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$\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\Vect{Vect}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Hom{Hom}$The infinitude of the in …

If $\otimes : V \times V \to V$ has a left adjoint and $V$ has finite products then $\otimes$ preserves them in the sense that the natural map $$(X \times Y) \otimes (Z \times W) \to (X \otimes Z) \ti …

One way to think about what the monoidal structure on vector spaces is doing is that it is telling us that vector spaces do not really form a category, or not "just" a category: they form a multicateg …

Let me see if I understand what Jacob says in the comments. I think his argument can be summarized as: the Brauer 3-group is étale-locally an Eilenberg-MacLane spectrum, hence étale-locally an $\mathb …

In the comments you ask: The question is the following: if I have an "abstract" category (or even a subcategory in the category of vector spaces), how can I understand that this is the category of …

Lemma: In a closed symmetric monoidal category where the unit object $1$ is tiny (meaning $\text{Hom}(1, -)$ preserves colimits), every dualizable object is tiny. Proof. If $x$ is dualizable, the …

Supervector spaces look a lot like the category of representations of $\mathbb{Z}/2\mathbb{Z}$ - the even part corresponds to the copies of the trivial representation and the odd part corresponds to t …

As a warmup, an $\mathbb{N}$-graded ring is a monoid object in the symmetric monoidal category of $\mathbb{N}$-graded abelian groups under the convolution tensor product, which you can think of as Day …

This is not an answer, just an explanation of the mistake I made in my original (now deleted) answer. It is very tempting to conjecture that an object $X$ has a right dual $X^{\ast}$ iff there is an …

If $M$ is any monoidal category, the presheaf category $[M^{op}, \text{Set}]$ inherits a monoidal structure given by Day convolution. It is uniquely determined by the condition that it restricts to th …

If $L$ and $R$ are a left and right adjoint, then doctrinal adjunction asserts that $L$ is oplax monoidal iff $R$ is lax monoidal. (I'm being a bit imprecise here, treating monoidality as if it were a …

Look at Section 3 of Leinster's Higher Operads, Higher Categories, where the term used is "unbiased monoidal category."

In a nonabelian setting the correct notion of kernel is given by the kernel pair, and the correct notion of cokernel is given by the cokernel pair. For example, in any category, a morphism $f : a \to …

No. What follows appears to be a counterexample for $C = \text{Vect}$ (I don't understand where in your argument you prove fullness). Let $M = \text{Vect}^{op}, C = \text{Vect}$, and let $h : \text{V …

Let $(C, \otimes)$ be a symmetric monoidal category (maybe braided is also okay). Then the category $\text{Mon}(C)$ of monoid objects in $C$ is also a symmetric monoidal category with the same monoida …