Let $\mathcal{C}$ be a [monoidal category](http://en.wikipedia.org/wiki/Monoidal_category) and Mon$\_{\mathcal{C}}$ [the category of monoids](http://en.wikipedia.org/wiki/Monoid_%28category_theory%29#Categories_of_monoids) (also called algebra objects) on $\mathcal{C}$. 


**Questions**: are there definitions of image and kernel for a monoid morphism, and of quotient of monoids? Is there a generalization of the [first isomorphism theorem of groups](http://en.wikipedia.org/wiki/Isomorphism_theorem#First_isomorphism_theorem)?


*Remark*: if my question has a negative answer in general, I'm interested in the case where $\mathcal{C}$ is the category of bifinite (dualizable) completely reducible $R$-$R$-bimodules, with $R$ the hyperfinite ${\rm II}\_1$ factor.



Image, kernel, quotient and first isomorphism theorem, in a category of monoid objects
In a nonabelian setting the correct notion of kernel is given by the [kernel pair](http://ncatlab.org/nlab/show/kernel+pair), and the correct notion of cokernel is given by the [cokernel pair](http://ncatlab.org/nlab/show/cokernel+pair). For example, in any category, a morphism $f : a \to b$ is a monomorphism iff its kernel pair exists and is trivial, and dually $f$ is an epimorphism iff its cokernel pair exists and is trivial. By comparison, the naive notions of kernel and cokernel already fail to have this property for monoids in $\text{Set}$. 


There are several reasonable notions of [image](http://ncatlab.org/nlab/show/image) depending on what you want to do. Let me highlight two that are particularly nice because they can be defined in terms of finite limits and colimits: the regular image $\text{im}(f)$ is the equalizer of the cokernel pair of $f$, while the regular coimage $\text{coim}(f)$ is the coequalizer of the kernel pair. (These are nonabelian generalizations of "kernel of the cokernel" and "cokernel of the kernel" respectively.)


In a category with finite limits and colimits, both of these exist and allow us to factor a morphism $f : a \to b$ as a composite


$$a \to \text{coim}(f) \to \text{im}(f) \to b$$


where the first map is a [regular epimorphism](http://ncatlab.org/nlab/show/regular+epimorphism) and is universal with respect to that property, and the last map is a regular monomorphism and is universal with respect to that property. 


*Example.* Let $C = \text{Top}$. Then a monomorphism and an epimorphism are just an injective and a surjective map, respectively. The kernel pair of a map $f : X \to Y$ is the pullback


$$X \times\_Y X = \{ (x, y) \in X \times X : f(x) = f(y) \}$$


which you should think of as the equivalence relation on $X$ determined by $f$. Dually the cokernel pair is the pushout $Y \sqcup\_X Y$. A regular monomorphism is an embedding while a regular epimorphism is a quotient map. The canonical factorization of a map $f : X \to Y$ as a composite


$$X \to \text{coim}(f) \to \text{im}(f) \to Y$$


takes the following form: $\text{coim}(f)$ is the set-theoretic image of $f$ topologized as a quotient of $X$, while $\text{im}(f)$ is the set-theoretic image of $f$ topologized as a subspace of $Y$.


The first isomorphism theorem for groups can be reintepreted in this language as the following claim:



> 
> If $f : G \to H$ is a homomorphism of groups, then $\text{coim}(f) \to \text{im}(f)$ is an isomorphism.
> 
> 
> 


(The correspondence here is that $\text{im}(f)$ is the set-theoretic image while $\text{coim}(f)$ is the quotient of $G$ by the kernel of $f$. To really sell the first identification you need to first prove that every monomorphism of groups is regular, though.) 


A morphism $f$ with this property is sometimes called a [strict morphism](http://ncatlab.org/nlab/show/strict+morphism). The condition that every morphism is strict, which in particular is implicitly one of the axioms defining an abelian category, is somewhat rare: for example, it already does not hold in the category of commutative rings. 


*Example.* Let $R$ be a commutative ring and let $f : R \to S^{-1} R$ be a localization. Then the coimage-image factorization


$$R \to \text{coim}(f) \to \text{im}(f) \to S^{-1} R$$


takes the following form: the first and last maps are both identities. In particular, the middle map is just $f$ itself, so it is almost never an isomorphism. 


And yet there is still a first isomorphism theorem for rings! The issue here is that the regular image doesn't agree with the set-theoretic image for rings; said another way, regular monomorphisms don't agree with monomorphisms for rings. If you like, you can consider a somewhat different generalization of the first isomorphism theorem as stated above, where you replace $\text{im}(f)$ with whatever you think the "correct" notion of image is. But $\text{coim}(f)$ is, to my mind, definitely the correct generalization of "quotient of the codomain by the kernel." 


