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.
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$\begingroup$ Do you mean functors up to equivalence? Otherwise it's a 2-category, not a category. $\endgroup$– Noah SnyderCommented Mar 17, 2014 at 20:02
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$\begingroup$ 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. $\endgroup$– Manuel BärenzCommented Mar 17, 2014 at 20:14
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$\begingroup$ 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? $\endgroup$– Noah SnyderCommented Mar 17, 2014 at 22:15
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$\begingroup$ 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. $\endgroup$– Manuel BärenzCommented Mar 17, 2014 at 22:57
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1$\begingroup$ @NoahSnyder, so I guess I'm fine with strict monoidal spherical categories, those should form a category, right? $\endgroup$– Manuel BärenzCommented Mar 18, 2014 at 16:17
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1 Answer
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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 an $X: \mathcal{C}$ such that $Y$ is a subobject of $FX$.
Theorem: Any tensor functor of fusion categories factors as a dominant functor and a full inclusion.
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.
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$\begingroup$ Does anyone know a reference for this? $\endgroup$ Commented Sep 7, 2015 at 18:40