Is there a notion of "normal subcategory" analgous to the notion of normal subgroup?

Question

The idea of a replete subcategory to me seems very analogous to the idea of a characteristic subgroup, as both are, in some sense, subobjects invariant under a notion of equivalence (categorical equivalence for replete subcategories, automorphisms for characteristic subgroups). This led me to the question: Is there a notion in category theory that is similarly analogous to that of a normal subgroup in group theory?

Answer 1

Despite the negativity of comments on this question, considering how the concept of normal subgroup extends to other categories is a very fruitful pastime, and I'll give a flavour of a few of generalizations. Since it seems to have been the motivation for asking this, I'll also explain how they apply to the ($1$-)category of categories.

The category of groups has the rather special property of admitting zero morphisms: between any pair of groups, there is a homomorphism sending everything to the identity element. A subgroup $N \leq G$ is normal if there is some homomorphism $h: G \to H$ such that the equalizer of $h$ and the zero morphism $0: G \to H$ is precisely the subgroup $N$.

Of course, not every category has zero morphisms. The most direct generalization in a category with equalizers is that of regular subobject, where we just ask for expressibility of the subobject as an equalizer of some pair of morphisms. $$N \hookrightarrow G \rightrightarrows H$$ Characterizing regular subcategories of categories is a little tricky (and trying to find references to them in the literature is tricky because regular categories refer to something else!) They are subcategories, as you would expect, but whenever we have morphisms $g: A \to C$ and $u:B \to C$ in a regular subcategory and morphisms $f:A \to B$, $v: C \to B$ with $u \circ f = g$ and $v \circ u = \mathrm{id}_B$, then we must also have $f$ in the regular subcategory, since knowing that $Fg = Gg$ and $Fu = Gu$ for a pair of functors $F,G$ means that $Fu \circ Ff = F(g) = G(g) = Gu \circ Gf = Fu \circ Gf$, whence $Ff = Gf$; the dual property is also necessary{*}. This in particular means that whenever a morphism of a regular subcategory has a two-sided inverse in the larger category, the regular subcategory must contain that inverse. However, it is a bit weaker than being closed under conjugation: when we view the alternating group $A_5$ as a one-object category, the equalizer of the identity homomorphism with the homomorphism obtained by conjugation with the element $(1 2)(3 4)$ is a non-trivial subgroup (which is clearly not a normal subgroup). In particular, there are regular subobjects in the category of groups which are not normal subgroups, so this is a weak generalization.

Alternatively, we could observe that in the category of groups, the zero morphisms are those which factor through the trivial group, which is a 'zero object' in the category of groups, being both initial and terminal. As such, another way to express normal subgroups of $G$ are those which can be expressed as a kernel: the pullback along some group homomorphism $G \to H$ of the unique homomorphism $0 \to H$. $$\require{AMScd} \begin{CD} N @>>> G;\\ @VVV @VVV \\ 0 @>>{!}> H; \end{CD}$$ This only uses the fact that $0$ is an initial object, so we can extend this definition to any category with an initial object and pullbacks. Unfortunately, this doesn't extend well to the 1-category of categories, since this has a strict initial object: the empty category. If we pull back the unique morphism from the empty category, we will always get the empty subcategory...

All is not lost, however. We can also note that a subgroup is normal if and only if it is the fiber of its cokernel, or in other words if the pushout of the inclusion of the subgroup along the unique morphism to the zero object produces a pullback square. $$\require{AMScd} \begin{CD} N @>>> G;\\ @V{!}VV @VVV \\ 1 @>>> H; \end{CD}$$ This time we are using the fact that the zero object is a terminal object, and this definition makes sense as soon as we have a terminal object and pushouts. This is stronger than the definition of regular subobject I gave earlier, since $N \hookrightarrow G$ is automatically the equalizer of the given morphism $G \to H$ and the constructed morphism $G \to 1 \to H$ in this scenario. Note that we couldn't just define normal subobjects to be pullbacks of a morphism $1 \to H$ since in general there may be several (or none) of these to choose from; using a pushout in this definition gives us a canonical choice. This definition coincides with the usual one for groups, even when we view groups as one-object categories living in the 1-category of categories. A normal subcategory in this sense has the 2-out-of-3 property and has morphisms which are closed under conjugation by any isomorphisms in the larger category between its objects{*}.

There are further characterizations one could generalize, like closure under inner automorphisms, although that involves exploiting some 2-categorical structure (to extend the notion of inner automorphism to categories, where conjugation by a single element doesn't directly make sense, we would use the fact that in the 2-category of groups these correspond to automorphisms which are naturally isomorphic to the identity automorphism). As always in category theory, which generalization is the right one depends on what you want to do with it, but I encourage you to have fun seeing what else you can find!

{*} Note that I have not proven, nor do I know, whether the properties I give are sufficient as well as necessary.