arXiv subjects
Publications and source records attributed to Takuo Matsuoka.
The purpose of these notes is to collect in one place some facts on the category of finite totally ordered sets and some related categories. More specifically, we collect some results on them which will be useful for the study of iteratedly meta theories of algebra in the style of our work arXiv:1601.00301 (arXiv:1601.00301), which is a kind of higher order universal algebra.
In various subjects including mathematics, one can hope to use mathematical thinking well when the right kinds of algebraic structure to consider can be discovered or spotted. Therefore, it would help to understand kinds of algebraic structure in some great generality. In tradition, certain general kinds of algebraic structure are studied through the theory of operads, of algebraic theories, of properads (possibly with "colours" or "sorts") or of the like. We understand this as use of algebra for studying a certain meta aspect of the subject of algebra, namely, studying kinds of algebraic structure in general (rather than structures of specific kinds themselves). In higher categorical contexts, more various algebraic structures can be considered (starting in fact with operads considered with arbitrarily varying colours) than can be covered with mere higher categorified versions of the traditional tools. In this survey, we develop a systematic view on quite general algebraic structures with high categorical dimensionality. We do this by extending algebra for the meta aspect of algebra, to a higher order, or highly meta, theory of algebra, where the theoretic order turns out to match the categorical dimension. We will discuss some examples including a natural generalization of the notion of topological field theory (TFT) obtained in the resulting new system. It turns out a generalized TFT can have a very different characterization from a conventional TFT does.
We study the Koszul duality between augmented $E_n$-algebras and augmented $E_n$-coalgebras in a symmetric monoidal stable infinity $1$-category equipped with a filtration in a suitable sense. We obtain that the Koszul duality constructions restrict to an equivalence between augmented algebras and coalgebras which have some positivity and completeness with respect to the filtration. We also obtain that the Koszul duality construction is functorial between carefully constructed generalized Morita categories consisting of those algebras/coalgebras in each dimension.
The notion of (symmetric) coloured operad or "multicategory" can be obtained from the notion of commutative algebra through a certain general process which we call "theorization" (where our term comes from an analogy with William Lawvere's notion of algebraic theory). By exploiting the inductivity in the structure of higher associativity, we obtain the notion of "$n$-theory" for every integer $n\ge 0$, which inductively "theorizes" $n$ times, the notion of commutative algebra. As a result, (coloured) morphism between $n$-theories is a "graded" and "enriched" generalization of ($n-1$)-theory. The inductive hierarchy of those "higher theories" extends in particular, the hierarchy of higher categories. Indeed, theorization turns out to produce more general kinds of structure than the process of categorification in the sense of Louis Crane does. In a part of low "theoretic" order of this hierarchy, graded and enriched $1$- and $0$-theories vastly generalize symmetric, braided, and many other kinds of enriched multicategories and their algebras in various places. We make various constructions of/with higher theories, and obtain some fundamental notions and facts. We also find iterated theorizations of more general kinds of algebraic structure including (coloured) properad of Bruno Vallette and various kinds of topological field theory (TFT). We show that a "TFT" in the extended context can reflect a datum of a very different type from a TFT in the conventional sense, despite close formal similarity of the notions. This work is intended to illustrate use of simple understanding of higher coherence for associativity.
We study descent properties of Jacob Lurie's topological chiral homology. We prove that this homology theory satisfies descent for a factorizing cover, as defined by Kevin Costello and Owen Gwilliam. We also obtain a generalization of Lurie's approach to this homology theory, which leads to a product formula for the infinity $1$-category of factorization algebras, and its twisted generalization.
We give an introduction to the topics of our forthcoming work, in which we introduce and study new mathematical objects which we call "higher theories" of algebras, where inspiration for the term comes from William Lawvere's notion of "algebraic theory". Indeed, our "theories" are `higher order' generalizations of coloured operad or multicategory, where we see an operad as analogous to Lawvere's theory. Higher theories are obtained by iterating a certain process, which we call "theorization", generalizing categorification in the sense of Louis Crane. The hierarchy of all iterated theorizations contains in particular, the hierarchy of all higher categories. As an expanded introduction to the mentioned work, we here introduce the notion of theorization, discuss basic ideas, notions, examples, facts and problems about theorization, and describe how these lead to our work, and what will be achieved.
Generalising Jacob Lurie's idea on the relation between the Verdier duality and the iterated loop space theory, we study the Koszul duality for locally constant factorisation algebras. We formulate an analogue of Lurie's "nonabelian Poincare duality" theorem (which is closely related to earlier results of Graeme Segal, of Dusa McDuff, and of Paolo Salvatore) in a symmetric monoidal stable infinity category carefully, using John Francis' notion of excision. Its proof depends on our earlier study of the Koszul duality for E_n-algebras. As a consequence, we obtain a Verdier type equivalence for factorisation algebras by a Koszul duality construction.
Generalizing Jacob Lurie's idea on the relation between the Verdier duality and the iterated loop space theory, we study the Koszul duality for locally constant factorization algebras. We formulate an analogue of Lurie's "nonabelian Poincare duality" theorem (which is closely related to earlier results of Graeme Segal, of Dusa McDuff, and of Paolo Salvatore) in a symmetric monoidal stable infinity category carefully, using John Francis' notion of excision. Its proof is done by first studying the Koszul duality for E_n-algebras in detail. As a consequence, we obtain a Verdier type equivalence for factorization algebras by a Koszul duality construction. At a foundational level, we study descent properties of Lurie's topological chiral homology. We prove that this homology theory satisfies descent for a factorizing cover, as defined by Kevin Costello and Owen Gwilliam. We also obtain a generalization of Lurie's approach to this homology theory, which leads to a product formula for the infinity category of factorization algebras, and its twisted generalization.