Let’s start by enunciating a recurring idea in mathematics: objects do not exist in isolation, but tend to occur in families. This might sound a bit vague at first blush, so let us consider the case of covering spaces/maps

\[E \xrightarrow{f} B\]

which can be thought of as a family of sets parameterized by a space \(B\): these assemble as the following collection of sets \(\{F^{-1}(b)\}_{b\in B}\), where for each point \(b\in B\), the fiber \(f^{-1}(b)\subset E\) is a set.

This construction furnishes an isomorphism

\[\mathsf{Cov}(B)\simeq [B,\mathcal{S}\mathsf{et}^{\simeq}]\]

from the set of covering spaces of \(B\) and the set of homotopy classes of maps \(B\rightarrow \mathcal{S}\mathsf{et}^{\simeq}\).

We should then ask ourselves: If \(\mathcal{B}\) is a “space” whose morphisms are not all invertible - that is if \(\mathcal{B}\) is an \(\infty\)-category -, then what, exactly is classified by a map \(\mathcal B \rightarrow \mathcal{S}\mathsf{et}\)?