Put in brackets for readability
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@ -109,7 +109,7 @@ definitions: If we let $p$ be a witness to the identity law, which formally is:
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\label{eq:identity}
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\var{IsIdentity} \defeq
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\prod_{A\ B \tp \Object} \prod_{f \tp \Arrow\ A\ B}
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\id \lll f \equiv f \x f \lll \id \equiv f
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\left(\id \lll f \equiv f\right) \x \left(f \lll \id \equiv f\right)
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\end{equation}
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%
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Then we can construct the identity isomorphism $\idIso \tp \identity,
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@ -160,8 +160,13 @@ And laws:
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%% \tag{associativity}
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h \lll (g \lll f) ≡ (h \lll g) \lll f \\
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%% \tag{identity}
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\identity \lll f ≡ f \x
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\left(
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\identity \lll f ≡ f
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\right)
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\x
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\left(
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f \lll \identity ≡ f
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\right)
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\\
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\label{eq:arrows-are-sets}
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%% \tag{arrows are sets}
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@ -201,7 +206,7 @@ So the proof goes like this: We `eliminate' the 3 function abstractions by
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applying $\propPi$ three times. So our proof obligation becomes:
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%
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$$
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\isProp\ \left( \id \comp f \equiv f \x f \comp \id \equiv f \right)
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\isProp\ \left( \left( \id \comp f \equiv f \right) \x \left( f \comp \id \equiv f \right) \right)
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$$
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%
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Then we eliminate the (non-dependent) sigma-type by applying $\propSig$ giving
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@ -360,7 +365,7 @@ The usual notion of a function $f \tp A \to B$ having an inverses is:
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%
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\begin{equation}
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\label{eq:isomorphism}
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\sum_{g \tp B \to A} f \comp g \equiv \identity \x g \comp f \equiv \identity
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\sum_{g \tp B \to A} \left( f \comp g \equiv \identity \right) \x \left( g \comp f \equiv \identity \right)
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\end{equation}
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%
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This is defined in \cite[p. 129]{hott-2013} where it is referred to as the a
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@ -608,7 +613,7 @@ An inhabitant of $A \approxeq B$ is simply an arrow $f \tp \Arrow\ A\ B$
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and its inverse $g \tp \Arrow\ B\ A$. In the opposite category $g$ will
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play the role of the isomorphism and $f$ will be the inverse. Similarly we can
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go in the opposite direction. I name these maps $\shufflef \tp (A \approxeq
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B) \to (A \wideoverbar{\approxeq} B)$ and $\shuffle^{-1} \tp (A
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B) \to (A \wideoverbar{\approxeq} B)$ and $\shufflef^{-1} \tp (A
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\wideoverbar{\approxeq} B) \to (A \approxeq B)$ respectively.
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As the inverse of $\wideoverbar{\idToIso}$ I will pick $\wideoverbar{\isoToId}
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@ -648,8 +653,8 @@ $$
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$$
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%
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As we have seen the laws in $\left(\bC^{\var{Op}}\right)^{\var{Op}}$ get quite
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involved.\footnote{We have not even seen the full story because we have used
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this `interface' for equivalences.} Luckily since being-a-category is a mere
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involved\footnote{We have not even seen the full story because we have used
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this `interface' for equivalences.}. Luckily since being-a-category is a mere
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proposition, we need not concern ourselves with this bit when proving the above.
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We can use the equality principle for categories that let us prove an equality
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just by giving an equality on the data-part. So, given a category $\bC$ all we
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@ -798,13 +803,13 @@ inverse to $f$. The path $p$ is inhabited by:
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%
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\begin{align*}
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x
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& \equiv x \comp \identity \\
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& = x \comp \identity \\
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& \equiv x \comp (f \comp y)
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&& \text{$y$ is an inverse to $f$} \\
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& \equiv (x \comp f) \comp y \\
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& \equiv \identity \comp y
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&& \text{$x$ is an inverse to $f$} \\
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& \equiv y
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& = y
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\end{align*}
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%
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For the other (dependent) path we can prove that being-an-inverse-of is a
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@ -1061,10 +1066,11 @@ the implementation for the details).
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\emph{Proposition} \ref{eq:univ-1} is isomorphic to \ref{eq:univ-2}:
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This proof of this has been omitted but can be found in the module:
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\begin{center}
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%
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\begin{center}%
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\sourcelink{Cat.Categories.Span}
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\end{center}
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%
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\emph{Proposition} \ref{eq:univ-2} is isomorphic to \ref{eq:univ-3}: For this I
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will show two corollaries of \ref{eq:coeCod}: For an isomorphism $(\iota,
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\inv{\iota}, \var{inv}) \tp A \cong B$, arrows $f \tp \Arrow\ A\ X$, $g \tp
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@ -1108,7 +1114,7 @@ To show that this choice fits the bill I must now verify that it satisfies
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\ref{eq:pairArrowLaw}, which in this case becomes:
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%
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\begin{align}
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y_{\mathcal{A}} \lll f ≡ x_{\mathcal{A}} × y_{\mathcal{B}} \lll f ≡ x_{\mathcal{B}}
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(y_{\mathcal{A}} \lll f ≡ x_{\mathcal{A}}) × (y_{\mathcal{B}} \lll f ≡ x_{\mathcal{B}})
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\end{align}
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%
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Which, since $f$ is an isomorphism and $p_{\mathcal{A}}$ (resp.\ $p_{\mathcal{B}}$)
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@ -1217,10 +1223,11 @@ indeed a product. That is, for an object $Y$ and two arrows $y_𝒜 \tp
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%
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\begin{align}
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\label{eq:pairCondRev}
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\begin{split}
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x_𝒜 \lll f & ≡ y_𝒜 \\
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x_ℬ \lll f & ≡ y_ℬ
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\end{split}
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%% \begin{split}
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( x_𝒜 \lll f ≡ y_𝒜 )
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\x
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( x_ℬ \lll f ≡ y_ℬ )
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%% \end{split}
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\end{align}
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%
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Since $X, x_𝒜, x_ℬ$ is a terminal object there is a \emph{unique} arrow from
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@ -1250,8 +1257,9 @@ of proofs:
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Here $\Phi$ is given as:
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$$
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\prod_{f \tp \Arrow\ Y\ X}
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x_𝒜 \lll f ≡ y_𝒜
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× x_ℬ \lll f ≡ y_ℬ
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( x_𝒜 \lll f ≡ y_𝒜 )
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×
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( x_ℬ \lll f ≡ y_ℬ )
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$$
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%
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$p$ follows from the universal property of $y_𝒜 \x y_ℬ$. For the latter we will
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@ -1260,8 +1268,9 @@ more general result:
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%
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$$
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\prod_{f \tp \Arrow\ Y\ X} \isProp\ (
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x_𝒜 \lll f ≡ y_𝒜
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× x_ℬ \lll f ≡ y_ℬ
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( x_𝒜 \lll f ≡ y_𝒜 )
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×
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( x_ℬ \lll f ≡ y_ℬ )
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)
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$$
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%
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@ -90,8 +90,8 @@
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\newunicodechar{⟨}{\textfallback{⟨}}
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\newunicodechar{⟩}{\textfallback{⟩}}
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\newunicodechar{∎}{\textfallback{∎}}
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\newunicodechar{𝒜}{\textfallback{?}}
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\newunicodechar{ℬ}{\textfallback{?}}
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%% \newunicodechar{𝒜}{\textfallback{𝒜}}
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%% \newunicodechar{ℬ}{\textfallback{ℬ}}
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%% \newunicodechar{≊}{\textfallback{≊}}
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\makeatletter
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\newcommand*{\rom}[1]{\expandafter\@slowroman\romannumeral #1@}
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