# Is the category of spherical fusion categories regular? (i.e. is image factorisation possible?)

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 factorisation possible in this category, similar to finite groups, where every homomorphism can be written as a composition of an epimorphism and a monomorphism.

• Do you mean functors up to equivalence? Otherwise it's a 2-category, not a category. Mar 17, 2014 at 20:02
• Is the composite of two spherical functors not spherical again? (I'm not referring to spherical fusion cats, bimodule cats and module functors here.) Also I don't know when to call two functors equivalent. Mar 17, 2014 at 20:14
• There's a 2-category of tensor categories, tensor functors, and tensor natural transformations. You're talking about a 1-category, so do you mean tensor functors up to natural isomorphism? Mar 17, 2014 at 22:15
• Ah yes, otherwise two monoidal functors don't compose to a monoidal functor, right? Sorry, in that case I'm not sure what my question is. What is a suitable definition of image factorisation in 2-categories? I probably mean strict monoidal then. Mar 17, 2014 at 22:57
• @NoahSnyder, so I guess I'm fine with strict monoidal spherical categories, those should form a category, right? Mar 18, 2014 at 16:17

A tensor functor $F: \mathcal{C} \to \mathcal{D}$ is called dominant (sometimes called "surjective") if for any $Y: \mathcal{D}$, there is an $X: \mathcal{C}$ such that $Y$ is a subobject of $FX$.
Proof: First, define the category $\operatorname{Im}F$. Its objects are all objects of $\mathcal{D}$ that are isomorphic to a subobject of an $FX$, where $X$ is any object of $\mathcal{C}$. The morphisms of $\operatorname{Im}F$ are such that it is a full subcategory of $\mathcal{D}$.
By construction, $F$ restricted to $\operatorname{Im}F$ is dominant. Also $\operatorname{Im}F$ is a full subcategory, so it inherits all additional structure like the pivotal/spherical structure, a braiding or a ribbon structure.