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I think the following is true, and what I was looking for. A tensor functor $F: \mathcal{C} \to \mathcal{D}$ is called dominant (sometimes called "surjective") if for any $Y: \mathcal{D}$, there is a …

Generalising Omar's answer, braided monoidal categories are (a special case of) tricategories with one 1-morphism. Symmetric monoidal categories are tetracategories (whatever that is) with one 2-morph …

Consider the category where objects are strict spherical fusion categories and morphisms are strict spherical functors (preserving cups and caps). I am wondering whether there is some kind of image fa …

In my opinion, dagger categories are not evil, but they are not categories with extra structure (in the ordinary sense) either. (I think this is what Qiaochu Yuan meant in his comment.) More specifica …

I don't exactly understand the question, but I think you should familiarise yourself with the notion of a fully dualisable object in a (possibly higher) symmetric monoidal category. For example, a vec …

Fusion categories can be seen as generalisations of the representation category of finite groups. I'm interested in spherical fusion categories. I'm trying to find "interesting" functors from a spheri …

I've been thinking about this since longer already and just realised a really easy example. I was a bit of a blockhead in thinking that for any inclusion functor $F$, we must have that $F\Omega_\mathc …

I'm surprised nobody has mentioned Jeff Eggers "On involutive monoidal categories", which generalises Peter Selingers work, I guess. It's an excellent article, and it connects to other special cases. …

Imagine a semisimple abelian category $\mathcal{C}$, for example representations of a finite group. Take two (nonsimple) objects $X, Y$ that are subobjects of a common object $Z$, and decompose them i …

One can define the category of subobjects of $Z$, where the objects are subobjects of $Z$ and the morphisms have to commute with the monomorphisms of the subobjects. In this category, $S$ is simply th …

Tannaka duality is essentially applying Yoneda twice. So a special case of Tannaka duality that doesn't require the Yoneda lemma would be Pontryagin duality.

A simplicial complex is a PL structure for, and thus also homeomorphic to, a manifold if the link of every vertex is a simplicial sphere, for which there exists a definition. (I know that for high dim …

I have heard that one can do algebraic geometry internal to symmetric monoidal categories. Topological quantum field theories also exist internal to symmetric monoidal categories, and the usual defini …

It is well known that a set of natural transformations can be expressed as an end: $$\int_{A \in \mathcal{A}} \mathcal{B}(FA, GA) =_{\operatorname{Set}} \operatorname{Nat}(F, G)$$ This holds for func …

Edit: I found a serious flaw in the question and my answer, and I had to change a lot. The basic question is still there, but the details are a lot different. Premodular categories In braided spheri …