This is a report on aspects of the theory and use of monoidal categories. The first section introduces the main concepts through the example of the category of vector spaces. String notation is explained and shown to lead naturally to a link between knot theory and monoidal categories. The second section reviews the light thrown on aspects of representation theory by the machinery of monoidal category theory, machinery such as braidings and convolution. The category theory of Mackey functors is reviewed in the third section. Some recent material and a conjecture concerning monoidal centres is included. The fourth and final section looks at ways in which monoidal categories are, and might be, used for new invariants of low-dimensional manifolds and for the field theory of theoretical physics.

We analyse some aspects of the notion of \emph{algebraic exponentiation} introduced by the second author [16] and satisfied by the category $Gp$ of groups. We show how this notion provides a new approach to the categorical-algebraic question of the centralization. We explore, in the category $Gp$, the unusual universal properties and constructions determined by this notion, and we show how it is the origin of various properties of this category.

In this paper we study loops, neardomains and nearfields from a categorical point of view. By choosing the right kind of morphisms, we can show that the category of neardomains is equivalent to the category of sharply 2-transitive groups. The other categories are also shown to be equivalent with categories whose objects are sets of permutations with suitable extra properties. Up to now the equivalence between neardomains and sharply 2-transitive groups was only known when both categories were equipped with the obvious isomorphisms as morphisms. We thank Hubert Kiechle for this observation.

For any small involutive quantaloid $\cal Q$ we define, in terms of symmetric quantaloid-enriched categories, an involutive quantaloid $\sf Rel(\cal Q)$ of $\cal Q$-sheaves and relations, and a category $\sf Sh(\cal Q)$ of $\cal Q$-sheaves and functions; the latter is equivalent to the category of symmetric maps in the former. We prove that $\sf Rel(\cal Q)$ is the category of relations in a topos if and only if $\cal Q$ is a modular, locally localic and weakly semi-simple quantaloid; in this case we call $\cal Q$ a Grothendieck quantaloid. It follows that $\sf Sh(\cal Q)$ is a Grothendieck topos whenever $\cal Q$ is a Grothendieck quantaloid. Any locale $L$ is a Grothendieck quantale, and $\sf Sh(L)$ is the topos of sheaves on $L$. Any small quantaloid of closed cribles is a Grothendieck quantaloid, and if $\cal Q$ is the quantaloid of closed cribles in a Grothendieck site $(\cal C,J)$ then $\sf Sh(\cal Q)$ is equivalent to the topos $\Sh(\cal C,J)$. Any inverse quantal frame is a Grothendieck quantale, and if $\cal O(G)$ is the inverse quantal frame naturally associated with an étale groupoid $G$ then $\sf Sh(\cal O(G))$ is the classifying topos of $G$.

We study a particular structure on a topos $\cal E$, related to the notion of a `class of étale maps' due to Joyal and Moerdijk and to Bénabou's notion of `calibration', which corresponds to giving for each object $A$ of $\cal E$ a `natural' comparison between the slice category ${\cal E}/A$ and a smaller `petit topos' associated with $A$. We show that there are many naturally-arising examples of such structures; but rather few of them satisfy the condition that the relation between the `gros' and `petit' toposes of every object is expressed by a local geometric morphism.

Möbius inversion, originally a tool in number theory, was generalized to posets for use in group theory and combinatorics. It was later generalized to categories in two different ways, both of which are useful. We provide a unifying abstract framework. This allows us to compare and contrast the two theories of Möbius inversion for categories, and advance each of them. Among several side benefits is an improved understanding of the following fact: the Euler characteristic of the classifying space of a (suitably finite) category depends only on its underlying graph.