  In this paper we show that if $\mathscr{C}$ is a tangent category then the
Ind-category $\operatorname{Ind}(\mathscr{C})$ is a tangent category as well
with a tangent structure which locally looks like the tangent structure on
$\mathscr{C}$. Afterwards we give a pseudolimit description of
$\operatorname{Ind}(\mathscr{C})_{/X}$ when $\mathscr{C}$ admits finite
products, show that the $\operatorname{Ind}$-tangent category of a
representable tangent category remains representable (in the sense that it has
a microlinear object), and we characterize the differential bundles in
$\operatorname{Ind}(\mathscr{C})$ when $\mathscr{C}$ is a Cartesian
differential category. Finally we compute the $\operatorname{Ind}$-tangent
category for the categories $\mathbf{CAlg}_{A}$ of commutative $A$-algebras,
$\mathbf{Sch}_{/S}$ of schemes over a base scheme $S$, $A$-$\mathbf{Poly}$ (the
Cartesian differential category of $A$-valued polynomials), and
$\mathbb{R}$-$\mathbf{Smooth}$ (the Cartesian differential category of
Euclidean spaces). In particular, during the computation of
$\operatorname{Ind}(\mathbf{Sch}_{/S})$ we give a definition of what it means
to have a formal tangent scheme over a base scheme $S$.
