Comma Category Research Articles

Abelian model structures on comma categories

UDC 512.64 Let A and B be bicomplete Abelian categories, which both have enough projectives and injectives and let T : A → B be a right exact functor. Under some mild conditions, we show that hereditary Abelian model structures on A and B can be amalgamated into a global hereditary Abelian model structure on the comma category ( T ↓ B ) . As an application of this result, we give an explicit description of a subcategory that consists of all trivial objects of the Gorenstein flat model structure on the category of modules over a triangular matrix ring.

On injective objects and existence of injective hulls in 𝑄-TOP/(𝑌, 𝜎)

In this paper, motivated by Cagliari and Mantovani, we have obtained a characterization of injective objects (with respect to the class of embeddings in the category 𝑄-TOP of 𝑄-topological spaces) in the comma category 𝑄-TOP/(𝑌,𝜎), when (𝑌,𝜎) is a stratified 𝑄-topological space, with the help of their 𝑇0-reflection. Further, we have proved that for any 𝑄-topological space (𝑌,𝜎), the existence of an injective hull of ((𝑋, 𝜏), 𝑓 ) in the comma category 𝑄-TOP/(𝑌, 𝜎) is equivalent to the existence of an injective hull of its 𝑇0-reflection ((𝑋 ̃,𝜏 ̃), 𝑓 ̃) in the comma category Q-TOP/(𝑌 ̃, 𝜎 ̃ ) (and in the comma category 𝑄-TOP0/(𝑌 ̃, 𝜎 ̃ ), where 𝑄-TOP0 denotes the category of 𝑇0-𝑄-topological spaces).

Functoriality of the Schmidt construction

Abstract After proving, in a purely categorial way, that the inclusion functor $\textrm {In}_{\textbf {Alg}(\varSigma )}$ from $\textbf {Alg}(\varSigma )$, the category of many-sorted $\varSigma $-algebras, to $\textbf {PAlg}(\varSigma )$, the category of many-sorted partial $\varSigma $-algebras, has a left adjoint $\textbf {F}_{\varSigma }$, the (absolutely) free completion functor, we recall, in connection with the functor $\textbf {F}_{\varSigma }$, the generalized recursion theorem of Schmidt, which we will also call the Schmidt construction. Next, we define a category $\textbf {Cmpl}(\varSigma )$, of $\varSigma $-completions, and prove that $\textbf {F}_{\varSigma }$, labelled with its domain category and the unit of the adjunction of which it is a part, is a weakly initial object in it. Following this, we associate to an ordered pair $(\boldsymbol {\alpha },f)$, where $\boldsymbol {\alpha }=(K,\gamma ,\alpha )$ is a morphism of $\varSigma $-completions from ${{\mathscr {F}}}=(\textbf {C},F,\eta )$ to $\mathscr {G}= (\textbf {D},G,\rho )$ and $f$ a homomorphism of $\textbf {D}$ from the partial $\varSigma $-algebra $\textbf {A}$ to the partial $\varSigma $-algebra $\textbf {B}$, a homomorphism $\varUpsilon ^{\mathscr {G},0}_{\boldsymbol {\alpha }}(f)\colon \textbf {Sch}_{\boldsymbol {\alpha }}(f)\longrightarrow \textbf {B}$. We then prove that there exists an endofunctor, $\varUpsilon ^{\mathscr {G},0}_{\boldsymbol {\alpha }}$, of $\textbf {Mor}_{\textrm {tw}}(\textbf {D})$, the twisted morphism category of $\textbf {D}$, thus showing the naturalness of the previous construction. Afterwards, we prove that, for every $\varSigma $-completion $\mathscr {G}=(\textbf {D},G,\rho )$, there exists a functor $\varUpsilon ^{\mathscr {G}}$ from the comma category $(\textbf {Cmpl}(\varSigma )\!\downarrow \!\mathscr {G})$ to $\textbf {End}(\textbf {Mor}_{\textrm {tw}}(\textbf {D}))$, the category of endofunctors of $\textbf {Mor}_{\textrm {tw}}(\textbf {D})$, such that $\varUpsilon ^{\mathscr {G},0}$, the object mapping of $\varUpsilon ^{\mathscr {G}}$, sends a morphism of $\varSigma $-completion of $\textbf {Cmpl}(\varSigma )$ with codomain $\mathscr {G}$, to the endofunctor $\varUpsilon ^{\mathscr {G},0}_{\boldsymbol {\alpha }}$.

Gorenstein projective objects in comma categories

Let \(\mathcal {A}\) and \(\mathcal {B}\) be abelian categories and \({\mathbf {F}} :\mathcal {A}\rightarrow \mathcal {B}\) an additive and right exact functor which is perfect, and let \(({\mathbf {F}},\mathcal {B})\) be the left comma category. We give an equivalent characterization of Gorenstein projective objects in \(({\mathbf {F}},\mathcal {B})\) in terms of Gorenstein projective objects in \(\mathcal {B}\) and \(\mathcal {A}\). We prove that there exists a left recollement of the stable category of the subcategory of \(({\mathbf {F}},\mathcal {B})\) consisting of Gorenstein projective objects modulo projectives relative to the same kind of stable categories in \(\mathcal {B}\) and \(\mathcal {A}\). Moreover, this left recollement can be filled into a recollement when \(\mathcal {B}\) is Gorenstein and \({\mathbf {F}}\) preserves projectives.

Special precovering classes in comma categories

Let T be a right exact functor from an abelian category ℬ into another abelian category \({\mathscr A}\). Then there exists a functor p from the product category \({\mathscr A} \times {\mathscr B}\) to the comma category (\(\left( {T \downarrow {\mathscr A}} \right)\)). In this paper, we study the property of the extension closure of some classes of objects in (\(\left( {T \downarrow {\mathscr A}} \right)\)), the exactness of the functor p and the detailed description of orthogonal classes of a given class \({\rm{P}}\left( {{\cal X},{\cal Y}} \right)\) in (\(\left( {T \downarrow {\mathscr A}} \right)\)). Moreover, we characterize when special precovering classes in abelian categories \({\mathscr A}\) and ℬ can induce special precovering classes in (\(\left( {T \downarrow {\mathscr A}} \right)\)). As an application, we prove that under suitable cases, the class of Gorenstein projective left Λ-modules over a triangular matrix ring \({\rm{\Lambda }} = \left( {\matrix{ R & M \cr 0 & S \cr } } \right)\) is special precovering if and only if both the classes of Gorenstein projective left R-modules and left S-modules are special precovering. Consequently, we produce a large variety of examples of rings such that the class of Gorenstein projective modules is special precovering over them.

Recollements, comma categories and morphic enhancements

For each recollement of triangulated categories, there is an epivalence between the middle category and the comma category associated with a triangle functor from the category on the right to the category on the left. For a morphic enhancement of a triangulated category $\mathcal {T}$, there are three explicit ideals of the enhancing category, whose corresponding factor categories are all equivalent to the module category over $\mathcal {T}$. Examples related to inflation categories and weighted projective lines are discussed.

Local Presentability of Certain Comma Categories

It follows from standard results that if \(\mathcal {A}\) and \(\mathcal {C}\) are locally \(\lambda \)-presentable categories and \(F : \mathcal {A}\rightarrow \mathcal {C}\) is a \(\lambda \)-accessible functor, then the comma category \(\mathsf {Id}_\mathcal {C}{\downarrow }{}F\) is locally \(\lambda \)-presentable. We show that, under the same hypotheses, \(F{\downarrow }{}\mathsf {Id}_\mathcal {C}\) is also locally \(\lambda \)-presentable.

A novel approach to sheaves on diffeological spaces

In this paper, we introduce sheaves on diffeological spaces as sheaves on the site of plots. We also define quasi-sheaves by presheaves that respect the limit over covering generating families. It is shown that every sheaf on diffeological spaces is a quasi-sheaf. Moreover, every sheaf on a diffeological space gives a sheaf on the D-topological structure. Finally, we characterize a class of sheaves, so-called comma sheaves, and prove that it is equivalent to the comma category of diffeological spaces.

Fraïssé Limits in Comma Categories

Fraisse’s theorem characterizing the existence of universal homogeneous structures is a cornerstone of model theory. A categorical version of these results was developed by Droste and Gobel. Such an abstract version of Fraisse theory allows to construct unusual objects that are far away from the usual structures. In this paper we are going to derive sufficient conditions for a comma category to contain universal homogeneous objects. Using this criterion, we characterize homogeneous structures that possess universal homogeneous endomorphisms. The existence of such endomorphisms helps to reduce questions about the full endomorphism monoid to the self-embedding monoid of the structure. As another application we characterize the retracts of homogeneous structures that are induced by universal homogeneous retractions. This extends previous results by Bonato, Delic, Mudrinski, Dolinka, and Kubiś.

Weak injectivity in the category of normal fuzzy hypergroups

Based on injectivity, we introduce some definitions in the category NFHG of normal fuzzy hypergroups. And we show that a complete normal fuzzy hypergroup is a weakly injective object in NFHG. Also we investigate weak injectivity in the comma category NFHG/K.

Monads on [formula omitted] and their lax extensions to [formula omitted

For a small quantaloid Q, we consider 2-monads on the 2-category Q-Cat and their lax extensions to the 2-category Q-Dist of small Q-categories and their distributors, in particular those lax extensions that are normal, also called flat, in the sense that they map identity distributors to identity distributors. In fact, unlike in the discrete case, a 2-monad on Q-Cat may admit only one normal lax extension. Every ordinary monad on the comma category Set/obQ with a lax extension to Q-Rel, the discrete counterpart of Q-Dist, gives rise to such a 2-monad on Q-Cat, and we describe this process globally as a coreflective embedding. The Q-presheaf and the double Q-presheaf monads are important examples of 2-monads on Q-Cat allowing normal lax extensions to Q-Dist, and so are their submonads, obtained by the restriction to conical (co)presheaves. These are known as the Q-Hausdorff and double Q-Hausdorff monads, which we define here in full generality, thus generalizing some previous work in the case when Q is a quantale, or just the Lawvere quantale [0,∞]. Their discretization leads naturally to various lax extensions of the relevant Set-monads used in monoidal topology.

Linear representations of formal loops

A representation of an object in a category is an abelian group in the corresponding comma category. In this paper, we derive the formulas describing linear representations of objects in the category of formal loops and formal loop homomorphisms and apply them to obtain a new approach to the representation theory of formal Moufang loops and Malcev algebras based on Moufang elements. Certain ‘non-associative Moufang symmetry’ of groups is revealed.

Stratified categorical fixed-basis fuzzy topological spaces and their duality

For an abstract category C, a class M of C-monomorphisms and a fixed C-object L, we introduce stratified C-M-L-spaces to be categorical counterparts of stratified fixed-basis fuzzy topological spaces in C, and consider their category SC-M-L-Top. As two main results of this paper, it is shown that SC-M-L-Top is dually adjoint to the comma category L↓C, and this adjunction can be restricted to a dual equivalence between the full category of L↓C with comma-spatial objects and the full category of SC-M-L-Top with comma-sober objects. The present paper also describes applications and relationships of these results to stratified fixed-basis fuzzy topological spaces. In this respect, a considerable part of this paper is devoted to stratified L-quasi-topological spaces and their duality.

Finitary -adhesive categories

Finitary$\mathcal{M}$-adhesive categories are$\mathcal{M}$-adhesive categories with finite objects only, where$\mathcal{M}$-adhesive categories are a slight generalisation of weak adhesive high-level replacement (HLR) categories. We say an object is finite if it has a finite number of$\mathcal{M}$-subobjects. In this paper, we show that in finitary$\mathcal{M}$-adhesive categories we not only have all the well-known HLR properties of weak adhesive HLR categories, which are already valid for$\mathcal{M}$-adhesive categories, but also all the additional HLR requirements needed to prove classical results including the Local Church-Rosser, Parallelism, Concurrency, Embedding, Extension and Local Confluence Theorems, where the last of these is based on critical pairs. More precisely, we are able to show that finitary$\mathcal{M}$-adhesive categories have a unique$\mathcal{E}$-$\mathcal{M}$factorisation and initial pushouts, and the existence of an$\mathcal{M}$-initial object implies we also have finite coproducts and a unique$\mathcal{E}$′-$\mathcal{M}$pair factorisation. Moreover, we can show that the finitary restriction of each$\mathcal{M}$-adhesive category is a finitary$\mathcal{M}$-adhesive category, and finitarity is preserved under functor and comma category constructions based on$\mathcal{M}$-adhesive categories. This means that all the classical results are also valid for corresponding finitary$\mathcal{M}$-adhesive transformation systems including several kinds of finitary graph and Petri net transformation systems. Finally, we discuss how some of the results can be extended to non-$\mathcal{M}$-adhesive categories.

Delta lenses and opfibrations

We compare the delta lenses, also known as d-lenses, of Diskin et al. with the c-lenses, known to be equivalent to opfibrations, already studied by the authors. Contrary to expectation a c-lens is a d-lens but not conversely. This result is surprising because d-lenses appear to provide the same information as c-lenses, and some more besides, suggesting that the implication would be the reverse -- a d-lens would appear to be a special kind of c-lens. The source of the surprise can be traced to the way the two concepts deal differently with morphisms in a certain base comma category $(G,1_\bV)$. Both c-lenses and d-lenses are important because they extend the notion of lens to take account of the information available in known transitions between states and this has important implications in practice.

Categorical Landstad duality for actions

Let G be a locally compact group. We show that the category A(G) of actions of G on C � -algebras (with equivari- ant nondegenerate ∗-homomorphisms into multiplier algebras) is equivalent, via a full-crossed-product functor, to a comma category of maximal coactions of G under the comultiplication (C � (G),�G); and also that A(G) is equivalent, via a reduced-crossed-product functor, to a comma category of normal coactions under the comul- tiplication. This extends classical Landstad duality to a category equivalence, and allows us to identify those C � -algebras which are isomorphic to crossed products by G as precisely those which form part of an object in the appropriate comma category.

A categorical manifesto

This paper tries to explain why and how category theory is useful in computing science, by giving guidelines for applying seven basic categorical concepts: category, functor, natural transformation, limit, adjoint, colimit and comma category. Some examples, intuition, and references are given for each concept, but completeness is not attempted. Some additional categorical concepts and some suggestions for further research are also mentioned. The paper concludes with some philosophical discussion.

On the Maunder Type Theorems in the Ex-homotopy Category

We study the homotopy theory of comma category and define the cell structure and Postnikov system in the ex-homotopy category. By using these structures we give the four types of spectral sequences and show that Maunder type theorems hold for these spectral sequences.

Closure operators I

Closure operators in an ( E , M )-category X are introduced as concrete endofunctors of the comma category whose objects are the elements of M . Various kinds of closure operators are studied. There is a Galois equivalence between the conglomerate of idempotent and weakly hereditary closure operators of X and the conglomerate of subclasses of M which are part of a factorization system. There is a one-to-one correspondence between the class of regular closure operators and the class of strongly epireflective subcategories of X . Every closure operators admits an idempotent hull and a weakly hereditary core. Various examples of additive closure operators in Top are given. For abelian categories standard closure operators are considered. It is shown that there is a one-to-one correspondence between the class of standard closure operators and the class of preradicals. Idempotent, weakly hereditary, standard closure operators correspond to idempotent radicals (= torsion theories).

Localization and Codensity Triples

We consider a complete category α and a functor H: J→α such that, for each object A of α, the comma category (A, H) has a small initial subcategory. We form the codensity triple (S, η, μ) on α, where S(A) = lim [H(I) u: A→H(I), I e J], and let x: Q→S be the equalizer of ηS and Sη.