# Intersections versus Multiple Pullbacks

In an arbitrary category $\mathcal C$, a subobject of an object $X$ is a monomorphism $m\colon X'\to X$.

The intersection of a class of subobjects $\langle m_i\colon X_i\to X\rangle_{i\in I}$ is defined to be the subobject $m\colon X'\to X$ such that

(1) $m$ factors through each $m_i$,

(2) if a morphism $f\colon Y\to X$ factors through each $m_i$, then it factors through $m$.

Now, the intersection to me seems to be the multiple pullback of the sink $\langle m_i\colon X_i \to X\rangle$, i.e., the limit over it. Indeed, by definition the limit is a morphism $m\colon X'\to X$ such that $m$ factors through each $m_i$ (condition (1) above), and if we are given a cone $\langle f_i\colon Y\to X_i \rangle_{i\in I}$ by composition we have $f\colon Y\to X$ factoring through each $m_i$ which gives us a factorization through $m$ by the universal property. The only thing left to check is $m\colon X'\to X$ is a monomorphism. But this follows again by the universal property of the limit.

My point of confusion is on pg. 211 in The Joy of Cats which gives the definition of a complete category as one which has all small limits, and a strongly complete category as one which is complete and has intersections.

If a category is complete, it has multiple pullbacks and hence, has intersections. So isn't the definition redundant, i.e., complete categories and strongly complete categories are the same? I must be making a mistake in some definition.

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In the definition of an intersection (Def. 11.23 in Joy of Cats) it is not assumed that the indexing set $I$ is small. Therefore if a category has small limits, it doesn't need to have all intersections (only small ones). In well-powered categories this doesn't make any difference (and most categories are well-powered).