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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 29th 2018
   • (edited Jan 29th 2018)
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   Author: David_Corfield
   Format: MarkdownItexSteve Awodey gives a nice [account](http://schd.ws/hosted_files/clmps2015/d4/Lambek.pdf) of his passage from Lambek and Moerdijk's work on sheaf representations of toposes to the work with his student Breiner on [[logical schemes]]. From David R. on [g+](https://plus.google.com/+DavidRoberts/posts/6qvkvAihMPh) I hear that Jacob Lurie is running a course on categorical logic right up to Ultracategories and the "Strong Conceptual Completeness" theorem of Makkai. Now Awodey tells us about how in his work with Forssell >"We replace Makkai’s ultraproduct structure on the groupoids of models by a Stone-style logical topology." (Slide 26) This was a step on his way to the work with Breiner who [observes](https://www.andrew.cmu.edu/user/awodey/students/breiner.pdf) >"We reframe Makkai & Reyes’ conceptual completeness theorem as a theorem about schemes...The theorem follows immediately from our scheme construction." Isn't it reasonable to take Awodey's approach as more natural category theoretically? Given we know that this interest of Lurie in [[conceptual completeness]] relates to an $(\infty,1)$-version found in spectral algebraic geometry (see [here](https://mathoverflow.net/questions/226931/variant-of-conceptual-completeness#comment600118_231768)), wouldn't we expect that logical schemes would provide a good basis to construct a logical variant of derived schemes. Should we not be seeing some nice HoTT syntax-semantics dualities emerging? If we can take a pretopos as a first-order theory, what can we say similarly in the case of the $\infty$-pretopoi of A. 9 of [[Spectral Algebraic Geometry]]? Won't they have to be HoTT-like theories (but somehow first-order), which perhaps can be recaptured through some $\infty$-stacks on an $\infty$-groupoid of models? The focus in SAG seems to be on recovering $\infty$-toposes, rather than pretoposes, (Theorem A.9.0.6). Does Makkai-Reyes offer advantages over Awodey-Breiner there?

   Steve Awodey gives a nice account of his passage from Lambek and Moerdijk’s work on sheaf representations of toposes to the work with his student Breiner on logical schemes.

   From David R. on g+ I hear that Jacob Lurie is running a course on categorical logic right up to Ultracategories and the “Strong Conceptual Completeness” theorem of Makkai.

   Now Awodey tells us about how in his work with Forssell

   “We replace Makkai’s ultraproduct structure on the groupoids of models by a Stone-style logical topology.” (Slide 26)

   This was a step on his way to the work with Breiner who observes

   “We reframe Makkai & Reyes’ conceptual completeness theorem as a theorem about schemes…The theorem follows immediately from our scheme construction.”

   Isn’t it reasonable to take Awodey’s approach as more natural category theoretically?

   Given we know that this interest of Lurie in conceptual completeness relates to an $(\infty,1)$-version found in spectral algebraic geometry (see here), wouldn’t we expect that logical schemes would provide a good basis to construct a logical variant of derived schemes.

   Should we not be seeing some nice HoTT syntax-semantics dualities emerging? If we can take a pretopos as a first-order theory, what can we say similarly in the case of the $\infty$-pretopoi of A. 9 of Spectral Algebraic Geometry? Won’t they have to be HoTT-like theories (but somehow first-order), which perhaps can be recaptured through some $\infty$-stacks on an $\infty$-groupoid of models? The focus in SAG seems to be on recovering $\infty$-toposes, rather than pretoposes, (Theorem A.9.0.6). Does Makkai-Reyes offer advantages over Awodey-Breiner there?

2. • CommentRowNumber2.
   • CommentAuthorjesse
   • CommentTimeJan 29th 2018
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   Author: jesse
   Format: MarkdownItexOne "unnatural" aspect of the Awodey-Forssell duality is that the equivalence between pretoposes and groupoids of models is only gotten by artificially restricting which continuous groupoid homomorphisms are allowed on the groupoid side (if I remember correctly, these are _defined_ to be ones which induce an inverse image functor between the toposes of equivariant sheaves sending coherent objects to coherent objects...) However, ultrafunctors between the ultracategories of models of T' and T correspond to interpretations T to T', full stop. According to Steve, one of the points of Spencer Breiner's thesis was to show that this was a feature instead of a bug: this definition is equivalent to requiring that the continuous groupoid homomorphism induce a map between their notion of structure sheaves. However, I haven't read Spencer's thesis as closely as I've read the Awodey-Forssell paper, so I can't yet give an opinion about that.

   One “unnatural” aspect of the Awodey-Forssell duality is that the equivalence between pretoposes and groupoids of models is only gotten by artificially restricting which continuous groupoid homomorphisms are allowed on the groupoid side (if I remember correctly, these are defined to be ones which induce an inverse image functor between the toposes of equivariant sheaves sending coherent objects to coherent objects…)

   However, ultrafunctors between the ultracategories of models of T’ and T correspond to interpretations T to T’, full stop.

   According to Steve, one of the points of Spencer Breiner’s thesis was to show that this was a feature instead of a bug: this definition is equivalent to requiring that the continuous groupoid homomorphism induce a map between their notion of structure sheaves. However, I haven’t read Spencer’s thesis as closely as I’ve read the Awodey-Forssell paper, so I can’t yet give an opinion about that.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 29th 2018
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   Author: David_Corfield
   Format: MarkdownItexRight. So that's being pointed out on slide 26 of Steve's [talk](http://schd.ws/hosted_files/clmps2015/d4/Lambek.pdf) and the resolution comes from the structure sheaves, as you say, part of what they call "affine logical schemes" (slide 31). Then Boolean pretoposes and affine logical schemes are contravariantly equivalent.

   Right. So that’s being pointed out on slide 26 of Steve’s talk and the resolution comes from the structure sheaves, as you say, part of what they call “affine logical schemes” (slide 31). Then Boolean pretoposes and affine logical schemes are contravariantly equivalent.