jonmsterling.com
+29
-14
lics.tex
+29
-14
lics.tex
···From \citet{riehl-shulman:2017} we recall that a type $C$ is called \emph{Segal complete} when it is internally right orthogonal to the comparison map $\Horn\hookrightarrow\Spx{\Two}$. To be precise, this means that the restriction map···is an equivalence. The Segal complete types form an \emph{accessible} reflective subuniverse---because the localising class $\mathcal{L}_{\textit{Segal}} \equiv \{\Horn\hookrightarrow\Spx{\Two}\}$ is very small indeed.···A type $C$ is said to be \emph{upper Segal complete} when it is orthogonal to the inclusion $\IsT{i}_\bot \hookrightarrow \II/i$ for generic $i:\II$. Dually, $C$ is said to be \emph{lower Segal complete} when it is orthogonal to the inclusion···(Evidently, a type is upper Segal complete with respect to $\J$ if and only if it is lower Segal complete with respect to the dual lattice $\J\Op$.)-When $\J$ is conservative, a type $C$ is upper Segal complete if and only if it is orthogonal to the little comparison map $\sigma_{\IsT{i}}\colon \IsT{i}_\bot\hookrightarrow \Lift\IsT{i}$ for generic $i:\II$.-Dually, when $\J\Op$ is conservative, $C$ is lower Segal complete if and only if it is orthogonal to the dual comparison map $\Flip{\sigma}_{\IsF{i}}\colon \IsF{i}^\top\hookrightarrow \CoLift\IsF{i}$ for generic $i:\II$. Therefore, when $\Sigma$ is stably quasi-coherent, the upper and lower Segal conditions are precisely equivalent to being orthogonal to each $\sigma_{\IsT{i}}$ and $\Flip{\sigma}_{\IsF{i}}$ respectively.We shall slightly adapt the definition of \emph{Sierpi\'nski completeness} from \citet{pugh-sterling-2025} to avoid relying on global assumptions.···for generic $\IsT{0}$-connected $X$. Evidently a type is Sierpi\'nski complete with respect to $\J$ if and only if it is inverted Sierpi\'nski complete with respect to $\J\Op$.If $\J$ is conservative, then any Sierpi\'nski complete type with respect to $\J$ is upper Segal complete.···When $\J\Op$ is conservative, dually, any inverted Sierpi\'nski complete type with respect to $\J$ is \emph{lower} Segal complete. Therefore, when $\J$ is stably quasi-coherent, Sierpi\'nski completeness (\emph{resp.}\ inverted Sierpi\'nski completeness) implies upper (\emph{resp.}\ lower) Segal completeness.···The thrust of \cref{thm:sierpinski-completeness-mapping-cylinders} is to say that in any subuniverse in which partial map classifiers exist and satisfy the universal property of the Sierpi\'nski cone, it \emph{also} follows that logical mapping cylinders (if they exist) must satisfy the universal property of (geometrical) mapping cylinders.
···From \citet{riehl-shulman:2017} we recall that a type $C$ is called \emph{Segal complete} when it is internally right orthogonal to the comparison map $\Horn\hookrightarrow\Spx{\Two}$. To be precise, this means that the restriction map···is an equivalence. The Segal complete types form an \emph{accessible} reflective subuniverse---because the localising class $\mathcal{L}_{\textit{Segal}} \equiv \{\Horn\hookrightarrow\Spx{\Two}\}$ is very small indeed.···A type $C$ is said to be \emph{upper Segal complete} when it is orthogonal to the inclusion $\IsT{i}_\bot \hookrightarrow \II/i$ for generic $i:\II$. Dually, $C$ is said to be \emph{lower Segal complete} when it is orthogonal to the inclusion···(Evidently, a type is upper Segal complete with respect to $\J$ if and only if it is lower Segal complete with respect to the dual lattice $\J\Op$.)We shall slightly adapt the definition of \emph{Sierpi\'nski completeness} from \citet{pugh-sterling-2025} to avoid relying on global assumptions.···for generic $\IsT{0}$-connected $X$. Evidently a type is Sierpi\'nski complete with respect to $\J$ if and only if it is inverted Sierpi\'nski complete with respect to $\J\Op$.+When $\J$ is conservative, a type $C$ is upper Segal complete if and only if it is orthogonal to the little comparison map $\sigma_{\IsT{i}}\colon \IsT{i}_\bot\hookrightarrow \Lift\IsT{i}$ for generic $i:\II$.+Dually, when $\J\Op$ is conservative, $C$ is lower Segal complete if and only if it is orthogonal to the dual comparison map $\Flip{\sigma}_{\IsF{i}}\colon \IsF{i}^\top\hookrightarrow \CoLift\IsF{i}$ for generic $i:\II$. Therefore, when $\Sigma$ is stably quasi-coherent, the upper and lower Segal conditions are precisely equivalent to being orthogonal to each $\sigma_{\IsT{i}}$ and $\Flip{\sigma}_{\IsF{i}}$ respectively.If $\J$ is conservative, then any Sierpi\'nski complete type with respect to $\J$ is upper Segal complete.···When $\J\Op$ is conservative, dually, any inverted Sierpi\'nski complete type with respect to $\J$ is \emph{lower} Segal complete. Therefore, when $\J$ is stably quasi-coherent, Sierpi\'nski completeness (\emph{resp.}\ inverted Sierpi\'nski completeness) implies upper (\emph{resp.}\ lower) Segal completeness.···The thrust of \cref{thm:sierpinski-completeness-mapping-cylinders} is to say that in any subuniverse in which partial map classifiers exist and satisfy the universal property of the Sierpi\'nski cone, it \emph{also} follows that logical mapping cylinders (if they exist) must satisfy the universal property of (geometrical) mapping cylinders.+We now complete the analysis of Sierpi\'nski completeness initiated by \citet{pugh-sterling-2025}.