Trivial Objects Research Articles

A categorical approach to largest and smallest connectednesses and disconnectednesses

Motivated by the existence of largest and smallest proper connectednesses and disconnectednesses in the categories of topological spaces (Top) and undirected graphs with loops (Graph), we look for necessary and sufficient conditions for the existence of such classes in a general categorical setting. This leads us to consider certain classes of simple objects. We start by giving the necessary theory of radical classes (connectednesses) and semisimple classes (disconnectednesses). In the next section we define certain subclasses of the class of all simple objects. These classes are then used in the third section to obtain largest and smallest radical and semisimple classes. Lastly we generalize the results of Wiegandt [8] concerning the hereditariness and cohereditariness of connectednesses and disconnectednesses. K will always be an arbitrary category and E and M will denote subclasses of the class of K-epimorphisms and K-monomorphisms respectively. Throughout this paper we shall assume that both E and M are closed under composition with isomorphisms and that the class of all K-identities is contained in EflM. Note that these two assumptions imply that the class of all K-isomorphisms is contained in E flM. A~BCE will mean there is a morphism ~: A-~B in Kwith c~EE. A ~ B E M will have a similar meaning. Let T be the class of all trivial objects in K (cf. [2]), i.e. T is that class of objects which satisfies the following conditions:

ON THE EXISTENCE AND NON-EXISTENCE OF COMPLEMENTARY RADICAL AND SEMISIMPLE CLASSES

ABSTRACT In certain categories of mathematical structures, non-trivial complementary radical classes (torsion classes or connectednesses) can be found. The question is why this is true for some but not for all categories. The answer depends on the embedding of trivial objects into nontrivial objects and is given by our main result: Any ‘reasonable’ category has no non-trivial complementary radical and semisimple classes if and only if for every trivial object T and every non-trivial object A there is a morphism T → A. Roughly, a ‘reasonable’ category in our sense is one with at least one object into which a terminal object can be embedded and has finite products, coproducts or lexicographic products.

On Brown–McCoy radical classes in categories

What does a simple ring with unity, a topological T0-space and a graph that has at most one loop but may possess edges, have in common? In this note we show that they all are Brown–McCoy semisimple. Suliński has generalised the well-known Brown–McCoy radical class of associative rings (cf. [1]) to a category which satisfies certain conditions. In [3] he defines a simple object, a modular class of objects and the Brown–McCoy radical class as the upper radical class determined by a modular class in a category which, among others, has a zero object and kernels. To include categories like that of topological spaces and graphs, we use the concepts of a trivial object and a fibre. We then follow Suliński and define a simple object, a modular class of objects and then the Brown–McCoy radical class as the upper radical class determined by a modular class.

The Geometry of the Modern City: G. W. M. Reynolds and The Mysteries of London

In Book vii of The Prelude Wordsworth chronicles the months he lived in London and concludes with two images for the modern city. Following his feeling that “The face of every one / That passes by me is a mystery,” he sees a blind beggar with his life's history on a card around his neck; this is followed by the spectacle of Bartholomew's Fair. The poet's response to these experiences of urban mystery and multitudinousness is well-known:O Blank confusion! true epitomeOf what the mighty City is herselfTo thousands upon thousands of her sonsLiving amid the same perpetual whirlOf trivial objects, melted and reducedTo one identity, by differencesThat have no law, no meaning, and no end -(Book VII, 11. 722–28)

Span and stably trivial bundles

E. Thomas introduced the notion of span of a differentiable manifold (or of a vector bundle). The notion of span can be extended in an obvious way to $PL$-microbundles, topological microbundles and spherical fibrations. In the case of a vector bundle or a microbundle the dimension of the fibre will be referred to as its rank. A spherical fibration with fibre homotopically equivalent to $S^{k-1}$ will be said to be of rank $k$. In this paper we study stably trivial objects of rank $k$ over a $CW$-complex of dimension $\leqq k$ from each of the above collections. Then we determine the span of such stably trivial objects over $CW$-complexes of a ``special type'' yielding generalizations of the Bredon-Kosinski, Thomas theorem on the span of a closed differentiable $\pi$-manifold [3], [19]. Though originally $PL$-microbundles were defined only over simplicial complexes, in this paper by a $PL$-microbundle of rank $k$ over a $CW$-complex $X$ we mean an element of the set $[X,BPL(k)]$ of homotopy classes of maps of $X$ into $BPL(k)$.

Models for mock homotopy categories of projectives

Let $R$ be a ring and Ch($R$) the category of chain complexes of $R$-modules. We put an abelian model structure on Ch($R$) whose homotopy category is equivalent to $K(Proj)$, the homotopy category of all complexes of projectives. However, the cofibrant objects are not complexes of projectives, but rather all complexes of flat modules. The trivial objects are what Positselski calls contraacyclic complexes and so the homotopy category coincides with his contraderived category. We in fact construct this model on the category of chain complexes of quasi-coherent sheaves on any scheme $X$ admitting a flat generator. In this case the homotopy category recovers what Murfet calls the mock homotopy category of projectives. In the same way we construct a model for the (mock) projective stable derived category, and we use model category methods to recover the recollement of Murfet. Finally, we consider generalizations by replacing the flat cotorsion pair with other complete hereditary cotorsion pairs in Grothendieck categories.