New entries torsion theory and quotient category.

]]>An entry which defines both the local category and the local Grothendieck category, two notions which generalize the notion of a category of modules over a commutative local ring.

]]>I looked at a number of entries like zero morphism, direct sum, biproduct, kernel etc. I am writing some lecture on additive categories.

Here is some confusion t seems. $Ab$-categories are categories enriched over the category of Abelian groups with monoidal product being the tensor product. These entries talk about how to define zero morphism, kernel (as representing object) etc. for such as a special case of enrichement over pointed sets ! But it does not look right to me. Pointed sets are assumed in zero morphism to be monoidal with respect to the smash product of sets. But the tensor product of Abelian groups after forgetting the Abelian structure to pointed sets, unless I am halucinating, disagrees with the smash product of the underlying pointed sets! So the general treatment which would be good enough for enrichement over pointed sets, Abelian groups and few others should be done more carefully, it seems. Any thoughts ?

It is also unclear how standard is our definition at additive and abelian categories that preadditive categores are the $Ab$-categories with zero object. MacLane in Homology (1963) lists the axioms for additive categories and then remarks that if one does not assume direct sums (but he defines just binary direct sums, so it does not include the existence of zero object) that this is called preadditive. In this reading it means we agree with MacLane. Weibel does not use the term preadditive in his book. Popescu (and wikipedia which follows it) as well as Borceux call preadditive the same as $Ab$-enriched.

]]>I have started a stub localization of an abelian category. Added a list of related terms at topologizing subcategory.

]]>New entry defining ideal of topologizing subcategory (of an abelian category), wanted at conormal bundle. It is in fact a subfunctor of the identity functor and if we evaluate it on projective generator in the case of a module category then we get the usual ideal in the corresponding ring.

]]>I started an important entry differential monad. According to Lunts-Rosenberg MPI 1996-53 pdf differential calculus on schemes and noncommutative schemes can be derived from the yoga of coreflective topologizing subcategories in the abelian category of quasicoherent sheaves on the scheme, like the $\mathbb{T}$-filtration, and $\mathbb{T}$-part, in the case when the topologizing subcategory is the diagonal in the sense of the smallest subcategory of the category of additive endofunctors having right adjoint which contains the identity functor – in that case we say differential filtration and differential part. The regular differential operators are the elements of the differential part of the bimodule of endomorphisms. Similarly, one can define the conormal bundle etc.

]]>This was the query in topologizing subcategory which I summarized shortly:

]]>Mike: Where does the word ’topologizing’ come from?

Zoran Skoda: I am not completely sure anymore, but I think it is from ring theory, where people looked at the localizations at topologizing categories. There exist some topologies on various sets of ideals like Jacobson topology, so it is something of that sort in the language of subcategories instead of the language of filters of ideals. I’ll consult old references like Popescu, maybe I recall better. In any case it is pretty standard and has long history in usage: both classical and modern. No, it is not in Popescu…old related term is in fact talking about topologizing filters of ideals in a ring, so that must be the source…for example, the classical algebra by Faith, vol I, page 520 defines when the set of right ideals is topologizing. I am not good with that notion, but I can make an entry with quotation to be improved later.