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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Some results on locally finitely presentable categories

Authors: M. Makkai and A. M. Pitts
Journal: Trans. Amer. Math. Soc. 299 (1987), 473-496
MSC: Primary 03G30; Secondary 03C20, 03C52, 18B05
MathSciNet review: 869216
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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)}}$.

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