Elementary remarks on units in monoidal categories
Joachim Kock
备注：
Comments: LaTeX, 29 pages. Uses Paul Taylor's diagrams package. Does not compile properly with pdflatex. Version v2: introduction rewritten, minor expository improvements (no mathematical changes), submitted for publication
发表信息：
Math. Proc. Cambridge Phil. Soc. 144 (2008), 53-76.
We explore an alternative definition of unit in a monoidal category originally due to Saavedra: a Saavedra unit is a cancellative idempotent (in a 1-categorical sense). This notion is more economical than the usual notion in terms of left-right constraints, and is motivated by higher category theory. To start, we describe the semi-monoidal category of all possible unit structures on a given semi-monoidal category and observe that it is contractible (if nonempty). Then we prove that the two notions of units are equivalent in a strong functorial sense. Next, it is shown that the unit compatibility condition for a (strong) monoidal functor is precisely the condition for the functor to lift to the categories of units, and it is explained how the notion of Saavedra unit naturally leads to the equivalent non-algebraic notion of fair monoidal category, where the contractible multitude of units is considered as a whole instead of choosing one unit. To finish, the lax version of the unit comparison is considered. The paper is self-contained. All arguments are elementary, some of them of a certain beauty.
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Over the past two decades several different approaches to defining a geometry over ${\mathbb F}_1$ have been proposed. In this paper, relying on Toën and Vaquié's formalism, we investigate a new category ${\mathsf{Sch}}_{\widetilde{\mathsf B}}$ of schemes admitting a Zariski cover by affine schemes relative to the category of blueprints introduced by Lorscheid. A blueprint, that may be thought of as a pair consisting of a monoid $M$ and a relation on the semiring $M \otimes_{{\mathbb F}_1} \mathbb N$, is a monoid object in a certain symmetric monoidal category $\mathsf B$, which is shown to be…
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This book is an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic discussion of 2-/bicategories, pasting diagrams, lax functors, 2-/bilimits, the Duskin nerve, 2-nerve, adjunctions and monads in bicategories, 2-monads, biequivalences, the Bicategorical Yoneda Lemma, and the Coherence Theorem for bicategories. Grothendieck fibrations and the Grothendieck construction are discussed next, followed by tricategories, monoidal bicategories, the Gray tensor product, and double cate…
