray type monoidal product be an example of it? We obtain and prove that precisely the mono…
We introduce simple models for associative algebras and bimodules in the context of non-symmetric $\infty$-operads, and use these to construct an $(\infty,2)$-category of associative algebras, bimodules, and bimodule homomorphisms in a monoidal $\infty$-category. By working with $\infty$-operads over $Δ^{n,\text{op}}$ we iterate these definitions and generalize our construction to get an $(\infty,n+1)$-category of $E_{n}$-algebras and iterated bimodules in an $E_{n}$-monoidal $\infty$-category. Moreover, we show that if $\mathcal{C}$ is an $E_{n+k}$-monoidal $\infty$-category then the $(\infty…
Given a horizontal monoid M in a duoidal category F, we examine the relationship between bimonoid structures on M and monoidal structures on the category of right M-modules which lift the vertical monoidal structure of F. We obtain our result using a variant of the Tannaka adjunction. The approach taken utilizes hom-enriched categories rather than categories on which a monoidal category acts ("actegories"). The requirement of enrichment in F itself demands the existence of some internal homs, leading to the consideration of convolution for duoidal categories. Proving that certain hom-functors …
In this paper Hom-Lie algebras, Lie color algebras, Lie superalgebras and other type of generalized Lie algebras are recovered by means of an iterated construction, known as monadic decomposition of functors, which is based on Eilenberg-Moore categories. To this aim we introduce the notion of Milnor-Moore category as a monoidal category for which a Milnor-Moore type Theorem holds. We also show how to lift the property of being a Milnor-Moore category whenever a suitable monoidal functor is given and we apply this technique to provide examples.
Symmetric monoidal closed categories may be related to one another not only by the functors between them but also by enrichment of one in another, and it was known to G. M. Kelly in the 1960s that there is a very close connection between these phenomena. In this first part of a two-part series on this subject, we show that the assignment to each symmetric monoidal closed category $V$ its associated $V$-enriched category $\underline{V}$ extends to a 2-functor valued in an op-2-fibred 2-category of symmetric monoidal closed categories enriched over various bases. For a fixed $V$, we show that th…
The notion of 2--monoidal category used here was introduced by B.~Vallette in 2007 for applications in the operadic context. The starting point for this article was a remark by Yu. Manin that in the category of quadratic algebras (that is, "quantum linear spaces") one can also define 2--monoidal structure(s) with rather unusual properties. Here we give a detailed exposition of these constructions, together with their generalisations to the case of quadratic operads. Their parallel exposition was motivated by the following remark. Several important operads/cooperads such as genus zero quantum c…
We define the notion of an additive model category, and we prove that any additive, stable, combinatorial model category has a natural enrichment over symmetric spectra based on simplicial abelian groups. As a consequence, every object in such a model category has a naturally associated endomorphism ring inside this spectra category. We establish the basic properties of this enrichment. We also develop some enriched model category theory. In particular, we have a notion of an adjoint pair of functors being a 'module' over another such pair. Such things are called "adjoint modules". We develop …
For a braided fusion category $\mathcal{V}$, a $\mathcal{V}$-fusion category is a fusion category $\mathcal{C}$ equipped with a braided monoidal functor $\mathcal{F}:\mathcal{V} \to Z(\mathcal{C})$. Given a fixed $\mathcal{V}$-fusion category $(\mathcal{C}, \mathcal{F})$ and a fixed $G$-graded extension $\mathcal{C}\subseteq \mathcal{D}$ as an ordinary fusion category, we characterize the enrichments $\widetilde{\mathcal{F}}:\mathcal{V} \to Z(\mathcal{D})$ of $\mathcal{D}$ which are compatible with the enrichment of $\mathcal{C}$. We show that G-crossed extensions of a braided fusion category …
We define (iterated) coisotropic correspondences between derived Poisson stacks, and construct symmetric monoidal higher categories of derived Poisson stacks where the $i$-morphisms are given by $i$-fold coisotropic correspondences. Assuming an expected equivalence of different models of higher Morita categories, we prove that all derived Poisson stacks are fully dualizable, and so determine framed extended TQFTs by the Cobordism Hypothesis. Along the way we also prove that the higher Morita category of $E_{n}$-algebras with respect to coproducts is equivalent to the higher category of iterate…
We develop a theory of infinity properads enriched in a general symmetric monoidal infinity category. These are defined as presheaves, satisfying a Segal condition and a Rezk completeness condition, over certain categories of graphs. In particular, we introduce a new category of level graphs which also allow us to give a framework for algebras over an enriched infinity properad. We show that one can vary the category of graphs without changing the underlying theory. We also show that infinity properads cannot always be rectified, indicating that a conjecture of the second author and Robertson …
We axiomatically define (pre-)Hilbert categories. The axioms resemble those for monoidal Abelian categories with the addition of an involutive functor. We then prove embedding theorems: any locally small pre-Hilbert category whose monoidal unit is a simple generator embeds (weakly) monoidally into the category of pre-Hilbert spaces and adjointable maps, preserving adjoint morphisms and all finite (co)limits. An intermediate result that is important in its own right is that the scalars in such a category necessarily form an involutive field. In case of a Hilbert category, the embedding extends …
This paper proves coherence results for categories with a natural transformation called \emph{intermutation} made of arrows from $(A\wedge B)\vee(C\wedge D)$ to ${(A\vee C)\wedge(B\vee D)}$, for $\wedge$ and $\vee$ being two biendofunctors. Intermutation occurs in iterated, or $n$-fold, monoidal categories, which were introduced in connection with $n$-fold loop spaces, and for which a related, but different, coherence result was obtained previously by Balteanu, Fiedorowicz, Schw\" anzl and Vogt. The results of the present paper strengthen up to a point this previous result, and show that two-f…
In this dissertation we examine enrichment relations between categories of dual structure and we sketch an abstract framework where the theory of fibrations and enriched category theory are appropriately united. We initially work in the context of a monoidal category, where we study an enrichment of the category of monoids in the category of comonoids under certain assumptions. This is induced by the existence of the universal measuring comonoid, a notion originally defined by Sweedler in vector spaces. We then consider the fibred category of modules over arbitrary monoids, and we establish it…
The well-known Lawvere category R of extended real positive numbers comes with a monoidal closed structure where the tensor product is the sum. But R has another such structure, given by multiplication, which is *-autonomous. Normed sets, with a norm in R, inherit thus two symmetric monoidal closed structures, and categories enriched on one of them have a 'subadditive' or 'submultiplicative' norm, respectively. Typically, the first case occurs when the norm expresses a cost, the second with Lipschitz norms. This paper is a preparation for a sequel, devoted to 'weighted algebraic topology', an …
Following Lawvere's description of metric spaces using enriched category theory, we introduce a change in the base of enrichment that allows description of some aspects of (relativistic) causal spaces. All such spaces are Cauchy complete, in the sense of enriched category theory. Furthermore, we give sufficient conditions on a base monoidal category for which enriched categories are Cauchy complete if and only if their underlying categories are (their idempotent arrows split).
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 set up a general theory of weak or homotopy-coherent enrichment in an arbitrary monoidal $\infty$-category $\mathcal{V}$. Our theory of enriched $\infty$-categories has many desirable properties; for instance, if the enriching $\infty$-category $\mathcal{V}$ is presentably symmetric monoidal then $\mathrm{Cat}^\mathcal{V}_\infty$ is as well. These features render the theory useful even when an $\infty$-category of enriched $\infty$-categories comes from a model category (as is often the case in examples of interest, e.g. dg-categories, spectral categories, and $(\infty,n)$-categories). This…
