Given a category $\mathcal{C}$, we can use copresheaves $H \in [\mathcal{C},Set]$ or presehaves $H \in [\mathcal{C}^{\mathrm{op}},Set]$ to state left or right universal properties. Existence of a representing object for H will amount to the existence, up to a canonical isomorphism, of an object $c\in \mathcal{C}$ satisfying the property encoded in $H$. Examples:
- Fix $x,y\in \mathcal{C}$. Their coproduct $x+y$ exists if and only if the following $H$ copresheaf is representable: $$H : z \mapsto \mathcal{C}(x,z) \times \mathcal{C}(y,z)$$ This of course generalizes to all individual colimits, and we can play the dual game (use presheaves instead) to define what a limit is.
- Assume $\mathcal{C}$ has finite limits. We can encode the property of being a subobject classifier using the following presheaf on $\mathcal{C}$: $$\mathrm{Sub}(z) = \lbrace \iota: s\hookrightarrow z\rbrace$$ On morphisms, $\mathrm{Sub}$ uses pullbacks.
Myriads of interesting properties can be encoded this way: as (co)presheaves on $\mathcal{C}$. (Btw I could not think of a way to phrase example (2) as a limit, I wonder if it's possible?). The point of the present question is that not all properties can be encoded as (co)presheaves on $\mathcal{C}$, i.e. you sometimes need to change $\mathcal{C}$ a bit. For instance:
- Assume $\mathcal{C}$ has finite coproducts, and is cartesian closed. Consider the endofunctor $F : x \mapsto 1 + x$. We say that $\mathcal{C}$ has a natural number object (NNO) if the category of functor-algebras $\mathrm{alg}(F)$ admits an initial object. In other words, $\mathcal{C}$ has a NNO if the following copresheaf (on $\mathrm{alg}(F)$!) is representable: $$H \in [\mathrm{alg}(F),Set] : {-} \mapsto \lbrace * \rbrace$$ If it exsits, the initial object of $\mathrm{alg}(F)$ consists of a triple $(\mathbb{N},z,s)$, i.e. an object $\mathbb{N}$ of $\mathcal{C}$ endowed with a zero and a successor function. We can recover this object by applying the obvious forgetful functor to the triple. $$ U : \mathrm{alg}(F) \to \mathcal{C}$$ Hence we had to formulate the NNO property "over" $\mathcal{C}$, in $\mathrm{alg}(F)$.
- My other example regards images of morphisms. Assume $\mathcal{C}$ is a category (no extra structure). Consider a morphism $f : x \to y$ in $\mathcal{C}$. Again we will formulate the universal property of $\mathrm{im}(f)$ "over" $\mathcal{C}$. We first define $\mathcal{M}|y$ as the full subcategory of the slice $\mathcal{C}/y$ consisting of monomorphisms with target $y$. "Being an image of f" means being a representing object for the following copresheaf $H$: $$H \in [\mathcal{M}|y,Set] : (i : m\hookrightarrow y) \mapsto \mathcal{C}/y\,(f,i)$$ Such a representing object consists of a couple $(\mathrm{im}(f), f|^{\mathrm{im}})$ of an object $\mathrm{im}(f)$ of $\mathcal{C}$ embedding in $y$, and a "corestricted" map $f|^{\mathrm{im}}$. Once again, we can recover the object of interest $\mathrm{im}(f)$ by applying a forgetful functor: $$ U : \mathcal{M}|y \to \mathcal{C}/y \to \mathcal{C}$$
I believe there is no way to phrase the two above examples by merely using copresheaves on $\mathcal{C}$ (this is particularly clear for the NNO). You have to state those universal properties "over" $\mathcal{C}$, in a different category $\mathcal{E}$: $$ U : \mathcal{E} \to \mathcal{C} \,\, ;\,\, H\in [\mathcal{E}^{(op)},Set]$$
Questions
- Would there exist a notion of category closed under universal properties over it? (some kind of generalization of (co)completeness)
- Can you think of a negative example: a case where an object has a negative (right) universal property that can not be stated as representability of a presheaf on $\mathcal{C}$ itself?