Lawvere Theories Research Articles

Boolean algebras, Morita invariance and the algebraic K-theory of Lawvere theories

AbstractThe algebraic K-theory of Lawvere theories is a conceptual device to elucidate the stable homology of the symmetry groups of algebraic structures such as the permutation groups and the automorphism groups of free groups. In this paper, we fully address the question of how Morita equivalence classes of Lawvere theories interact with algebraic K-theory. On the one hand, we show that the higher algebraic K-theory is invariant under passage to matrix theories. On the other hand, we show that the higher algebraic K-theory is not fully Morita invariant because of the behavior of idempotents in non-additive contexts: We compute the K-theory of all Lawvere theories Morita equivalent to the theory of Boolean algebras.

Varieties of ordered algebras as categories

A variety is a category of ordered (finitary) algebras presented by inequations between terms. We characterize categories enriched over the category of posets which are equivalent to a variety. This is quite analogous to Lawvere’s classical characterization of varieties of ordinary algebras. We also study the relationship of varieties to discrete Lawvere theories, and varieties as concrete categories over $$\mathbf{ Pos }$$ .

GENERALISATIONS OF LODAY’S ASSEMBLY MAPS FOR LAWVERE’S ALGEBRAIC THEORIES

AbstractLoday’s assembly maps approximate the K-theory of group rings by the K-theory of the coefficient ring and the corresponding homology of the group. We present a generalisation that places both ingredients on the same footing. Building on Elmendorf–Mandell’s multiplicativity results and our earlier work, we show that the K-theory of Lawvere theories is lax monoidal. This result makes it possible to present our theory in a user-friendly way without using higher-categorical language. It also allows us to extend the idea to new contexts and set up a nonabelian interpolation scheme, raising novel questions. Numerous examples illustrate the scope of our extension.

Bifold Algebras and Commutants for Enriched Algebraic Theories

Commuting pairs of algebraic structures on a set have been studied by several authors and may be described equivalently as algebras for the tensor product of Lawvere theories, or more basically as certain bifunctors that here we call bifold algebras. The much less studied notion of commutant for Lawvere theories was first introduced by Wraith and generalizes the notion of centralizer clone in universal algebra. Working in the general setting of enriched algebraic theories for a system of arities, we study the interaction of the concepts of bifold algebra and commutant. We show that the notion of commutant arises via a universal construction in a two-sided fibration of bifold algebras over various theories. On this basis, we study special classes of bifold algebras that are related to commutants, introducing the notions of commutant bifold algebra and balanced bifold algebra. We establish several adjunctions and equivalences among these categories of bifold algebras and related categories of algebras over various theories, including commutative, contracommutative, saturated, and balanced algebras. We also survey and develop examples of commutant bifold algebras, including examples that employ Pontryagin duality and a theorem of Ehrenfeucht and Łoś on reflexive abelian groups. Along the way, we develop a functorial treatment of fundamental aspects of bifold algebras and commutants, including tensor products of theories and the equivalence of bifold algebras and commuting pairs of algebras. Because we work relative to a (possibly large) system of arities in a closed category $${\mathscr {V}}$$ , our main results are applicable to arbitrary $${\mathscr {V}}$$ -monads on a finitely complete $${\mathscr {V}}$$ , the enriched theories of Borceux and Day, the enriched Lawvere theories of Power relative to a regular cardinal, and other notions of algebraic theory.

Categorical Stochastic Processes and Likelihood

We take a category-theoretic perspective on the relationship between probabilistic modeling and gradient based optimization. We define two extensions of function composition to stochastic process subordination: one based on a co-Kleisli category and one based on the parameterization of a category with a Lawvere theory. We show how these extensions relate to the category of Markov kernels Stoch through a pushforward procedure.We extend stochastic processes to parametric statistical models and define a way to compose the likelihood functions of these models. We demonstrate how the maximum likelihood estimation procedure defines a family of identity-on-objects functors from categories of statistical models to the category of supervised learning algorithms Learn.Code to accompany this paper can be found on GitHub (https://github.com/dshieble/Categorical_Stochastic_Processes_and_Likelihood).

Enriched Lawvere Theories for Operational Semantics

Enriched Lawvere theories are a generalization of Lawvere theories that allow us to describe the operational semantics of formal systems. For example, a graph enriched Lawvere theory describes structures that have a graph of operations of each arity, where the vertices are operations and the edges are rewrites between operations. Enriched theories can be used to equip systems with operational semantics, and maps between enriching categories can serve to translate between different forms of operational and denotational semantics. The Grothendieck construction lets us study all models of all enriched theories in all contexts in a single category. We illustrate these ideas with the SKI-combinator calculus, a variable-free version of the lambda calculus.

Petri nets based on Lawvere theories

AbstractWe give a definition of Q-net, a generalization of Petri nets based on a Lawvere theory Q, for which many existing variants of Petri nets are a special case. This definition is functorial with respect to change in Lawvere theory, and we exploit this to explore the relationships between different kinds of Q-nets. To justify our definition of Q-net, we construct a family of adjunctions for each Lawvere theory explicating the way in which Q-nets present free models of Q in Cat. This gives a functorial description of the operational semantics for an arbitrary category of Q-nets. We show how this can be used to construct the semantics for Petri nets, pre-nets, integer nets, and elementary net systems.

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

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

Distributive laws for Lawvere theories

Distributive laws give a way of combining two algebraic structures expressed as monads; in this paper we propose a theory of distributive laws for combining algebraic structures expressed as Lawvere theories. We propose four approaches, involving profunctors, monoidal profunctors, an extension of the free finite-product category 2-monad from Cat to Prof, and factorisation systems respectively. We exhibit comparison functors between CAT and each of these new frameworks to show that the distributive laws between the Lawvere theories correspond in a suitable way to distributive laws between their associated finitary monads. The different but equivalent formulations then provide, between them, a framework conducive to generalisation, but also an explicit description of the composite theories arising from distributive laws.

Higher Lawvere theories

We survey Lawvere theories at the level of ∞-categories, as an alternative framework for higher algebra (rather than ∞-operads). From a pedagogical perspective, they make many key definitions and constructions less technical. They also play a prominent role in equivariant homotopy theory and its relatives.Our main result establishes a universal property for the ∞-category of Lawvere theories, which completely characterizes the relationship between a Lawvere theory and its ∞-category of models. Many familiar properties of Lawvere theories follow directly.As a consequence, we establish a correspondence between enriched and module Lawvere theories, which implies that the Burnside category is a classifying object for additive categories. This completes a proof from our earlier paper on the commutative algebra of categories.

Lawvere theories and C-systems

In this paper we consider the class of l-bijective C-systems, i.e., C-systems for which the length function is a bijection. The main result of the paper is a construction of an isomorphism between two categories - the category of l-bijective C-systems and the category of Lawvere theories.

Algebraic Theories and Commutativity in a Sheaf Topos

For any site of definition $$\mathcal {C}$$ C of a Grothendieck topos $$\mathcal {E}$$ E, we define a notion of a $$\mathcal {C}$$ C-ary Lawvere theory $$\tau : \mathscr {C} \rightarrow \mathscr {T}$$ τ:C→T whose category of models is a stack over $$\mathcal {E}$$ E. Our definitions coincide with Lawvere’s finitary theories when $$\mathcal {C}=\aleph _0$$ C=ℵ0 and $$\mathcal {E} = {{\,\mathrm{\mathbf {Set}}\,}}$$ E=Set. We construct a fibered category $${{\,\mathrm{\mathbf {Mod}}\,}}^{\mathscr {T}}$$ ModT of models as a stack over $$\mathcal {E}$$ E and prove that it is $$\mathcal {E}$$ E-complete and $$\mathcal {E}$$ E-cocomplete. We show that there is a free-forget adjunction $$F \dashv U: {{\,\mathrm{\mathbf {Mod}}\,}}^{\mathscr {T}} \leftrightarrows \mathscr {E}$$ F⊣U:ModT⇆E. If $$\tau $$ τ is a commutative theory in a certain sense, then we obtain a “locally monoidal closed” structure on the category of models, which enhances the free-forget adjunction to an adjunction of symmetric monoidal $$\mathcal {E}$$ E-categories. Our results give a general recipe for constructing a monoidal $$\mathcal {E}$$ E-cosmos in which one can do enriched $$\mathcal {E}$$ E-category theory. As an application, we describe a convenient category of linear spaces generated by the theory of Lebesgue integration.

Modalities, cohesion, and information flow

It is informally understood that the purpose of modal type constructors in programming calculi is to control the flow of information between types. In order to lend rigorous support to this idea, we study the category of classified sets, a variant of a denotational semantics for information flow proposed by Abadi et al. We use classified sets to prove multiple noninterference theorems for modalities of a monadic and comonadic flavour. The common machinery behind our theorems stems from the the fact that classified sets are a (weak) model of Lawvere's theory of axiomatic cohesion. In the process, we show how cohesion can be used for reasoning about multi-modal settings. This leads to the conclusion that cohesion is a particularly useful setting for the study of both information flow, but also modalities in type theory and programming languages at large.

Gabriel-Morita Theory for Excisive Model Categories

We develop a Gabriel-Morita theory for strong monads on pointed monoidal model categories. Assuming that the model category is excisive, i.e. the derived suspension functor is conservative, we show that if the monad T preserves cofibre sequences up to homotopy and has a weakly invertible strength, then the category of T-algebras is Quillen equivalent to the category of T(I)-modules where I is the monoidal unit. This recovers Schwede’s theorem on connective stable homotopy over a pointed Lawvere theory as special case.

On the commutative algebra of categories

We discuss what it means for a symmetric monoidal category to be a module over a commutative semiring category. Each of the categories of (1) cartesian monoidal categories, (2) semiadditive categories, and (3) connective spectra can be recovered in this way as categories of modules over a commutative semiring category (or $\infty$-category in the last case). This language provides a simultaneous generalization of the formalism of algebraic theories (operads, PROPs, Lawvere theories) and stable homotopy theory, with essentially a variant of algebraic K-theory bridging between the two.

Commutants for Enriched Algebraic Theories and Monads

We define and study a notion of commutant for $$\mathscr {V}$$ -enriched $${\mathscr {J}}$$ -algebraic theories for a system of arities $${\mathscr {J}}$$ , recovering the usual notion of commutant or centralizer of a subring as a special case alongside Wraith’s notion of commutant for Lawvere theories as well as a notion of commutant for $$\mathscr {V}$$ -monads on a symmetric monoidal closed category $$\mathscr {V}$$ . This entails a thorough study of commutation and Kronecker products of operations in $${\mathscr {J}}$$ -theories. In view of the equivalence between $${\mathscr {J}}$$ -theories and $${\mathscr {J}}$$ -ary monads we reconcile this notion of commutation with Kock’s notion of commutation of cospans of monads and, in particular, the notion of commutative monad. We obtain notions of $${\mathscr {J}}$$ -ary commutant and absolute commutant for $${\mathscr {J}}$$ -ary monads, and we show that for finitary monads on $$\text {Set}$$ the resulting notions of finitary commutant and absolute commutant coincide. We examine the relation of the notion of commutant to both the notion of codensity monad and the notion of algebraic structure in the sense of Lawvere.

Convex Spaces, Affine Spaces, and Commutants for Algebraic Theories

Certain axiomatic notions of affine space over a ring and convex space over a preordered ring are examples of the notion of $$\mathscr {T}$$ -algebra for an algebraic theory $$\mathscr {T}$$ in the sense of Lawvere. Herein we study the notion of commutant for Lawvere theories that was defined by Wraith and generalizes the notion of centralizer clone. We focus on the Lawvere theory of left R -affine spaces for a ring or rig R, proving that this theory can be described as a commutant of the theory of pointed right R-modules. Further, we show that for a wide class of rigs R that includes all rings, these theories are commutants of one another in the full finitary theory of R in the category of sets. We define left R -convex spaces for a preordered ring R as left affine spaces over the positive part $$R_+$$ of R. We show that for any firmly archimedean preordered algebra R over the dyadic rationals, the theories of left R-convex spaces and pointed right $$R_+$$ -modules are commutants of one another within the full finitary theory of $$R_+$$ in the category of sets. Applied to the ring of real numbers $$\mathbb {R}$$ , this result shows that the connection between convex spaces and pointed $$\mathbb {R}_+$$ -modules that is implicit in the integral representation of probability measures is a perfect ‘duality’ of algebraic theories.

Sound and Complete Equational Reasoning over Comodels

Comodels of Lawvere theories, i.e. models in Setop, model state spaces with algebraic access operations. Standard equational reasoning is known to be sound but incomplete for comodels. We give two sound and complete calculi for equational reasoning over comodels: an inductive calculus for equality-on-the-nose, and a coinductive/inductive calculus for equality modulo bisimulation which captures bisimulations syntactically through non-wellfounded proofs.

Stateful Runners of Effectful Computations

What structure is required of a set so that computations in a given notion of computation can be run statefully with this set as the state set? For running nondeterministic computations statefully, a resolver structure is needed; for interactive I/O computations, a “responder-listener” structure is necessary; to be able to serve stateful computations, the set must carry the structure of a lens. We show that, in general, to be a stateful runner of computations for a monad corresponding to a Lawvere theory (defined as a set equipped with a monad morphism between the given monad and the state monad for this set) is the same as to be a comodel of the theory, i.e., a coalgebra of the corresponding comonad. We work out a number of instances of this observation and also compare runners to handlers.

Nominal Lawvere Theories: A category theoretic account of equational theories with names

Names, or object-level variables, are a ubiquitous feature in programming languages and other computational applications. Reasoning with names, and related constructs like binding and freshness, often poses conceptual and technical challenges. Nominal Equational Logic (NEL) is a logic for reasoning about equations in the presence of freshness side conditions. This paper gives a category theoretic account of NEL theories, by analogy with Lawvere's classic correspondence between equational theories and small categories with finite products. This development reveals the abstract structure behind reasoning with equations modulo freshness.