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We describe the analogous construction in the language of syntactic categories\/sites. As an application we identify <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\textbf{Set}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>Set<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-valued regular functors on the syntactic category with a certain class of topos-valued models (we will refer to them as \"<jats:italic>Sh<\/jats:italic>(<jats:italic>B<\/jats:italic>)-valued models\"). For the coherent fragment <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$L_{\\omega \\omega }^g \\subseteq L_{\\omega \\omega }$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msubsup>\n                      <mml:mi>L<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mi>\u03c9<\/mml:mi>\n                        <mml:mi>\u03c9<\/mml:mi>\n                      <\/mml:mrow>\n                      <mml:mi>g<\/mml:mi>\n                    <\/mml:msubsup>\n                    <mml:mo>\u2286<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>L<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mi>\u03c9<\/mml:mi>\n                        <mml:mi>\u03c9<\/mml:mi>\n                      <\/mml:mrow>\n                    <\/mml:msub>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> this was proved by Jacob Lurie, our discussion gives a new proof, together with a generalization to <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$L_{\\kappa \\kappa }^g$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msubsup>\n                    <mml:mi>L<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mi>\u03ba<\/mml:mi>\n                      <mml:mi>\u03ba<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mi>g<\/mml:mi>\n                  <\/mml:msubsup>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> when <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\kappa $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03ba<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> is weakly compact. We present some further applications: first, a <jats:italic>Sh<\/jats:italic>(<jats:italic>B<\/jats:italic>)-valued completeness theorem for <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$L_{\\kappa \\kappa }^g$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msubsup>\n                    <mml:mi>L<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mi>\u03ba<\/mml:mi>\n                      <mml:mi>\u03ba<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mi>g<\/mml:mi>\n                  <\/mml:msubsup>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> (<jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\kappa $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03ba<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> is weakly compact), second, that <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\mathcal {C}\\rightarrow \\textbf{Set} $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>C<\/mml:mi>\n                    <mml:mo>\u2192<\/mml:mo>\n                    <mml:mi>Set<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> regular functors (on coherent categories with disjoint coproducts) admit an elementary map to a product of coherent functors.<\/jats:p>","DOI":"10.1007\/s10485-025-09804-4","type":"journal-article","created":{"date-parts":[[2025,3,11]],"date-time":"2025-03-11T13:08:23Z","timestamp":1741698503000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Sh(B)-Valued Models of $$(\\kappa ,\\kappa )$$-Coherent Categories"],"prefix":"10.1007","volume":"33","author":[{"given":"Krist\u00f3f","family":"Kanalas","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,3,11]]},"reference":[{"key":"9804_CR1","series-title":"London Mathematical Society","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511600579","volume-title":"Locally Presentable and Accessible Categories","author":"J Ad\u00e1mek","year":"1994","unstructured":"Ad\u00e1mek, J., Rosick\u00fd, J.: Locally Presentable and Accessible Categories. 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