Left Adjoint Functors Research Articles

Semiseparable Functors and Conditions up to Retracts

In a previous paper we introduced the concept of semiseparable functor. Here we continue our study of these functors in connection with idempotent (Cauchy) completion. To this aim, we introduce and investigate the notions of (co)reflection and bireflection up to retracts. We show that the (co)comparison functor attached to an adjunction whose associated (co)monad is separable is a coreflection (reflection) up to retracts. This fact allows us to prove that a right (left) adjoint functor is semiseparable if and only if the associated (co)monad is separable and the (co)comparison functor is a bireflection up to retracts, extending a characterization pursued by X.-W. Chen in the separable case. Finally, we provide a semi-analogue of a result obtained by P. Balmer in the framework of pre-triangulated categories.

Lax monoidal adjunctions, two‐variable fibrations and the calculus of mates

AbstractWe provide a calculus of mates for functors to the ‐category of ‐categories and extend Lurie's unstraightening equivalences to show that (op)lax natural transformations correspond to maps of (co)cartesian fibrations that do not necessarily preserve (co)cartesian edges. As a sample application, we obtain an equivalence between lax symmetric monoidal structures on right adjoint functors and oplax symmetric monoidal structures on the left adjoint functors between symmetric monoidal ‐categories that is compatible with both horizontal and vertical composition of such structures. As the technical heart of the paper, we study various new types of fibrations over a product of two ‐categories. In particular, we show how they can be dualised over one of the two factors and how they encode functors out of the Gray tensor product of ‐categories.

A note on a free group. The decomposition of a free group functor through the category of heaps

This note aims to introduce a left adjoint functor to the functor which assigns a heap to a group. The adjunction is monadic. It is explained how one can decompose a free group functor through the previously introduced adjoint and employ it to describe a slightly different construction of free groups.

Universal Enveloping Algebras of Lie–Rinehart Algebras as a Left Adjoint Functor

We prove how the universal enveloping algebra constructions for Lie-Rinehart algebras and anchored Lie algebras are naturally left adjoint functors. This provides a conceptual motivation for the universal properties these constructions satisfy. As a supplement, the categorical approach offers new insights into the definitions of Lie-Rinehart algebra morphisms, of modules over Lie-Rinehart algebras and of the infinitesimal gauge algebra of a module.

Applications of hyperhomology to adjoint functors

Hyperhomology is applied to give explicit constructions of left or right adjoint functors of some inclusions between unbounded homotopy categories of additive categories arising from cotorsion pairs in abelian categories. These constructions are independent of Brown representability and do not involve cardinality in set theory. Applications are then given to injective cotorsion pairs in abelian categories and some typical cotorsion pairs in module categories of rings.

Construction of free differential algebras by extending Gröbner-Shirshov bases

As a fundamental notion, the free differential algebra on a set is concretely constructed as the polynomial algebra on the differential variables. Such a construction is not known for the more general notion of the free differential algebra on an algebra, from the left adjoint functor of the forgetful functor from differential algebras to algebras, instead of sets. In this paper we show that a generator-relation presentation of a base algebra can be extended to the free differential algebra on this base algebra. More precisely, a Gröbner-Shirshov basis property of the base algebra can be extended to the free differential algebra on this base algebra, allowing a Poincaré-Birkhoff-Witt type basis for these more general free differential algebras. Examples are provided as illustrations.

Cauchy completion of fuzzy quasi-uniform spaces

In this paper, we study the completion of fuzzy quasi-uniform spaces from a categorical point of view. Firstly, we introduce the concept of prorelations and describe fuzzy quasi-uniform spaces as enriched categories. Then we construct the Yoneda embedding in fuzzy quasi-uniform spaces through promodules, and prove the validness of Yoneda Lemma for right adjoint promodules. Finally, we study the Cauchy completion of fuzzy quasi-uniform spaces by the Yoneda embedding. We show that the inclusion functor from the category of T0 separated complete fuzzy quasi-uniform spaces to the category of fuzzy quasiuniform spaces has a left adjoint functor. The monad related to this adjunction is just the T0 completion monad of fuzzy quasi-uniform spaces.

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].

The Path Algebra as a Left Adjoint Functor

We consider an intermediate category between the category of finite quivers and a certain category of pseudocompact associative algebras whose objects include all pointed finite dimensional algebras. We define the completed path algebra and the Gabriel quiver as functors. We give an explicit quotient of the category of algebras on which these functors form an adjoint pair. We show that these functors respect ideals, obtaining in this way an equivalence between related categories.

Bousfield Localisation and Colocalisation of One-Dimensional Model Structures

We give an account of Bousfield localisation and colocalisation for one-dimensional model categories---ones enriched over the model category of $0$-types. A distinguishing feature of our treatment is that it builds localisations and colocalisations using only the constructions of projective and injective transfer of model structures along right and left adjoint functors, and without any reference to Smith's theorem.

Presentably symmetric monoidal ∞–categories are represented by symmetric monoidal model categories

We prove the theorem stated in the title. More precisely, we show the stronger statement that every symmetric monoidal left adjoint functor between presentably symmetric monoidal infinity-categories is represented by a strong symmetric monoidal left Quillen functor between simplicial, combinatorial and left proper symmetric monoidal model categories.

Quantales, Generalised Premetrics and Free Locales

Premetrics and premetrisable spaces have been long studied and their topological interrelationships are well-understood. Consider the category ${\bf Pre}$ of premetric spaces and $\epsilon$-$\delta$ continuous functions as morphisms. The absence of the triangle inequality implies that the faithful functor ${\bf Pre} \to {\bf Top}$ - where a premetric space is sent to the topological space it generates - is not full. Moreover, the sequential nature of topological spaces generated from objects in ${\bf Pre}$ indicates that this functor is not surjective on objects either. Developed from work by Flagg and Weiss, we illustrate an extension ${\bf Pre}\hookrightarrow {\bf P} $ together with a faithful and surjective on objects left adjoint functor ${\bf P} \to {\bf Top}$ as an extension of ${\bf Pre} \to {\bf Top}$. We show this represents an optimal scenario given that ${\bf Pre} \to {\bf Top}$ preserves coproducts only. The objects in ${\bf P}$ are metric-like objects valued on value distributive lattices whose limits and colimits we show to be generated by free locales on discrete sets.

Affine hom-complexes

For two general polytopal complexes the set of face-wise affine maps between them is shown to be a polytopal complex in an algorithmic way. The resulting algorithm for computing the affine hom-complex is analyzed in detail. There is also a natural tensor product of polytopal complexes, which is the left adjoint functor for Hom. This extends the corresponding facts from single polytopes, systematic study of which was initiated in [6], [12]. Explicit examples of computations of the resulting structures are included. In the special case of simplicial complexes, the affine hom-complex is a functorial subcomplex of Kozlov’s combinatorial hom-complex [14], which generalizes Lovász’ well-known construction [15] for graphs.

Dynamical Systems in Categories

In this article we establish a bridge between dynamical systems, including topological and measurable dynamical systems as well as continuous skew product flows and nonautonomous dynamical systems; and coalgebras in categories having all finite products. We introduce a straightforward unifying definition of abstract dynamical system on finite product categories. Furthermore, we prove that such systems are in a unique correspondence with monadic algebras whose signature functor takes products with the time space. We discuss that the categories of topological spaces, metrisable and uniformisable spaces have exponential objects w.r.t. locally compact Hausdorff, strongly σ-compact or arbitrary time spaces as exponents, respectively. Exploiting the adjunction between taking products and exponential objects, we demonstrate a one-to-one correspondence between monadic algebras (given by dynamical systems) for the left-adjoint functor and comonadic coalgebras for the other. This, finally, provides a new, alternative perspective on dynamical systems.

Poincaré duality for Ext–groups between strict polynomial functors

We study the relation between left and right adjoint functors to the precomposition functor. As a consequence, we obtain various dualities for the Ext–groups in the category of strict polynomial functors.

The concept lattice functors

This paper is concerned with the relationship between contexts, closure spaces, and complete lattices. It is shown that, for a unital quantale L, both formal concept lattices and property oriented concept lattices are functorial from the category L-Ctx of L-contexts and infomorphisms to the category L-Sup of complete L-lattices and suprema-preserving maps. Moreover, the formal concept lattice functor can be written as the composition of a right adjoint functor from L-Ctx to the category L-Cls of L-closure spaces and continuous functions and a left adjoint functor from L-Cls to L-Sup.

Hodge realizations of 1-motives and the derived Albanese

AbstractWe prove that the embedding of the derived category of 1-motives up to isogeny into the triangulated category of effective Voevodsky motives, as well as its left adjoint functor LAlbℚ, commute with the Hodge realization. This result yields a new proof of the rational form of Deligne's conjecture on 1-motives.

Good tilting modules and recollements of derived module categories

Let T be an infinitely generated tilting module of projective dimension at most one over an arbitrary associative ring A, and let B be the endomorphism ring of T. We prove that if T is good, then there exists a ring C, a homological ring epimorphism B→C and a recollement among the (unbounded) derived module categories 𝒟C of C, 𝒟B of B and 𝒟A of A. In particular, the kernel of the total left-derived functor T⊗B𝕃- is triangle equivalent to the derived module category 𝒟C. Conversely, if T⊗B𝕃- admits a fully faithful left adjoint functor, then T is good. Moreover, if T arises from an injective ring epimorphism, then C is isomorphic to the coproduct of two relevant rings. In the case of commutative rings, the ring C can be strengthened as the tensor product of two commutative rings. Consequently, we produce a large variety of examples (from Dedekind domains and p-adic number theory, or Kronecker algebra) to show that two different stratifications of the derived module category of a ring by derived module categories of rings may have completely different derived composition factors (even up to ordering and up to derived equivalence), or different lengths. This shows that the Jordan–Hölder theorem fails even for stratifications by derived module categories.

Adjoints and Monads Related to Compact Lattices and Compact Lawson Idempotent Semimodules

For compact Hausdorff Lawson lattices L that are idempotent semirings with lower semicontinuous multiplications, free compact Hausdorff Lawson L-idempotent semimodules are constructed over compacta and over compact Hausdorff Lawson lower semilattices. It is shown that these free objects are spaces of L-valued regular measures. Related left adjoint functors and monads are described.

Skew Boolean algebras derived from generalized Boolean algebras

Every skew Boolean algebra S has a maximal generalized Boolean algebra image given by S/\({\mathcal{D}}\) where \({\mathcal{D}}\) is the Green’s relation defined initially on semigroups. In this paper we study skew Boolean algebras \(\omega({\bf B})\) constructed from generalized Boolean algebras B by a twisted product construction for which \(\omega({\bf B})/{\mathcal{D}} \cong {\bf B}\). In particular we study the congruence lattice of \(\omega({\bf B})\) with an eye to viewing \(\omega({\bf B})\) as a minimal skew Boolean cover of B. This construction is the object part of a functor \(\omega: {\bf GB} \rightarrow {\bf LSB}\) from the category GB of generalized Boolean algebras to the category LSB of left-handed skew Boolean algebras. Thus we also look at its left adjoint functor \(\Omega: {\bf LSB} \rightarrow {\bf GB}\).