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what about $\mathbb{Z}_2$ vector spaces over a field $k$? $(k,0)$ is the unit in this category and $(0,k)\otimes (0,k) = (k,0)$.

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In general if you don't want to start with the monoidal structure, you start with a closed category. In a closed category $\mathsf C$, you have a bifunctor $$[-,-] : \mathsf C^{op} \times \mathsf C \to \mathsf C,$$ called the internal hom, and various other data that are somewhat "dual" to the axioms of a monoidal category (I put dual in quotes because this ...

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This is the classical situation covered by the Eckmann–Hilton argument. The point is that the multiplication on $M$ defined by $\mu$ is a homomorphism for the original multiplication $\cdot$ on $M$ and that's enough for the argument given on the Wikipedia page to apply. It seems a bit pointless to repeat it here. A number of references are given on ...

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"Uniqueness will fail for noncommutative monoids, so here we must use commutativity of our monoids, " This is not correct. You get a morphism in $V$ also for non-commutative monoids, but it won't be a monoid morphism. This is where commutativity is used. Uniqueness holds in general: If $h : M \otimes N \to X$ is a monoid morphism with $h \circ (M \otimes ... 6 What's the definition of an additive monoidal category? Is it that tensor product distributes over addition of morphisms? If so, use the fact that a functor between additive categories preserves addition of morphisms iff it preserves biproducts (see for example this blog post). 5 I hope what follows will clear up your confusion. A lax monoidal functor consists of the following data (satisfying some axioms that I won't spell out): One functor$\color{red}{F : \mathsf{C} \to \mathsf{D}}$. This means that for all objects$A \in \mathsf{C}$you have an object$F(A) \in \mathsf{D}$, and for all morphisms$f : A \to B$in$\mathsf{C}$... 5 The$\lambda$-structure is given by taking exterior powers. This is the main motivation I know for defining$\lambda$-rings in the first place. (You need an action of$S_n$on an$n^{th}$tensor power$V^{\otimes n}$to define the exterior power, which is what being symmetric monoidal gets you; in the braided monoidal case you only get an action of$B_n$.) 5 http://arxiv.org/abs/math/0401347 4 Let$\mathcal{C}$be the following category: The objects are pairs$(I, X, p)$where$I$and$X$are sets and$p : X \to I$is a surjection. The morphisms$(I, X, p) \to (J, Y, q)$are maps$f : X \to Y$such that$q \circ f = p$. (That means we must have$I \subseteq J$to have a morphism.) Composition and identities are inherited from$\mathbf{Set}$. I ... 4 Edit, 3/25/15: In an earlier version of this answer I claimed that group objects can be defined in any monoidal category. In fact this isn't true; one of the axioms requires access to a diagonal map, and so it's standard to only define group objects in cartesian monoidal categories. But since talking about actions only uses the monoid structure this doesn't ... 4 You are right that the Yoneda lemma plays the key role in the proof. By the definition of convolution: $$(F \otimes G) \otimes H \approx \int^{C, D} \left(\int^{A, B} F(A) \times G(B) \times \hom(C, A \otimes B)\right) \times H(D) \times \hom(-, C \otimes D)$$ Because products in$\mathbf{Set}$preserve coends, the above is isomorphic to: $$\int^{A, B, C, ... 4 A module is an abelian group. (It's more useful to think of a module as the analog of a vector space, but with the set of scalars coming from a ring instead of a field. Usually, one arrives at this notion of a module in terms of "the action of a ring on a set" where the set is a module.) A monoid is a relaxation of the definition of a group. A monoid has ... 4 There is a chain of forgetful functors which progressively forgets the various operations in the structure:$$\mathrm{Mod_R}\to\mathrm{Ab}\to\mathrm{AbMon}\to\mathrm{Set}$$The interesting thing is that you can go in the opposite direction too with free functors$$\mathrm{Set}\to\mathrm{AbMon}\to\mathrm{Ab}\to\mathrm{Mod_R}$$Each forgetful functor U ... 4 Your observation only means that there is no object (X,\phi) in the Drinfeld center which has X simple and non-isomorphic to V_e (e being the identity element of G) But let C\subseteq G be a conjugacy class, and let V_C=\bigoplus_{g\in C}V_g. Then you should be able to find a natural isomorphism ... 4 Use the Yoneda lemma! By adjunction there is a natural isomorphism$$\mathcal{C}(X \otimes I, C) \to \mathcal{C}(X, [I, C])$$for all objects X in \mathcal{C}. Being an isomorphism is preserved by all functors, so the natural transformation$$\mathcal{C}(X, C) \to \mathcal{C}(X \otimes I, C)$$induced by the right unitor \rho_X : X \otimes I \to X is ... 4 The term "linear monoidal category" doesn't mean "linear and monoidal category", but rather "monoidal (linear category)", i.e. a (weak) monoid in the monoidal bicategory of linear categories (over some fixed base ring R). The monoidal product of two linear categories C,D has as objects pairs of objects of C,D with hom-modules \hom((a,b),(c,d)) = ... 3 The main point is that composition of arrows in bicategories (tensor of objects in monoidal categories) wants to be associative by nature, but it is associative only up to isomorphism. However, we should really speak about multiple composition such as$$e\otimes f\otimes g\otimes h $$in a sense, without parenthesis! This can be made precise in more ways. ... 3 Cartesian closure applies to cartesian categories, i.e. categories which are (symmetric) monoidal with respect to the (binary) product bifunctor (basically any finitely complete category is cartesian). Cartesian closed categories are those categories where each functor A\times- has a right adjoint (-)^A realizing the binatural bijection$$ {\cal ... 3 I'm a PhD student in operator algebraic quantum groups, which is a generalisation of topological groups effectively. The other side of quantum groups are the purely algebraic versions, that is to say Hopf algebras. So quantum groups begin with the study of functions on a group (as it's a generalisation), in the Hopf algebra case we have polynomials, for ... 3 It's standard to assume that a "linear monoidal category" means that the linear and monoidal structures are compatible. For example, see Def 0.1.2 of Catègories Tensorielles, Def 1.12.3 of Tensor Categories or Def 3.2.4 of Dualizable Tensor Categories. If I ran across a paper that didn't make this compatibility explicit, I'd assume that the authors were ... 3 The additional structure is that there is a natural further quotient of the isomorphism classes you can take where you impose the additional relation$[X] - [Y] + [Z] = 0$for every distinguished triangle$X \to Y \to Z \to \Sigma X$. As Martin says in the comments, any commutative monoid gives a discrete braided monoidal category. (Recall that a discrete ... 3 The only dualisable object in a cartesian monoidal category is the terminal object. Thus, a cartesian compact closed category must be trivial. Indeed, suppose$A$has a dual$A^*$. Since the unit object is terminal, the counit$\epsilon : A \times A^* \to 1$is forced. Consider the unit$\eta : 1 \to A^* \times A$. This decomposes into components as a ... 3 I think you want to be a little more careful, as a priori there might be multiple ways of writing a given object as F(X). This isn't a really problem though because if you have some object A, you know basically how to write it as F(X): just let X = G(A)! So we just have $$G(A \bullet B) \cong G(F(G(A))\bullet F(G(B))) \cong GF(G(A) \otimes G(B)) \cong G(A) ... 3 In general it is very rare for a braided monoidal category to commute "on the nose". For a very concrete example, the usual category of sets and functions is cartesian monoidal, so braided (even symmetric) monoidal, but in general A \times B \neq B \times A. As for the braids, there is a PRO (monoidal category with \mathbb{N} as the set of objects and ... 3 Since you are interested in higher categories, you should see the Eckmann-HIlton argument as a special case of the use of the interchange law, or exchange law. A general formulation of this is for double categories: this is a set, or class, C, with two category structures \circ_1, \circ_2 each of which is a morphism for the other. This amounts to the ... 3 You can think of it as a extranatural transformation or a dinatural transformation. In fact the equation you wrote out is precisely the dinaturality condition in question. 3 The pentagon identity is a commutative diagram of 2-morphisms, each one induced by \alpha_M. It looks as follows. Because I'm lazy, I will omit the associators from \mathcal{C}.$$\begin{array}{cc} \mu \circ ((\mu \circ (\mu \otimes\mathrm{id}_M)) \otimes \mathrm{id}_M) & \rightarrow & \mu \circ ((\mu \circ (\mathrm{id}_M \otimes \mu)) ... 3 The term you want is "distributes over," not "commutes with." (Whatever "$A$commutes with$B$" means it should be symmetric in$A$and$B$, which the condition you want is not.) Such monoidal categories are called distributive. Examples include any closed monoidal category with coproducts because in this case$A \otimes (-)\$ has a right adjoint and hence ...

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