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George Peschke

Publications and source records attributed to George Peschke.

5 recordsLinked to original sources
arXiv2024

A Homological View of Categorical Algebra

We provide a foundation for working with homological and homotopical methods in categorical algebra. This involves two mutually complementary components, namely (a) the strategic selection of suitable axiomatic frameworks, some well known and some new, and (b) the development of categorical tools for effective reasoning and computing within those frameworks. The selection of axiomatic frameworks begins `from the ground up' with z-exact categories. These are pointed categories in which every morphism has a kernel and cokernel. Then we progress all the way to abelian categories, en route meeting contexts such as Borceux--Bourn homological categories and Janelidze--Márki--Tholen semiabelian categories. We clarify the relationship between these axiomatic frameworks by direct comparison, but also by explaining how concrete examples fit into the selection. The outcome is a fine-grained set of criteria by which one can map varieties of algebras (in the sense of Universal Algebra) and topological models of algebraic theories into the various frameworks. The categorical tools for effective computation deal mostly with situations involving universal factorizations of morphisms, with exact sequences, and with the homology of chain complexes. Further, the categorical tools for computation include the `basic diagram lemmas' of homological algebra, that is the (Short) $5$-Lemma, the $(3\times 3)$-Lemma, the Snake Lemma, and applications thereof. We find that these tools are even available in some surprisingly weak categorical environments, such as the category of pointed sets. In discovering such features, we make systematic use of what we call the self-dual axis of a category.

math.CT↗
arXiv2022

Exact couples and their spectral sequences

Given a bigraded exact couple of modules over some ring, we determine the meaning of the $E^{\infty}$-terms of its associated spectral sequence: Let $L^{\ast}$ and $L_{\ast}$ denote the limit and colimit abutting objects of the exact couple, filtered by the kernel and image objects to the associated cone and cocone diagrams. Then the unstable E-infinite extension theorem states how adjacent filtration quotients of the colimit filtration are extended by $E^{\infty}$ objects over corresponding adjacent filtration quotients of the kernel filtration. The stable E-infinity extension theorem is based on the fact that the derivation process of the exact couple admits a transfinite recursion which is beyond the scope of the traditional spectral sequence perspective. The transfinite recursion always stabilizes at some ordinal. The resulting stable $E$-objects are (a) always subobjects of $E^{\infty}$ and (b) extend adjacent filtration quotients of the colimit filtration over corresponding adjacent filtration quotients of the kernel filtration {\em without} the need for lim-1 corrective terms. The E-infinity extension theorems enable conclusions about the filtered limit/colimit abutments even in cases where the spectral sequence is far from converging in any traditional sense. We develop such results in the context of 'comparing' the spectral sequences via the morphism that is induced by a morphism of underlying exact couples. We also contribute to 'reverse comparison' in a spectral sequence; that is using information about the universal abutment(s) of the underlying exact couple to extract information about one or more pages of the spectral sequence. These results overlap with Zeeman's comparison theorems in a generalizing fashion.

math.KT↗
arXiv2021

Diagrams, Fibrations, and the Decomposition of Colimits

The contributions of this paper are twofold. Within the framework of Grothendieck's fibrational category theory, we present a web of fundamental 2-adjunctions surrounding the formation of the category of all small diagrams in a given category and the formation of the Grothendieck category of a functor into the category of small categories. We demonstrate the utility of these adjunctions, in part by deriving three formulae for (co-)limits: a `twisted' generalization of the well-known Fubini formula, as first established by Chachólski and Scherer; a new `general colimit decomposition formula'; and a special case of the general formula, which actually initiated this work, and which was proved independently by Batanin and Berger. We give three proofs for this colimit decomposition formula, using methods that provide quite distinct insights. The `base' of our web of 2-adjunctions extends earlier work of the Ehresmann school and Guitart and promises to be of independent interest. It involves forming the diagram category of an arbitrary functor, seen as an object of the arrow category of the category of locally small categories, rather than that of a mere category. The left adjoint of the emerging generalized Guitart 2-adjunction factors through the 2-equivalence of split Grothendieck (co-)fibrations and strictly (co-)indexed categories, which we present here most generally by allowing 2-dimensional variation in the base categories.

math.CT↗
arXiv2015

The Yoneda isomorphism commutes with homology

We show that, for a right exact functor from an abelian category to abelian groups, Yoneda's isomorphism commutes with homology and, hence, with functor derivation. Then we extend this result to semiabelian domains. An interpretation in terms of satellites and higher central extensions follows. As an application, we develop semiabelian (higher) torsion theories and the associated theory of (higher) universal (central) extensions.

math.KT↗