Question about the definition of the inverse limit in category theory.

Question

Let $C$ be a category of some algebraic structures (groups, rings, algebras..). Then the direct limit of a directed system $(X_i, f_{ij})$ is the quotient $\bigsqcup\limits_{i \in I} X_i \backslash \sim$, where the equivalence relation $\sim$ is defined as follows: If $x_i \in X_i$ and $x_j \in X_j$, then $x_i \sim x_j \Leftrightarrow$ there exists some $k \in I$ with $i, j \leq k$ such that $f_{ik}(x_i) = f_{jk}(x_j)$. Can the inverse limit also be defined as a quotient by this equivalence relation ? Maybe as $\prod\limits_{i \in I} X_i \backslash \sim$ ? I am not sure.

Answer 1

Inverse limits and direct limits are in some sense dual, so since direct limits are quotients of a "sum", you may expect inverse limits to be a subalgebra of a product, since subalgebras are dual to quotients and products are dual to sums.

This is in fact what happens : if you have an inverse system of algebras $((X_i), (f_{ij}))$, $f_{ij}: X_i \to X_j$ when $j\leq i$, then the inverse limit may be described as $\{x\in \displaystyle\prod_{i\in I}X_i \mid \forall i\geq j, f_{ij}(x_i)= x_j\}$.

So you don't get a quotient of the product, but a subalgebra of the product.