The notion of proof-net category defined in this paper is closely related to graphs implicit in proof nets for the multiplicative fragment without constant propositions of linear logic. Analogous graphs occur in Kelly's and Mac Lane's coherence theorem for symmetric monoidal closed categories. A coherence theorem with respect to these graphs is proved for proof-net categories. Such a coherence theorem is also proved in the presence of arrows corresponding to the mix principle of linear logic. The notion of proof-net category catches the unit free fragment of the notion of star-autonomous category, a special kind of symmetric monoidal closed category.
This paper presents a coherence theorem for star-autonomous categories exactly analogous to Kelly's and Mac Lane's coherence theorem for symmetric monoidal closed categories. The proof of this theorem is based on a categorial cut-elimination result, which is presented in some detail.
This paper gives a simple presentation of the free star-autonomous category over a category, based on Eilenberg-Kelly-MacLane graphs and Trimble rewiring, yielding a full coherence theorem: the commutativity of diagrams of canonical maps is decidable.
Coherence with respect to Kelly-Mac Lane graphs is proved for categories that correspond to the multiplicative fragment without constant propositions of classical linear first-order predicate logic without or with mix. To obtain this result, coherence is first established for categories that correspond to the multiplicative conjunction-disjunction fragment with first-order quantifiers of classical linear logic, a fragment lacking negation. These results extend results published in previous two books by the authors, where coherence was established for categories of the corresponding fragments o…
A coherence result for symmetric monoidal closed categories with biproducts is shown in this paper. It is explained how to prove, by using the same technique, coherence for compact closed categories with biproducts and for dagger compact closed categories with dagger biproducts.
We introduce homotopical methods based on rewriting on higher-dimensional categories to prove coherence results in categories with an algebraic structure. We express the coherence problem for (symmetric) monoidal categories as an asphericity problem for a track category and we use rewriting methods on polygraphs to solve it. The setting is extended to more general coherence problems, seen as 3-dimensional word problems in a track category, including the case of braided monoidal categories.
It is proved that MacLane's coherence results for monoidal and symmetric monoidal categories can be extended to some other categories with multiplication; namely, to relevant, affine and cartesian categories. All results are formulated in terms of natural transformations equipped with ``graphs'' (g-natural transformations), and corresponding morphism theorems are given as consequences. Using these results, some basic relations between the free categories of these classes are obtained.
Coherence is here demonstrated for sesquicartesian categories, which are categories with nonempty finite products and arbitrary finite sums, including the empty sum, where moreover the first and the second projection from the product of the initial object with itself are the same. (Every bicartesian closed category, and, in particular, the category Set, is such a category.) This coherence amounts to the existence of a faithful functor from categories of this sort freely generated by sets of objects to the category of relations on finite ordinals, and it yields a very easy decision procedure fo…
Coherence is demonstrated for categories with binary products and sums, but without the terminal and the initial object, and without distribution. This coherence amounts to the existence of a faithful functor from a free category with binary products and sums to the category of relations on finite ordinals. This result is obtained with the help of proof-theoretic normalizing techniques. When the terminal object is present, coherence may still be proved if of binary sums we keep just their bifunctorial properties. It is found that with the simplest understanding of coherence this is the best on…
A survey is given of results about coherence for categories with finite products and coproducts. For these results, which were published previously by the authors in several places, some formulations and proofs are here corrected, and matters are updated. The categories investigated in this paper formalize equality of proofs in classical and intuitionistic conjunctive-disjunctive logic without distribution of conjunction over disjunction.
This paper presents the proof of the coherence theorem for Ann-categories whose set of axioms and original basic properties were given in [9]. Let $$\A=(\A,{\Ah},c,(0,g,d),a,(1,l,r),{\Lh},{\Rh})$$ be an Ann-category. The coherence theorem states that in the category $ \A$, any morphism built from the above isomorphisms and the identification by composition and the two operations $\tx$, $\ts$ only depends on its source and its target. The first coherence theorems were built for monoidal and symmetric monoidal categories by Mac Lane [7]. After that, as shown in the References, there are many res…
Over the recent years, the theory of rewriting has been used and extended in order to provide systematic techniques to show coherence results for strict higher categories. Here, we investigate a further generalization to Gray categories, which are known to be equivalent to tricategories. This requires us to develop the theory of rewriting in the setting of precategories, which include Gray categories as particular cases, and are adapted to mechanized computations. We show that a finite rewriting system in precategories admits a finite number of critical pairs, which can be efficiently computed…
A symmetric monoidal category is a category equipped with an associative and commutative (binary) product and an object which is the unit for the product. In fact, those properties only hold up to natural isomorphisms which satisfy some coherence conditions. The coherence theorem asserts the commutativity of all linear diagrams involving the left and right unitors, the associator and the braiding. We prove the coherence for symmetric monoidal categories using a homotopical method based on rewriting. For that scope, we detail the con vergence proof of Lafont's string diagram rewriting system wh…
I motivate a variation (due to K. Szlachányi) of monoidal categories called skew-monoidal categories where the unital and associativity laws are not required to be isomorphisms, only natural transformations. Coherence has to be formulated differently than in the well-known monoidal case. In my (to my knowledge new) version, it becomes a statement of uniqueness of normalizing rewrites. I present a proof of this coherence theorem and also formalize it fully in the dependently typed programming language Agda.
Coherence theorems are fundamental to how we think about monoidal categories and their generalizations. In this paper we revisit Mac Lane's original proof of coherence for monoidal categories using the Grothendieck construction. This perspective makes the approach of Mac Lane's proof very amenable to generalization. We use the technique to give efficient proofs of many standard coherence theorems and new coherence results for bicategories with shadow and for their functors.
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…
We prove coherence theorems for bicategories, pseudofunctors and pseudonatural transformations. These theorems boil down to proving the coherence of some free $(4,2)$-categories. In the case of bicategories and pseudofunctors, existing rewriting techniques based on Squier's Theorem allow us to conclude. In the case of pseudonatural transformations this approach only proves the coherence of part of the structure, and we use a new rewriting result to conclude. To this end, we introduce the notions of white-categories and partial coherence.
The usual coherence theorem of MacLane for categories with multiplication assumes that a certain pentagonal diagram commutes in order to conclude that associativity isomorphisms are well defined in a certain practical sense. The practical aspects include creating associativity isomorphisms from a given one by tensoring with the identity on either the right or the left. We show, by reinspecting MacLane's original arguments, that if tensoring with the identity is restricted to one side, then the well definedness of constructed isomorphisms follows from naturality only, with no need of the commut…
Local coherence and recollement of abelian categories
Moscow Institute of Physics and Technology (MIPT)（莫斯科物理技术学院）；Center of Fundamental Mathematics（基础数学中心）；Faculty of Mathematics, HSE University（高等经济大学数学系）
Proofs of coherence in category theory, starting from Mac Lane's original proof of coherence for monoidal categories, are sometimes based on confluence techniques analogous to what one finds in the lambda calculus, or in term-rewriting systems in general. This applies to coherence results that assert that a category is a preorder, i.e. that ``all diagrams commute''. This note is about this analogy, paying particular attention to cases where the category for which coherence is proved is not a groupoid.
A new, self-contained, proof of a coherence result for categories equipped with two symmetric monoidal structures bridged by a natural transformation is given. It is shown that this coherence result is sufficient for $ω\timesω$-indexed family of iterated reduced bar constructions based on such a category.
Given an algebraic theory which can be described by a (possibly symmetric) operad $P$, we propose a definition of the \emph{weakening} (or \emph{categorification}) of the theory, in which equations that hold strictly for $P$-algebras hold only up to coherent isomorphism. This generalizes the theories of monoidal categories and symmetric monoidal categories, and several related notions defined in the literature. Using this definition, we generalize the result that every monoidal category is monoidally equivalent to a strict monoidal category, and show that the "strictification" functor has an i…
In this paper, we develop the proof theory of skew prounital closed categories. These are variants of the skew closed categories of Street where the unit is not represented. Skew closed categories in turn are a weakening of the closed categories of Eilenberg and Kelly where no structural law is required to be invertible. The presence of a monoidal structure in these categories is not required. We construct several equivalent presentations of the free skew prounital closed category on a given set of generating objects: a categorical calculus (Hilbert-style system), a cut-free sequent calculus a…
Categories of polymorphic lenses in computer science, and of open games in compositional game theory, have a curious structure that is reminiscent of compact closed categories, but differs in some crucial ways. Specifically they have a family of morphisms that behave like the counits of a compact closed category, but have no corresponding units; and they have a `partial' duality that behaves like transposition in a compact closed category when it is defined. We axiomatise this structure, which we refer to as a `teleological category'. We precisely define a diagrammatic language suitable for th…
We study what happens when coherence fails. Categories with a tensor product and a natural associativity isomorphism that does not necessarily satisfy the pentagon coherence requirements (called associative categories) are considered. Categorical versions of associahedra where naturality squares commute and pentagons do not, are constructed (called Catalan groupoids, $\An$). These groupoids are used in the construction of the free associative category. They are also used in the construction of the theory of associative categories (given as a 2-sketch). Generators and relations are given for th…
A relevant category is a symmetric monoidal closed category with a diagonal natural transformation that satisfies some coherence conditions. Every cartesian closed category is a relevant category in this sense. The denomination 'relevant' comes from the connection with relevant logic. It is shown that the category of sets with partial functions, which is isomorphic to the category of pointed sets, is a category that is relevant, but not cartesian closed.
Coherence theorems for covariant structures carried by a category have traditionally relied on the underlying term rewriting system of the structure being terminating and confluent. While this holds in a variety of cases, it is not a feature that is inherent to the coherence problem itself. This is demonstrated by the theory of iterated monoidal categories, which model iterated loop spaces and have a coherence theorem but fail to be confluent. We develop a framework for expressing coherence problems in terms of term rewriting systems equipped with a two dimensional congruence. Within this fram…
In this paper we give sufficient conditions for lifting an enhanced factorization system $ (\mathcal{E}, \mathcal{M}) $ on a $ 2 $-category $ \mathbf{D} $ to the functor $ 2 $-category $ \mathbf{D}^{\mathbf{C}} $, where $ \mathbf{C} $ is a small $ 2 $-category. Due to previous work of Lack, our work provides coherence results for $ 2 $-monads on functor $ 2 $-categories. These coherence results are of immediate interest to an ongoing collaboration of Guillou, May, Merling, and Osorno on equivariant infinite loop space theory.
2-Theories are a canonical way of describing categories with extra structure. 2-theory-morphisms are used when discussing how one structure can be replaced with another structure. This is central to categorical coherence theory. We place a Quillen model category structure on the category of 2-theories and 2-theory-morphisms where the weak equivalences are biequivalences of 2-theories. A biequivalence of 2-theories (Morita equivalence) induces and is induced by a biequivalence of 2-categories of algebras. This model category structure allows one to talk of the homotopy of 2-theories and discuss…
We define strict and weak duality involutions on 2-categories, and prove a coherence theorem that every bicategory with a weak duality involution is biequivalent to a 2-category with a strict duality involution. For this purpose we introduce "2-categories with contravariance", a sort of enhanced 2-category with a basic notion of "contravariant morphism", which can be regarded either as generalized multicategories or as enriched categories. This enables a universal characterization of duality involutions using absolute weighted colimits, leading to a conceptual proof of the coherence theorem.
This article tackles categorical coherence within a two-dimensional generalization of Lawvere's functorial semantics. 2-theories, a syntactical way of describing categories with structure, are presented. From the perspective here afforded, many coherence results become simple statements about the quasi-Yoneda lemma and 2-theory-morphisms. Given two 2-theories and a 2-theory-morphism between them, we explore the induced relationship between the corresponding 2-categories of algebras. The strength of the induced quasi-adjoints are classified by the strength of the 2-theory-morphism. These quasi-…
We introduce categorical models of $N_\infty$ spaces, which we call normed symmetric monoidal categories (NSMCs). These are ordinary symmetric monoidal categories equipped with compatible families of norm maps, and when specialized to a particular class of examples, they reveal a connection between the equivariant symmetric monoidal categories of Guillou-May-Merling-Osorno and those of Hill-Hopkins. We also give an operadic interpretation of the Mac Lane coherence theorem and generalize it to include NSMCs. Among other things, this theorem ensures that the classifying space of a NSMC is a $N_\…
Hypergraph categories have been rediscovered at least five times, under various names, including well-supported compact closed categories, dgs-monoidal categories, and dungeon categories. Perhaps the reason they keep being reinvented is two-fold: there are many applications---including to automata, databases, circuits, linear relations, graph rewriting, and belief propagation---and yet the standard definition is so involved and ornate as to be difficult to find in the literature. Indeed, a hypergraph category is, roughly speaking, a "symmetric monoidal category in which each object is equipped…
Differential categories are now an established abstract setting for differentiation. However not much attention has been given to the process which is inverse to differentiation: integration. This paper presents the parallel development for integration by axiomatizing an integral transformation, $s_A: !A \to !A \otimes A$, in a symmetric monoidal category with a coalgebra modality. When integration is combined with differentiation, the two fundamental theorems of calculus are expected to hold (in a suitable sense): a differential category with integration which satisfies these two theorem is c…
In here we define the concept of fibered symmetric bimonoidal categories. These are roughly speaking fibered categories D->C whose fibers are symmetric monoidal categories parametrized by C and such that both D and C have a further structure of a symmetric monoidal category that satisfy certain coherences that we describe. Our goal is to show that we can correspond to a fibered symmetric bimonoidal category an E_{\infty}-ring spectrum in a functorial way.
In this thesis I lift the Curry--Howard--Lambek correspondence between the simply-typed lambda calculus and cartesian closed categories to the bicategorical setting, then use the resulting type theory to prove a coherence result for cartesian closed bicategories. Cartesian closed bicategories---2-categories `up to isomorphism' equipped with similarly weak products and exponentials---arise in logic, categorical algebra, and game semantics. I show that there is at most one 2-cell between any parallel pair of 1-cells in the free cartesian closed bicategory on a set and hence---in terms of the dif…
A quadratic algebra is a homogeneous algebra generated by its elements of degree 1. Manin has endowed the category of quadratic algebras with two tensor products. These structures have been adapted to operads by Ginsburg and Kapranov. Berger has defined such tensor products for n-homogeneous algebras. The purpose of this paper is to define the notion of quadratic category, which is a category endowed with two tensor products. The Manin and Ginsburg-Kapranov constructions are examples of quadratic categories. We define also a Koszul complex, n-homogeneous operads and show how this notion can be…
Tangent categories were introduced by Rosicky as a categorical setting for differential structures in algebra and geometry; in recent work of Cockett, Crutwell and others, they have also been applied to the study of differential structure in computer science. In this paper, we prove that every tangent category admits an embedding into a representable tangent category---one whose tangent structure is given by exponentiating by a free-standing tangent vector, as in, for example, any model of Kock and Lawvere's synthetic differential geometry. The key step in our proof uses a coherence theorem fo…
We consider a theory of centers and homotopy centers of monoids in monoidal categories which themselves are enriched in duoidal categories. Duoidal categories (introduced by Aguillar and Mahajan under the name 2-monoidal categories) are categories with two monoidal structures which are related by some, not necessary invertible, coherence morphisms. Centers of monoids in this sense include many examples which are not `classical.' In particular, the 2-category of categories is an example of a center in our sense. Examples of homotopy center (analogue of the classical Hochschild complex) include …
Invertibility is an important concept in category theory. In higher category theory, it becomes less obvious what the correct notion of invertibility is, as extra coherence conditions can become necessary for invertible structures to have desirable properties. We define some properties we expect to hold in any reasonable definition of a weak $ω$-category. With these properties we define three notions of invertibility inspired by homotopy type theory. These are quasi-invertibility, where a two sided inverse is required, bi-invertibility, where a separate left and right inverse is given, and hal…
We present a new coherence theorem for comprehension categories, providing strict models of dependent type theory with all standard constructors, including dependent products, dependent sums, identity types, and other inductive types. Precisely, we take as input a "weak model": a comprehension category, equipped with structure corresponding to the desired logical constructions. We assume throughout that the base category is close to locally Cartesian closed: specifically, that products and certain exponentials exist. Beyond this, we require only that the logical structure should be *weakly sta…
The extension of ordinary category theory to $\infty$-categories at the start of the 21st century was a spectacular achievement pioneered by Joyal and Lurie with contributions from many others. Unfortunately, the technical arguments required to solve the infinite homotopy coherence problems inherent in these results make this theory difficult for non-experts to learn. This essay surveys two programs that seek to narrow the gap between $\infty$-category theory and ordinary 1-category theory. The first leverages similarities between the categories in which 1-categories and $\infty$-categories li…
We introduce a dependent type theory whose models are weak ω-categories, generalizing Brunerie's definition of ω-groupoids. Our type theory is based on the definition of ω-categories given by Maltsiniotis, himself inspired by Grothendieck's approach to the definition of ω-groupoids. In this setup, ω-categories are defined as presheaves preserving globular colimits over a certain category, called a coherator. The coherator encodes all operations required to be present in an ω-category: both the compositions of pasting schemes as well as their coherences. Our main contribution is to provide a ca…
General coherence theorems are constructed that yield explicit presentations of categorical and algebraic objects. The categorical structures involved are finitary discrete Lawvere 2-theories, though they are approached within the language of term rewriting theory. Two general coherence theorems are obtained. The first applies to terminating and confluent rewriting 2-theories. This result is exploited to construct systematic presentations for the higher Thompson groups and the Higman-Thompson groups. The presentations are categorically interesting as they arise from higher-arity analogues of t…
After recalling basic definitions and constructions for a finite group $G$ action on a $k$-linear category we give a concise proof of the following theorem of Elagin: if $\mathcal{C} = \langle \mathcal{A}, \mathcal{B} \rangle$ is a semiorthogonal decomposition of a triangulated category which is preserved by the action of $G$, and $\mathcal{C}^G$ is triangulated, then there is a semiorthogonal decomposition $\mathcal{C}^G = \langle \mathcal{A}^G, \mathcal{B}^G \rangle$. We also prove that any $G$-action on $\mathcal{C}$ is weakly equivalent to a strict $G$-action which is the analog of the Coh…
We introduce a tensor product for symmetric monoidal categories with the following properties. Let SMC denote the 2-category with objects small symmetric monoidal categories, arrows symmetric monoidal functors and 2-cells monoidal natural transformations. Our tensor product together with a suitable unit is part of a structure on SMC that is a 2-categorical version of the symmetric monoidal closed categories. This structure is surprisingly simple. In particular the arrows involved in the associativity and symmetry laws for the tensor and in the unit cancellation laws are 2-natural and satisfy c…
Many monoidal-type objects are known to be classified by maps from the Catalan simplicial set $\mathbb{C}$ to various nerves of categories and higher categories. There are, for example, three different nerves of the 2-category of categories $\mathsf{Cat}$ with the property that maps from $\mathbb{C}$ into them are in one to one correspondence with strict monoidal, monoidal, and skew monoidal categories. In this last case, the classification result can be understood as justifying the definition of skew monoidal categories intrinsically in terms of the structure of $\mathbb{C}$. In this paper, w…
Locally cartesian closed (lcc) categories are natural categorical models of extensional dependent type theory. This paper introduces the "gros" semantics in the category of lcc categories: Instead of constructing an interpretation in a given individual lcc category, we show that also the category of all lcc categories can be endowed with the structure of a model of dependent type theory. The original interpretation in an individual lcc category can then be recovered by slicing. As in the original interpretation, we face the issue of coherence: Categorical structure is usually preserved by func…
The skew monoidal categories of Szlachányi are a weakening of monoidal categories where the three structural laws of left and right unitality and associativity are not required to be isomorphisms but merely transformations in a particular direction. In previous work, we showed that the free skew monoidal category on a set of generating objects can be concretely presented as a sequent calculus. This calculus enjoys cut elimination and admits focusing, i.e. a subsystem of canonical derivations, which solves the coherence problem for skew monoidal categories. In this paper, we develop sequent cal…
Presentations of categories are a well-known algebraic tool to provide descriptions of categories by means of generators, for objects and morphisms, and relations on morphisms. We generalize here this notion, in order to consider situations where the objects are considered modulo an equivalence relation, which is described by equational generators. When those form a convergent (abstract) rewriting system on objects, there are three very natural constructions that can be used to define the category which is described by the presentation: one consists in turning equational generators into identi…
We briefly relate the existence of a middle-four interchange map in a category with two monoidal structures, to the standard Cockett and Seely notion of a weakly distributive category.
We study actions of discrete groups on 2-categories. The motivating examples are actions on the 2-category of representations of finite tensor categories and their relation with the extension theory of tensor categories by groups. Associated to a group action on a 2-category, we construct the 2-category of equivariant objects. We also introduce the G-equivariant notions of pseudofunctor, pseudonatural transformation and modification. Our first main result is a coherence theorem for 2-categories with an action of a group. For a 2-category B with an action of a group G, we construct a braided G-…
