Hochschild Cohomology, Monoid Objects and Monoidal Categories
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Hochschild Cohomology, Monoid Objects and Monoidal Categories Magnus Hellstrøm-Finnsen1 Received: 31 December 2019 / Revised: 14 July 2020 / Accepted: 18 July 2020 © The Author(s) 2020
Abstract This paper expands further on a category theoretical formulation of Hochschild cohomology for monoid objects in monoidal categories enriched over abelian groups, which has been studied in Hellstrøm-Finnsen (Commun Algebra 46(12):5202–5233, 2018). This topic was also presented at ISCRA, Isfahan, Iran, April 2019. The present paper aims to provide a more intuitive formulation of the Hochschild cochain complex and extend the definition to Hochschild cohomology with values in a bimodule object. In addition, an equivalent formulation of the Hochschild cochain complex in terms of a cosimplicial object in the category of abelian groups is provided. Keywords Monoidal categories · Hochschild cohomology Mathematics Subject Classification 18D10 · 18D20 · 18G60 · 16E40
1 Introduction Hochschild cohomology was initially studied by Hochschild in [1] and [2], and provides a cohomology theory for associative algebras. In [3], Gerstenhaber discovered that the cohomology ring has a rich structure, which later has been called a Gerstenhaber algebra. The rich structure provides interesting implications, not only restricted to mathematics, but also to physics and related fields. At Isfahan School and Conference on Representations of Algebras (ISCRA), April 2019, I reported from [4]. This article gives a description of Hochschild cohomology in terms of monoid objects (“ring-like objects”) in Ab-enriched monoidal categories. Monoidal categories were independently discovered by Bénabou and Maclane in the beginning of the 1960s (see [5–7]), and they provided an axiomatic system to describe
Communicated by Javad Asadollahi.
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Magnus Hellstrøm-Finnsen [email protected] Avdeling for ingeniørfag, Høgskolen i Østfold, Postboks 700, 1757 Halden, Norway
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Bulletin of the Iranian Mathematical Society
the categories with tensor product, like modules over a ring R with tensor product over R, i.e. ⊗ R , or abelian groups with the tensor product over the integers, i.e. ⊗Z , etc. In this paper, we will first improve the construction of the Hochschild cochain complex given in [4], by taking a more intuitive (and perhaps less combinatorial) approach to this complex. Thereafter, we approach Hochschild cohomology by a cosimplicial object in the category of abelian groups. We will discover that these formulations are equivalent.
2 Monoidal Categories, Monoid Objects and Module Objects First, we recall the definition of a monoidal category. Definition 2.1 A category K is said to be a monoidal category if it is equipped with a bifunctor ⊗:K ×K →K , called the tensor product or monoidal product, and an object 1 ∈ K , called the tensor unit or monoidal unit, together with three natural isomorphisms: • The associator, α : (? ⊗?) ⊗?? ⊗(? ⊗?), which has components: α X ,Y ,Z : (X ⊗ Y ) ⊗ Z → X ⊗(Y ⊗ Z ), for all objects X , Y and Z in K . • The lef
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