Let $C$ be the category of pairs $(A, I)$, where $A$ is a ring and $I$ is an ideal, whose morphisms are morphisms $(A, I) \rightarrow (B, J)$ are morphisms of rings $\phi : A \rightarrow B$ such that $\phi(I) \subset J$. (This is equivalent to the full subcategory of the arrow category of rings consisting of the surjective morphisms, if you think about it).
There is a functor $F$ from $C$ to the category $\text{Gr}(\text{Ring})$ of $\mathbb{N}_{\geq 0}$-graded rings, where $(A, I)$ is sent to $\oplus_{i = 0}^n I^i/I^{i+1}$, where $I^0 = A$ by convention.
Question: Does this functor have an adjoint $G : \text{Gr}(\text{Ring}) \rightarrow C$?
This would of course be nice to know since it would then preserve either limits or colimits.
It seems like $\oplus_{i =0}^\infty A_i \mapsto (A_0, \oplus_{ i =1}^\infty A_i)$ would be its adjoint.
Best Answer
This is a riff on the more general question "what is the universal property of associated graded?" I won't discuss the rings-and-ideals case at all, and will just restrict my attention to filtered vector spaces.
Classically a filtered vector space is an increasing sequence $V_0 \subseteq V_1 \subseteq V_2 \subseteq \dots $ and it has an associated graded vector space given in degree $i$ by $V_i/V_{i-1}$. One of the first MO questions (#263!) asked whether this functor is a left or a right adjoint, and the answer is that it is neither.
This turns out, however, to be an artifact of working with an insufficiently flexible notion of filtration; it turns out that requiring all the maps $V_i \to V_{i+1}$ to be inclusions is too restrictive and makes the category behave poorly, and we can fix this by working with a generalized notion of filtration given just by a sequence $V_0 \to V_1 \to V_2 \to \dots$ of not-necessarily-injective maps. With this fix, as described by Nicholas Schmidt, taking associated graded is left adjoint to the functor which sends a graded vector space $W_i$ to the sequence $W_0 \xrightarrow{0} W_1 \xrightarrow{0} W_2 \dots$ where all maps are zero, and in particular it now preserves colimits (which behave differently and better in this new category of filtered vector spaces).
There is a lovely algebro-geometric interpretation of a further generalization of this construction where we allow $\mathbb{Z}$-gradings, as follows. The category of $\mathbb{Z}$-graded vector spaces can be identified with the category of quasicoherent sheaves over the stack $B \mathbb{G}_m$ which classifies line bundles; this is a fancy way of saying that an action of $\mathbb{G}_m$ is the same thing as a $\mathbb{Z}$-grading. The category of "$\mathbb{Z}$-filtered" vector spaces, by which I mean sequences $\dots V_i \to V_{i+1} \dots$ possibly extending infinitely in both directions, can in turn be identified with the category of quasicoherent sheaves over the quotient stack $\mathbb{A}^1/\mathbb{G}_m$. There is a natural inclusion
$$i : B \mathbb{G}_m \to \mathbb{A}^1/\mathbb{G}_m$$
given by thinking of $B \mathbb{G}_m$ as $\bullet / \mathbb{G}_m$ and including $\bullet$ into $\mathbb{A}^1$ as the origin, and the associated graded functor turns out to be the pullback $i^{\ast}$ of quasicoherent sheaves along $i$, left adjoint to pushforward $i_{\ast}$.