Concise definition of subobjects

Higher category theory tells us that it is a bad idea to identify isomorphic things. Rather, the isomorphism should belong to some additional data. Also, categorification tells us that one should, whenever possible, look at a category directly, not just on its set of isomorphism classes. These are two well-known and accepted principles, right?

However, the definition of a subobject seems to contradict these principles. Why should two monomorphisms be identified with each other when they are isomorphic? What is wrong with the following definition: A subobject of an object $B$ is a monomorphism $A \to B$. With this definition, you can do categorical algebra as usual. It also works nicely in examples. For example, a subring of a ring $B$ is just an injective homomorphism of rings $A \to B$. I think this is more abstract but also more natural than the usual set-theoretic definition; for example with $\mathbb{C}=\mathbb{R}[x]/(x^2+1)$ it is not correct that $\mathbb{R}$ is a subring of $\mathbb{C}$ in the set-theoretic sense, but rather in the sense defined above, via the canonical homomorphism $\mathbb{R} \to \mathbb{C}$. For what purpose should I now look at the class of all monomorphisms $\mathbb{R} \to \mathbb{C}$ isomorphic to that? (This is a bad example since there is only one homomorphism $\mathbb{R} \to \mathbb{C}$ anyway, but I hope that my point is clear. Otherwise consider $\mathbb{Z}[i] \to \mathbb{C}$, $i \mapsto \pm i$.)

• Well still another principle on the next level is that just as in a category the correct notion is isomorphism of objects, not equality, in a 2-category the correct notion is equivalence of objects, not isomorphism. Now subobjects of an object $A$ in a category in your sense themselves form a category, i. e. an object of the 2-category of categories. Subobjects in the old sense form another category. And these two categories are equivalent. So... Oct 11 '14 at 21:50
• Yes, I know that. The process basically takes a preorder and makes it a partial order. I think people like partial orders more than preorders - is this the only reason? Oct 11 '14 at 21:52
• I assume the only real reason is that the concept of subobject was invented long before any categorical riff-raff, especially long before higher category theory. At the time it seemed especially perverse to consider any functor representable besides a functor into $Set$. From a modern viewpoint you lose nothing, since the groupoid of monomorphisms is discrete. Oct 11 '14 at 22:03
• @AntonFetisov I think your last sentence is wrong in two ways. Firstly, distinguishing discrete groupoids from homotopy discrete groupoids has a long history and many applications. For instance, the coherence theorem of monoidal categories says that certain groupoids are homotopy discrete; but they are certainly not discrete. And the Barratt-Eccles operad is the nerve of an operad in Gpd consisting of homotopy-discrete (indeed, contractible) but not discrete groupoids; if they were replaced by discrete ones then the whole point would be lost. Jul 15 '17 at 5:06
• Secondly, the whole point of HoTT is that it syntactically enforces homotopy invariance, and thus in particular is unable to distinguish internally between discrete and homotopy-discrete groupoids. Jul 15 '17 at 5:07

Of course it's not necessary to make this identification, but it's fairly harmless since the groupoid of monomorphisms into an object $$X$$ is equivalent to the discrete category of subobjects, and it can be a slight technical convenience, especially in relation to smallness conditions. For example, we say that a category is well-powered if for each object the class of its subobjects is "small" (is a set), and this is convenient for example when discussing certain forms of adjoint functor theorems, etc.
In topos theory, assuming that a topos $$E$$ is well-powered (e.g., a Grothendieck topos), one way of describing a subobject classifier $$\Omega$$ is that the contravariant subobject functor $$Sub: E^{op} \to Set$$ is representable as $$\hom(-, \Omega)$$. This description is conceptually convenient to some people's taste.
• Yes, I know all this, so here are my cons: 1) Many diagrams/categories/limits are actually not small, but only essentially small. So I would define a well-powered category to be one where $\mathrm{Sub}(A)$ is essentially small for every object $A$. (Or even better, define smallness in a better non-set-theoretic way so that this becomes essentially smallness). 2) This should be seen as a functor to categories, or at least, preorders (not partial orders). Oct 11 '14 at 22:03