Some results on locally finitely presentable categories

Abstract:

We prove that any full subcategory of a locally finitely presentable (l.f.p.) category having small limits and filtered colimits preserved by the inclusion functor is itself l.f.p. Here "full" may be weakened to "full with respect to isomorphisms." Further, we characterize those left exact functors $I:{\mathbf {C}} \to {\mathbf {D}}$ between small categories with finite limits for which the functor ${I^{\ast }}:{\mathbf {LEX}}({\mathbf {D}},{\text {Set)}} \to {\mathbf {LEX}}{\text {(}}{\mathbf {C}}{\text {,Set)}}$ induced by composition is full and faithful. As an application, we prove a theorem on sheaf representations, a consequence of which is that, for any site $\mathcal {C} = ({\mathbf {C}},J)$ on a category ${\mathbf {C}}$ with finite limits, defined by a subcanonical Grothendieck topology $J$, the closure in ${\mathbf {LEX}}({\mathbf {C}},{\text {Set)}}$ under small limits and filtered colimits of the models of $\mathcal {C}$ is the whole of ${\mathbf {LEX}}({\mathbf {C}},{\text {Set)}}$.