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Abstract: Abstract Let \({\mathbb {A}}\) be a 2-category with suitable opcomma objects and pushouts. We give a direct proof that, provided that the codensity monad of a morphism p exists and is preserved by a suitable morphism, the factorization given by the lax descent object of the two-dimensional cokernel diagram of p is up to isomorphism the same as the semantic factorization of p, either one existing if the other does. The result can be seen as a counterpart account to the celebrated Bénabou–Roubaud theorem. This leads in particular to a monadicity theorem, since it characterizes monadicity via descent. It should be noted that all the conditions on the codensity monad of p trivially hold whenever p has a left adjoint and, hence, in this case, we find monadicity to be a two-dimensional exact condition on p, namely, to be an effective faithful morphism of the 2-category \({\mathbb {A}}\) . PubDate: 2022-11-15

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Abstract: Abstract We study t-structures generated by sets of objects which satisfy a condition weaker than the compactness. We also study weight structures cogenerated by sets of objects satisfying the dual condition. Under some appropriate hypothesis, it turns out that the weight structure is right adjacent to the t-structure. PubDate: 2022-10-01 DOI: 10.1007/s10485-022-09676-y

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Abstract: Abstract In this paper, we introduce the notion of Grothendieck enriched categories for categories enriched over a sufficiently nice Grothendieck monoidal category \(\mathcal {V}\) , generalizing the classical notion of Grothendieck categories. Then we establish the Gabriel-Popescu type theorem for Grothendieck enriched categories. We also prove that the property of being Grothendieck enriched categories is preserved under the change of the base monoidal categories by a monoidal right adjoint functor. In particular, if we take as \(\mathcal {V}\) the monoidal category of complexes of abelian groups, we obtain the notion of Grothendieck dg categories. As an application of the main results, we see that the dg category of complexes of quasi-coherent sheaves on a quasi-compact and quasi-separated scheme is an example of Grothendieck dg categories. PubDate: 2022-10-01 DOI: 10.1007/s10485-022-09681-1

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Abstract: Abstract \(\mathfrak {KNJ}\) is the category of compact normal joinfit frames and frame homomorphisms and \(\mathfrak {KReg}\) is the coreflective subcategory of compact regular frames. This work investigates \(\mathfrak {KNJ}\) through its interaction with \(\mathfrak {KReg}\) via the coreflection \(\rho \) . A \(\mathfrak {KNJ}\) morphism \(\phi : F \longrightarrow M\) is \(\mathcal {P}\) -essential if \(\phi \) is skeletal and the map between the frames of polars, \(\mathcal {P}(\phi ): \mathcal {P}F \longrightarrow \mathcal {P}M\) defined by \(\mathcal {P}(\phi )(p)=\phi (p)^{\perp \perp }\) , is a boolean isomorphism. The \(\mathcal {P}\) -essential morphisms in \(\mathfrak {KNJ}\) are closely related to the essential embeddings in \(\mathfrak {KReg}\) . We provide a characterization of the \(\mathcal {P}\) -essential morphisms in \(\mathfrak {KNJ}\) and a connection to the essential embeddings in \(\mathfrak {KReg}\) . Further results about the preservation of joinfitness, the factorization of morphisms, and monomorphisms in \(\mathfrak {KNJ}\) are provided. Moreover, in the category of \(\mathfrak {KNJ}\) objects and skeletal frame homomorphisms, \(\mathfrak {KNJS}\) , we construct for \(F \in \mathfrak {KNJ}\) and \(\phi :\rho F \longrightarrow H\) (an arbitrary \(\mathfrak {KReg}\) essential embedding of \(\rho F\) ) the \(\mathfrak {KNJS}\) pushout of \(\rho _F: \rho F \longrightarrow F\) and \(\phi : \rho F \longrightarrow H\) . Lastly, we investigate the epimorphisms and epicomplete objects in \(\mathfrak {KNJS}\) . PubDate: 2022-10-01 DOI: 10.1007/s10485-022-09679-9

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Abstract: Abstract We prove that for each locally \(\alpha \) -presentable category \({\mathcal {K}}\) there exists a regular cardinal \(\gamma \) such that any \(\alpha \) -accessible functor out of \({\mathcal {K}}\) (into another locally \(\alpha \) -presentable category) is continuous if and only if it preserves \(\gamma \) -small limits; as a consequence we obtain a new adjoint functor theorem specific to the \(\alpha \) -accessible functors out of \({\mathcal {K}}\) . Afterwards we generalize these results to the enriched setting and deduce, among other things, that a small \({\mathcal {V}}\) -category is accessible if and only if it is Cauchy complete. PubDate: 2022-10-01 DOI: 10.1007/s10485-022-09677-x

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Abstract: Abstract Let \(\Lambda \) be an artin algebra of finite global dimension. We study when the composition of three irreducible morphisms between indecomposable complexes in \({{\mathbf {K}}^{b}(\mathrm {proj}\,\Lambda )}\) is a non-zero morphism in the fourth power of the radical. We apply such results to prove that the composition of three irreducible morphisms between indecomposable complexes in the bounded derived category of a gentle Nakayama algebra, not selfinjective, whose ordinary quiver is an oriented cycle, belongs to the fourth power of the radical if and only if it vanishes. PubDate: 2022-10-01 DOI: 10.1007/s10485-022-09682-0

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Abstract: Abstract We construct and study projective and Reedy model category structures for bimodules and infinitesimal bimodules over topological operads. Both model structures produce the same homotopy categories. For the model categories in question, we build explicit cofibrant and fibrant replacements. We show that these categories are right proper and under some conditions left proper. We also study the extension/restriction adjunctions. PubDate: 2022-10-01 DOI: 10.1007/s10485-022-09675-z

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Abstract: Abstract Let \(\mathcal {T}\) be a k-linear Hom-finite \((n+2)\) -angulated category with n-suspension functor \(\Sigma ^n\) , split idempotents, and Serre functor \(\mathbb {S}\) . Let T be an Oppermann–Thomas cluster tilting object in \(\mathcal {T}\) with endomorphism algebra \(\Gamma = \mathrm {End}_\mathcal {T}(T)\) . We introduce the notions of relative Oppermann–Thomas cluster tilting objects and support \(\tau _n\) -tilting pairs, and show that there is an bijection between the set of isomorphism classes of basic relative Oppermann–Thomas cluster tilting objects in \(\mathcal {T}\) and the set of isomorphism classes of basic support \(\tau _n\) -tilting pairs in an n-cluster tilting subcategory of \(\mathrm {mod}~\Gamma \) . As applications, we recover the Yang–Zhu bijection (Trans Am Math Soc 371:387–412, 2019) and Adachi–Iyama–Reiten bijection (Compos Math 150:415–452, 2014), and we give a natural partial order for relative Oppermann–Thomas cluster tilting objects. PubDate: 2022-10-01 DOI: 10.1007/s10485-022-09673-1

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Abstract: Abstract 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. Then, we contextualize the new notion by comparing it to another known generalization of enrichment: that of enrichment for indexed categories. It turns out that the two notions are closely related. PubDate: 2022-10-01 DOI: 10.1007/s10485-022-09678-w

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Abstract: Abstract Let \({p : \mathcal {E}\rightarrow \mathcal S}\) be a hyperconnected geometric morphism. For each X in the ‘gros’ topos \(\mathcal {E}\) , there is a hyperconnected geometric morphism \({p_X : \mathcal {E}/X \rightarrow \mathcal S(X)}\) from the slice over X to the ‘petit’ topos of maps (over X) with discrete fibers. We show that if p is essential then \(p_X\) is essential for every X. The proof involves the idea of collapsing a connected subspace to a ‘basepoint’, as in Algebraic Topology, but formulated in topos-theoretic terms. In case p is local, we characterize when \({p_X}\) is local for every X. This is a very restrictive property, typical of toposes of spaces of dimension \({\le 1}\) . PubDate: 2022-10-01 DOI: 10.1007/s10485-022-09680-2

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Abstract: Abstract In this paper, we provide a notion of \(\infty \) -bicategories fibred in \(\infty \) -bicategories which we call 2-Cartesian fibrations. Our definition is formulated using the language of marked biscaled simplicial sets: Those are scaled simplicial sets equipped with an additional collection of triangles containing the scaled 2-simplices, which we call lean triangles, in addition to a collection of edges containing all degenerate 1-simplices. We prove the existence of a left proper combinatorial simplicial model category whose fibrant objects are precisely the 2-Cartesian fibrations over a chosen scaled simplicial set S. Over the terminal scaled simplicial set, this provides a new model structure modeling \(\infty \) -bicategories, which we show is Quillen equivalent to Lurie’s scaled simplicial set model. We conclude by providing a characterization of 2-Cartesian fibrations over an \(\infty \) -bicategory. This characterization then allows us to identify those 2-Cartesian fibrations arising as the coherent nerve of a fibration of \({\text {Set}}^+_{\Delta }\) -enriched categories, thus showing that our definition recovers the preexisting notions of fibred 2-categories. PubDate: 2022-09-28

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Abstract: Abstract In a recent paper (Hoefnagel et al. in Theory Appl Categ 38:737–790, 2022), an algorithm has been presented for determining implications between a particular kind of category theoretic property represented by matrices—the so called ‘matrix properties’. In this paper we extend this algorithm to include matrix properties involving pointedness of a category, such as the properties of a category to be unital, strongly unital or subtractive, for example. Moreover, this extended algorithm can also be used to determine whether a given matrix property is the Bourn localization of another, thus leading to new characterizations of Mal’tsev, majority and arithmetical categories. Using a computer implementation of our algorithm, we can display all such properties given by matrices of fixed dimensions, grouped according to their Bourn localizations, as well as the implications between them. PubDate: 2022-09-21

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Abstract: Abstract In ordinary category theory, limits are known to be equivalent to terminal objects in the slice category of cones. In this paper, we prove that the 2-categorical analogues of this theorem relating 2-limits and 2-terminal objects in the various choices of slice 2-categories of 2-cones are false. Furthermore we show that, even when weakening the 2-cones to pseudo- or lax-natural transformations, or considering bi-type limits and bi-terminal objects, there is still no such correspondence. PubDate: 2022-09-08 DOI: 10.1007/s10485-022-09691-z

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Abstract: Abstract In this paper we provide a Stone style duality for monotone semilattices by using the topological duality developed in S. Celani, L.J. González (Appl Categ Struct 28:853–875, 2020) for semilattices together with a topological description of their canonical extension. As an application of this duality we obtain a characterization of the congruences of monotone semilattices by means of monotone lower-Vietoris-type topologies. PubDate: 2022-08-29 DOI: 10.1007/s10485-022-09690-0

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Abstract: This is the first in a series of papers in which we study representations of the Brauer category and its allies. We define a general notion of triangular category that abstracts key properties of the triangular decomposition of a semisimple complex Lie algebra, and develop a highest weight theory for them. We show that the Brauer category, the partition category, and a number of related diagram categories admit this structure. PubDate: 2022-08-13 DOI: 10.1007/s10485-022-09689-7

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Abstract: Abstract We consider the ordinary category \(\mathsf {Span}({\mathcal {C}})\) of (isomorphism classes of) spans of morphisms in a category \(\mathcal {C}\) with finite limits as needed, composed horizontally via pullback, and give a general criterion for a quotient of \(\mathsf {Span}({\mathcal {C}})\) to be an allegory. In particular, when \({\mathcal {C}}\) carries a pullback-stable, but not necessarily proper, \(({\mathcal {E}},{\mathcal {M}})\) -factorization system, we establish a quotient category \(\mathsf {Span}_{{\mathcal {E}}}({\mathcal {C}})\) that is isomorphic to the category \(\mathsf {Rel}_{{\mathcal {M}}}({\mathcal {C}})\) of \({\mathcal {M}}\) -relations in \({\mathcal {C}}\) , and show that it is a (unitary and tabular) allegory precisely when \({\mathcal {M}}\) is a class of monomorphisms in \({\mathcal {C}}\) . Without the restriction to monomorphisms, one can still find a least pullback-stable and composition-closed class \({\mathcal {E}}_{\bullet }\) containing \(\mathcal E\) such that \(\mathsf {Span}_{{\mathcal {E}}_{\bullet }}({\mathcal {C}})\) is a unitary and tabular allegory. In this way one obtains a left adjoint to the 2-functor that assigns to every unitary tabular allegory the regular category of its Lawverian maps. With the Freyd-Scedrov Representation Theorem for regular categories, we conclude that every finitely complete category with a stable factorization system has a reflection into the 2-category of all regular categories. PubDate: 2022-08-01 DOI: 10.1007/s10485-022-09687-9

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Abstract: Abstract We provide an explicit characterization of the covariant isotropy group of any Grothendieck topos, i.e. the group of (extended) inner automorphisms of any sheaf over a small site. In order to do so, we first extend previous techniques for computing covariant isotropy from locally finitely presentable categories to locally presentable categories. As a consequence, we also obtain an explicit characterization of the centre of a Grothendieck topos, i.e. the automorphism group of its identity functor. We conclude by providing a more categorical approach to show that these characterizations also extend to any extensive category. PubDate: 2022-08-01 DOI: 10.1007/s10485-022-09674-0

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Abstract: Abstract The present article aims to develop a categorical duality for the category of finite distributive join-semilattices and \(\wedge \) -homomorphisms (maps that preserve the joins and the meets, when they exist). This dual equivalence is a generalization of the famous categorical duality given by Birkhoff for finite distributive lattices. Moreover, we show that every finite distributive semilattice is a Hilbert algebra with supremum. We obtain some applications from the dual equivalence. We provide a dual description of the 1–1 and onto \(\wedge \) -homomorphisms, and we obtain a dual characterization of some subalgebras. Finally, we present a representation for the class of finite semi-boolean algebras. PubDate: 2022-08-01 DOI: 10.1007/s10485-021-09669-3

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Abstract: Abstract We describe how to obtain a global t-structure from a semiorthogonal decomposition with compatible t-structures on every component. This result is used to generalize a well-known theorem of Bondal on full strong exceptional sequences. PubDate: 2022-08-01 DOI: 10.1007/s10485-022-09672-2

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Abstract: Abstract One way of interpreting a left Kan extension is as taking a kind of “partial colimit”, whereby one replaces parts of a diagram by their colimits. We make this intuition precise by means of the partial evaluations sitting in the so-called bar construction of monads. The (pseudo)monads of interest for forming colimits are the monad of diagrams and the monad of small presheaves, both on the (huge) category CAT of locally small categories. Throughout, particular care is taken to handle size issues, which are notoriously delicate in the context of free cocompletion. We spell out, with all 2-dimensional details, the structure maps of these pseudomonads. Then, based on a detailed general proof of how the restriction-of-scalars construction of monads extends to the case of pseudoalgebras over pseudomonads, we consider a morphism of monads between them, which we call image. This morphism allows in particular to generalize the idea of confinal functors, i.e. of functors which leave colimits invariant in an absolute way. This generalization includes the concept of absolute colimit as a special case. The main result of this paper spells out how a pointwise left Kan extension of a diagram corresponds precisely to a partial evaluation of its colimit. This categorical result is analogous to what happens in the case of probability monads, where a conditional expectation of a random variable corresponds to a partial evaluation of its center of mass. PubDate: 2022-08-01 DOI: 10.1007/s10485-021-09671-9