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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeApr 17th 2017
   • (edited Apr 17th 2017)
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   Author: Todd_Trimble
   Format: MarkdownItexHave internal sup-lattices in quasitoposes been studied much? Of course they've been studied in toposes to a pretty fair extent. In a topos $E$, the covariant power object functor $P: E \to E$ (where for $f: X \to Y$, we define $P f = \exists_f: P X \to P Y$) carries a structure of monad whose unit is the singleton map $\sigma: X \to P X$ and whose multiplication is internal union $\bigcup_X: P P X \to P X$. In fact this is a lax idempotent or KZ monad in some suitable internal sense, and we define a sup-lattice to be a $P$-algebra. This is all well-studied, and figures heavily in the Joyal-Tierney monograph on a generalized Galois theory à la Grothendieck. It seems to me that a certain chunk of this can be developed in quasitoposes in parallel fashion. For example, for $f: X \to Y$ let me construct $P f$ by hand. There is a canonical regular subobject $\in_X \hookrightarrow X \times \Omega^X$ classified by the evaluation map $X \times \Omega^X \to \Omega$. Form the composite $$\in_X \hookrightarrow X \times \Omega^X \stackrel{f \times 1}{\to} Y \times \Omega^X$$ and then take its epi-(regular mono) factorization, using the fact that quasitoposes are *co*regular. Take the regular image $E_f \hookrightarrow Y \times \Omega^X$ and its classifying map $\chi_{E_f}: Y \times \Omega^X \to \Omega$. Curry that to a map $\Omega^X \to \Omega^Y$; this is what I call $P f$. Now I'll construct a map $\bigcup_X: P P X \to P X$. Form the diagram below where the square is a pullback $$\array{ E & \to & X \times P X \times P P X & \stackrel{\pi_{13}}{\to} & X \times P P X \\ \downarrow & pb & \downarrow \mathrlap{1 \times \delta \times 1} & & \\ \in_X \times \in_{P X} & \hookrightarrow & X \times P X \times P X \times P P X & & }$$ and take the regular image of the top composite, say $\Sigma \hookrightarrow X \times P P X$. Curry its classifying map $\chi_\Sigma: X \times P P X \to \Omega$ to get the map $\bigcup_X: P P X \to P X$. All this is parallel to similar constructions in topos theory; we're just being mindful to take regular images instead of just images at the appropriate steps. And so if I had to guess, I'd guess that there's not a great deal of difference: we get a KZ monad $P$ on a quasitopos just as we do in topos theory, and there are theorems like $P$-algebra maps are the same as internal left adjoints between algebras, and there's an adjoint functor theorem that if $X \to Y$ is a map between sup-lattices that preserves infs, then it has a left adjoint. Does anyone know about this?

   Have internal sup-lattices in quasitoposes been studied much?

   Of course they’ve been studied in toposes to a pretty fair extent. In a topos $E$, the covariant power object functor $P: E \to E$ (where for $f: X \to Y$, we define $P f = \exists_f: P X \to P Y$) carries a structure of monad whose unit is the singleton map $\sigma: X \to P X$ and whose multiplication is internal union $\bigcup_X: P P X \to P X$. In fact this is a lax idempotent or KZ monad in some suitable internal sense, and we define a sup-lattice to be a $P$-algebra. This is all well-studied, and figures heavily in the Joyal-Tierney monograph on a generalized Galois theory à la Grothendieck.

   It seems to me that a certain chunk of this can be developed in quasitoposes in parallel fashion. For example, for $f: X \to Y$ let me construct $P f$ by hand. There is a canonical regular subobject $\in_X \hookrightarrow X \times \Omega^X$ classified by the evaluation map $X \times \Omega^X \to \Omega$. Form the composite

   $$\in_X \hookrightarrow X \times \Omega^X \stackrel{f \times 1}{\to} Y \times \Omega^X$$

   and then take its epi-(regular mono) factorization, using the fact that quasitoposes are coregular. Take the regular image $E_f \hookrightarrow Y \times \Omega^X$ and its classifying map $\chi_{E_f}: Y \times \Omega^X \to \Omega$. Curry that to a map $\Omega^X \to \Omega^Y$; this is what I call $P f$.

   Now I’ll construct a map $\bigcup_X: P P X \to P X$. Form the diagram below where the square is a pullback

   $$\array{ E & \to & X \times P X \times P P X & \stackrel{\pi_{13}}{\to} & X \times P P X \\ \downarrow & pb & \downarrow \mathrlap{1 \times \delta \times 1} & & \\ \in_X \times \in_{P X} & \hookrightarrow & X \times P X \times P X \times P P X & & }$$

   and take the regular image of the top composite, say $\Sigma \hookrightarrow X \times P P X$. Curry its classifying map $\chi_\Sigma: X \times P P X \to \Omega$ to get the map $\bigcup_X: P P X \to P X$.

   All this is parallel to similar constructions in topos theory; we’re just being mindful to take regular images instead of just images at the appropriate steps. And so if I had to guess, I’d guess that there’s not a great deal of difference: we get a KZ monad $P$ on a quasitopos just as we do in topos theory, and there are theorems like $P$-algebra maps are the same as internal left adjoints between algebras, and there’s an adjoint functor theorem that if $X \to Y$ is a map between sup-lattices that preserves infs, then it has a left adjoint.

   Does anyone know about this?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeApr 18th 2017
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   Author: Mike Shulman
   Format: MarkdownItexGood question! I don't recall seeing this developed anywhere, but you could check for instance Wyler's book on quasitopoi. The only thing that I would worry about at first glance is to what extent the adjoint functor theorem depends on the function comprehension principle, which is false for the logic of regular subobjects in a quasitopos.

   Good question! I don’t recall seeing this developed anywhere, but you could check for instance Wyler’s book on quasitopoi. The only thing that I would worry about at first glance is to what extent the adjoint functor theorem depends on the function comprehension principle, which is false for the logic of regular subobjects in a quasitopos.

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeApr 19th 2017
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   Author: Todd_Trimble
   Format: MarkdownItexThanks for that heads-up! The adjoint functor theorem I had in mind is a very algebraic form which uses basically the structures available for lax idempotent monads, and so I think such a blunt instrument as the general adjoint functor theorem wouldn't be needed for the context I had in mind. The reason I'm asking has to do with an answer I've just written up to an [MO query](https://mathoverflow.net/questions/136478/can-the-lawvere-fixed-point-theorem-be-used-to-prove-the-brouwer-fixed-point-the/) by Qiaochu Yuan. Luckily, I saw in the meantime how to circumvent a lot of the complications in getting such a theory of sup-lattices up and running.

   Thanks for that heads-up! The adjoint functor theorem I had in mind is a very algebraic form which uses basically the structures available for lax idempotent monads, and so I think such a blunt instrument as the general adjoint functor theorem wouldn’t be needed for the context I had in mind.

   The reason I’m asking has to do with an answer I’ve just written up to an MO query by Qiaochu Yuan. Luckily, I saw in the meantime how to circumvent a lot of the complications in getting such a theory of sup-lattices up and running.