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## Categorical Aspects of Compact Quantum Groups

### Applied Categorical Structures (2015-06-01) 23: 381-413 , June 01, 2015

We show that either of the two reasonable choices for the category of compact quantum groups is nice enough to allow for a plethora of universal constructions, all obtained “by abstract nonsense” via the adjoint functor theorem. This approach both produces new objects (such as the coproduct of a family of compact quantum groups or the compact quantum group freely generated by a locally compact quantum space) and recovers in a uniform setting constructions which have appeared in the literature, such as the quantum Bohr compactification of a locally compact semigroup. We also provide Tannakian descriptions of these universal constructions, and characterize epimorphisms and monomorphisms in the category of compact quantum groups.

## 2-Filteredness and The Point of Every Galois Topos

### Applied Categorical Structures (2010-04-01) 18: 115-121 , April 01, 2010

A connected locally connected topos is a *Galois topos* if the Galois objects generate the topos. We show that the full subcategory of Galois objects in any connected locally connected topos is an inversely 2-filtered 2-category, and as an application of the construction of 2-filtered bi-limits of topoi, we show that every Galois topos has a point.

## The algebra of directed complexes

### Applied Categorical Structures (1993-09-01) 1: 247-284 , September 01, 1993

The theory of directed complexes is a higher-dimensional generalisation of the theory of directed graphs. In a directed graph, the simple directed paths form a subset of the free category which they generate; if the graph has no directed cycles, then the simple directed paths constitute the entire category. Generalising this, in a directed complex there is a class of split subsets which is contained in and generates a free ω-category; when a simple loop-freeness condition is satisfied, the split sets constitute the entire ω-category. The class of directed complexes is closed under the natural product and join constructions. The free ω-categories generated by directed complexes include the important examples associated to cubes and simplexes.

## Positively Convex Spaces

### Applied Categorical Structures (1998-09-01) 6: 333-344 , September 01, 1998

We give a construction of the left adjoint of the comparison functor $$\widehat\Delta :Ban_{\text{1}}^{\text{ + }} \to PC {\text{(resp}}{\text{. }}\widehat\Delta _{{\text{fin}}} {\text{: }}Vec_{\text{1}}^{\text{ + }} \to PC_{{\text{fin}}} {\text{)}}$$ in one step and we give a characterization of separated (finitely) positively convex spaces.

## Algebras of Higher Operads as Enriched Categories

### Applied Categorical Structures (2011-02-01) 19: 93-135 , February 01, 2011

One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we begin to adapt the machinery of globular operads (Batanin, Adv Math 136:39–103, 1998) to this task. We present a general construction of a tensor product on the category of *n*-globular sets from any normalised (*n* + 1)-operad *A*, in such a way that the algebras for *A* may be recaptured as enriched categories for the induced tensor product. This is an important step in reconciling the globular and simplicial approaches to higher category theory, because in the simplicial approaches one proceeds inductively following the idea that a weak (*n* + 1)-category is something like a category enriched in weak *n*-categories. In this paper we reveal how such an intuition may be formulated in terms of globular operads.

## An Axiomatic Approach for Degenerations in Triangulated Categories

### Applied Categorical Structures (2016-08-01) 24: 385-405 , August 01, 2016

We generalise Yoshino’s definition of a degeneration of two Cohen Macaulay modules to a definition of degeneration between two objects in a triangulated category. We derive some natural properties for the triangulated category and the degeneration under which the Yoshino-style degeneration is equivalent to the degeneration defined by a specific distinguished triangle analogous to Zwara’s characterisation of degeneration in module varieties.

## Universalities

### Applied Categorical Structures (1994-06-01) 2: 173-185 , June 01, 1994

We show that a quotient category of the category of all topological spaces and all open continuous mappings contains an isomorphic copy of every category as a full subcategory. We construct a functor*F : K → K* universal in the following sense: for every functor*H : H*_{1}*→ H*_{2} (*H*_{1},*H*_{2} arbitrary) there exist full one-to-one functors φ_{i} :*H*_{i}*→ K* such that*F* o φ_{1} = φ_{2} o*H* (the construction proceeds in a more general setting of enriched categories).

## Closure Operators with Respect to a Functor

### Applied Categorical Structures (2001-09-01) 9: 525-537 , September 01, 2001

A notion of closure operator with respect to a functor *U* is introduced. This allows us to describe a number of mathematical constructions that could not be described by means of the already existing notion of closure operator. Some basic results and examples are provided.

## Generalized Tilting Theory

### Applied Categorical Structures (2017-05-15): 1-60 , May 15, 2017

Given small dg categories A and B and a B-A-bimodule T, we give necessary and sufficient conditions for the associated derived functors of Hom and the tensor product to be fully faithful. Special emphasis is put on the case when RHom $$_\mathrm{A}$$ (T,?) is fully faithful and preserves compact objects, in which case nice recollements situations appear. It is also shown that, given an algebraic compactly generated triangulated category D, all compactly generated co-smashing triangulated subcategories which contain the compact objects appear as the image of such a RHom $$_\mathrm{A}$$ (T,?). The results are then applied to the case when A and B are ordinary algebras, comparing the situation with the well-stablished tilting theory of modules. In this way we recover and extend recent results by Bazzoni–Mantese–Tonolo, Chen-Xi and D. Yang.

## How many Adjunctions give Rise to the same Monad?

### Applied Categorical Structures (2016-12-03): 1-17 , December 03, 2016

Given an adjoint pair of functors *F*, *G*, the composite *GF* naturally gets the structure of a monad. The same monad may arise from many such adjoint pairs of functors, however. Can one describe all of the adjunctions giving rise to a given monad? In this paper we single out a class of adjunctions with especially good properties, and we develop methods for computing all such adjunctions, up to natural equivalence, which give rise to a given monad. To demonstrate these methods, we explicitly compute the finitary homological presentations of the free *A*-module monad on the category of sets, for *A* a Dedekind domain. We also prove a criterion, reminiscent of Beck’s monadicity theorem, for when there is essentially (in a precise sense) only a single adjunction that gives rise to a given monad.