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1. • CommentRowNumber1.
   • CommentAuthorkyod
   • CommentTimeNov 1st 2018
   • PermaLink
   Author: kyod
   Format: MarkdownItexDefinition of extensional $Pi$-type structure taken from _Natural models of homotopy type theory_ Should we develop how to get application, $\beta$ and $\eta$ here or should we leave it to the interpretation ? <a href="https://ncatlab.org/nlab/revision/Initiality+Project+-+Semantics+-+Pi-types/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Initiality+Project+-+Semantics+-+Pi-types">current</a>

   Definition of extensional $Pi$-type structure taken from Natural models of homotopy type theory

   Should we develop how to get application, $\beta$ and $\eta$ here or should we leave it to the interpretation ?

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeNov 1st 2018
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI think it would be best to give the definitions of this structure in as close a manner as possible to what we'll want for the interpretation.

   I think it would be best to give the definitions of this structure in as close a manner as possible to what we’ll want for the interpretation.

3. • CommentRowNumber3.
   • CommentAuthorAli Caglayan
   • CommentTimeNov 1st 2018
   • PermaLink
   Author: Ali Caglayan
   Format: MarkdownItexIm finding this hard to parse, its the same problem with Awodey’s paper too. Is this a difficult concept to grasp or to parse?

   Im finding this hard to parse, its the same problem with Awodey’s paper too. Is this a difficult concept to grasp or to parse?

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeNov 1st 2018
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexIt makes somewhat more sense when expressed in the internal extensional type theory of $[C^{op},Set]$.

   It makes somewhat more sense when expressed in the internal extensional type theory of $[C^{op},Set]$.

5. • CommentRowNumber5.
   • CommentAuthormaxsnew
   • CommentTimeNov 1st 2018
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   Author: maxsnew
   Format: MarkdownItexIt helped me to first take Proposition 5 in Awodey as the definition of the operation $-^{\Pi}$, and then revisit the concrete description as a polynomial functor. That is that for a preasheaf $X$ on $C$, $X^{\Pi}$ is given on objects by $X^{\Pi}(\Gamma) = \Sigma_{A : Ty(\Gamma)} X(\Gamma, A)$, i.e., it is the presheaf of "$X$s in an extended context". Then the typing of $\Pi$ and $\lambda$ make a lot of sense and unwinding the pullback tells you that $\lambda$ has $\Pi$ type plus the $\beta,\eta$ laws.

   It helped me to first take Proposition 5 in Awodey as the definition of the operation $-^{\Pi}$, and then revisit the concrete description as a polynomial functor.

   That is that for a preasheaf $X$ on $C$, $X^{\Pi}$ is given on objects by $X^{\Pi}(\Gamma) = \Sigma_{A : Ty(\Gamma)} X(\Gamma, A)$, i.e., it is the presheaf of “$X$s in an extended context”. Then the typing of $\Pi$ and $\lambda$ make a lot of sense and unwinding the pullback tells you that $\lambda$ has $\Pi$ type plus the $\beta,\eta$ laws.

6. • CommentRowNumber6.
   • CommentAuthorkyod
   • CommentTimeNov 2nd 2018
   • PermaLink
   Author: kyod
   Format: MarkdownItexRe #2 I think it matches rather closely : For instance if $T$ is a $\Pi$-type $\Pi(x:A).B$, then by inductive hypothesis, $[\![A]\!]_{type}^V : Tm^V \rightharpoonup Ty$, wlog. we can pick $x$ fresh for $V$, that is $x \notin V$ and $[\![B]\!]_{type}^{V\cup\{x\}} : Tm^{V \cup \{x\}} \rightharpoonup Ty$. By restricting the domain of $[\![B]\!]_{type}^{V\cup\{x\}}$ and currying we get $F : Tm^V \rightharpoonup Ty^\mathcal{A}$ where $\mathcal{A}$ is the fiber of $[\![A]\!]_{type}^V $ along $of : Tm \to Ty$ (when this makes sense, $F$ being undefined otherwise). Then $$[\![\Pi(x:A).B]\!]_{type}^{V} = \Pi \circ \langle [\![A]\!]_{type}^V, F\rangle : Tm^V \rightharpoonup Ty$$ I think it also works rather well for $\lambda$ and $App$, my only problem before starting to write the partial interpretation is that I have trouble with writing down a more precise definition of $F$ above.

   Re #2 I think it matches rather closely :

   For instance if $T$ is a $\Pi$-type $\Pi(x:A).B$, then by inductive hypothesis, $[\![A]\!]_{type}^V : Tm^V \rightharpoonup Ty$, wlog. we can pick $x$ fresh for $V$, that is $x \notin V$ and $[\![B]\!]_{type}^{V\cup\{x\}} : Tm^{V \cup \{x\}} \rightharpoonup Ty$. By restricting the domain of $[\![B]\!]_{type}^{V\cup\{x\}}$ and currying we get $F : Tm^V \rightharpoonup Ty^\mathcal{A}$ where $\mathcal{A}$ is the fiber of $[\![A]\!]_{type}^V$ along $of : Tm \to Ty$ (when this makes sense, $F$ being undefined otherwise). Then

   $$[\![\Pi(x:A).B]\!]_{type}^{V} = \Pi \circ \langle [\![A]\!]_{type}^V, F\rangle : Tm^V \rightharpoonup Ty$$

   I think it also works rather well for $\lambda$ and $App$, my only problem before starting to write the partial interpretation is that I have trouble with writing down a more precise definition of $F$ above.

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeNov 2nd 2018
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   Author: Mike Shulman
   Format: MarkdownItexYes, it's pretty close, but I think it would match even more closely if we state the categorical definition using application and abstraction morphisms. E.g. I'm thinking of translating Definition 3.4.2.1 of the [local universes paper](https://arxiv.org/abs/1411.1736) into the language of $[C^{op},Set]$.

   Yes, it’s pretty close, but I think it would match even more closely if we state the categorical definition using application and abstraction morphisms. E.g. I’m thinking of translating Definition 3.4.2.1 of the local universes paper into the language of $[C^{op},Set]$.

8. • CommentRowNumber8.
   • CommentAuthorkyod
   • CommentTimeNov 12th 2018
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   Author: kyod
   Format: MarkdownItexReformulating $Ty^\Pi$ and $Tm^\Pi$ using @maxsnew version that seems more understandable. Re #7 I looked at the definition in that paper but I couldn't come up with something that seemed better than the current one. Also I have the impression that providing a natural transformation for application correspond to giving eagerly the denotation of $$\Gamma, x:A, f : \Pi(x:A)B \vdash App^{x:A.B}(f, x) : B$$ <a href="https://ncatlab.org/nlab/revision/diff/Initiality+Project+-+Semantics+-+Pi-types/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/Initiality+Project+-+Semantics+-+Pi-types/2">v2</a>, <a href="https://ncatlab.org/nlab/show/Initiality+Project+-+Semantics+-+Pi-types">current</a>

   Reformulating $Ty^\Pi$ and $Tm^\Pi$ using @maxsnew version that seems more understandable.

   Re #7 I looked at the definition in that paper but I couldn’t come up with something that seemed better than the current one. Also I have the impression that providing a natural transformation for application correspond to giving eagerly the denotation of $$\Gamma, x:A, f : \Pi(x:A)B \vdash App^{x:A.B}(f, x) : B$$

   diff, v2, current

9. • CommentRowNumber9.
   • CommentAuthorAli Caglayan
   • CommentTimeDec 22nd 2018
   • (edited Dec 22nd 2018)
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   Author: Ali Caglayan
   Format: MarkdownItexHow can this be phrased talking about the category of elements? This is some very confusing category theory. If I am understanding correctly the functor $P^\Pi : [C^op,Set] \to [C^op,Set]$ is the composition of three functors. Firstly we have the pullback along $Tm \to 1$ usually denoted $Tm^*$, then we have the right adjoint to pullback along $of$, which I will call $\prod_of$ and finally composition along $Ty \to 1$ giving us our functor $P^\Pi$. Then the $\Pi$-structure on a CwF is two maps $\lambda$ and $\Pi$ that satisfy the square.

   How can this be phrased talking about the category of elements? This is some very confusing category theory.

   If I am understanding correctly the functor $P^\Pi : [C^op,Set] \to [C^op,Set]$ is the composition of three functors. Firstly we have the pullback along $Tm \to 1$ usually denoted $Tm^*$, then we have the right adjoint to pullback along $of$, which I will call $\prod_of$ and finally composition along $Ty \to 1$ giving us our functor $P^\Pi$. Then the $\Pi$-structure on a CwF is two maps $\lambda$ and $\Pi$ that satisfy the square.