Fix spacing after 'e.g.' and 'i.e.'
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@ -61,7 +61,7 @@ $I$. I will sometimes refer to $P$ as the \nomenindex{path space} of some path $
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\Path\ P\ a\ b$. By this token $P\ 0$ then corresponds to the type at the
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left-endpoint and $P\ 1$ as the type at the right-endpoint. The type is called
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$\Path$ because it is connected with paths in homotopy theory. The intuition
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behind this is that $\Path$ describes paths in $\MCU$ -- i.e. between types. For
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behind this is that $\Path$ describes paths in $\MCU$ -- i.e.\ between types. For
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a path $p$ for the point $p\ i$ the index $i$ describes how far along the path
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one has moved. An inhabitant of $\Path\ P\ a_0\ a_1$ is a (dependent-) function,
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$p$, from the index-space to the path space:
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@ -104,7 +104,7 @@ With this definition we can also recover reflexivity. That is, for any $A \tp
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Here the path space is $P \defeq \lambda i \to A$ and it satsifies $P\ i = A$
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definitionally. So to inhabit it, is to give a path $I \to A$ which is
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judgmentally $a$ at either endpoint. This is satisfied by the constant path;
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i.e. the path that stays at $a$ at any index $i$.
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i.e.\ the path that stays at $a$ at any index $i$.
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It is also surpisingly easy to show functional extensionality with which we can
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construct a path between $f$ and $g$ -- the functions defined in the
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@ -207,7 +207,7 @@ Then comes the set of homotopical sets:
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\end{equation}
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%
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I will not give an example of a set at this point. It turns out that proving
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e.g. $\isProp\ \bN$ is not so straight-forward (see \cite[\S3.1.4]{hott-2013}).
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e.g.\ $\isProp\ \bN$ is not so straight-forward (see \cite[\S3.1.4]{hott-2013}).
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There will be examples of sets later in this report. At this point it should be
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noted that the term ``set'' is somewhat conflated; there is the notion of sets
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from set-theory, in Agda types are denoted \texttt{Set}. I will use it
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@ -15,7 +15,7 @@ formalized this common result about monads:
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By transporting this to the Kleisli formulation we get a result that we can use
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to compute with. This is particularly useful because the Kleisli formulation
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will be more familiar to programmers e.g. those coming from a background in
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will be more familiar to programmers e.g.\ those coming from a background in
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Haskell. Whereas the theory usually talks about monoidal monads.
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\TODO{Mention that with postulates we cannot do this}
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@ -3,7 +3,7 @@ I've written this as an appendix because 1) the aim of the thesis changed
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drastically from the planning report/proposal 2) partly I'm not sure how to
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structure my thesis.
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My work so far has very much focused on the formalization, i.e. coding. It's
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My work so far has very much focused on the formalization, i.e.\ coding. It's
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unclear to me at this point what I should have in the final report. Here I will
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describe what I have managed to formalize so far and what outstanding challenges
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I'm facing.
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@ -202,7 +202,7 @@ set.
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This example illustrates nicely how we can use these combinators to reason about
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`canonical' types like $\sum$ and $\prod$. Similar combinators can be defined at
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the other homotopic levels. These combinators are however not applicable in
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situations where we want to reason about other types - e.g. types we have
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situations where we want to reason about other types - e.g.\ types we have
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defined ourselves. For instance, after we have proven that all the projections
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of pre categories are propositions, then we would like to bundle this up to show
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that the type of pre categories is also a proposition. Formally:
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@ -247,7 +247,7 @@ $$
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So to give the continuous function $I \to \IsPreCategory$, which is our goal, we
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introduce $i \tp I$ and proceed by constructing an element of $\IsPreCategory$
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by using the fact that all the projections are propositions to generate paths
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between all projections. Once we have such a path e.g. $p \tp a.\isIdentity
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between all projections. Once we have such a path e.g.\ $p \tp a.\isIdentity
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\equiv b.\isIdentity$ we can eliminate it with $i$ and thus obtain $p\ i \tp
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(p\ i).\isIdentity$. This element satisfies exactly that it corresponds to the
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corresponding projections at either endpoint. Thus the element we construct at
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@ -436,7 +436,7 @@ That is, we must demonstrate that there is an isomorphism (on types) between
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equalities and isomorphisms (on arrows). It is worthwhile to dwell on this for a
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few seconds. This type looks very similar to univalence for types and is
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therefore perhaps a bit more intuitive to grasp the implications of. Of course
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univalence for types (which is a proposition -- i.e. provable) does not imply
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univalence for types (which is a proposition -- i.e.\ provable) does not imply
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univalence of all pre category since morphisms in a category are not regular
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functions -- in stead they can be thought of as a generalization hereof. The univalence criterion therefore is simply a way of restricting arrows
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to behave similarly to maps.
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@ -551,7 +551,7 @@ same as the one in the underlying category (they have the same type). Function
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composition will be reverse function composition from the underlying category.
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I will refer to things in terms of the underlying category, unless they have an
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over-bar. So e.g. $\idToIso$ is a function in the underlying category and the
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over-bar. So e.g.\ $\idToIso$ is a function in the underlying category and the
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corresponding thing is denoted $\wideoverbar{\idToIso}$ in the opposite
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category.
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@ -1333,7 +1333,7 @@ f \rrr (\bind\ g)$ . (\TODO{Better way to typeset $\fish$?})
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\subsection{Equivalence of formulations}
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%
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The notation I have chosen here in the report
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overloads e.g. $\pure$ to both refer to a natural transformation and an arrow.
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overloads e.g.\ $\pure$ to both refer to a natural transformation and an arrow.
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This is of course not a coincidence as the arrow in the Kleisli formulation
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shall correspond exactly to the map on arrows from the natural transformation
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called $\pure$.
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@ -185,16 +185,16 @@ will make it possible to prove more things and to reuse proofs and to try and
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compare some aspects of this formalization with the existing ones.\TODO{How can
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I live up to this?}
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There are alternative approaches to working in a cubical setting where one can
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still have univalence and functional extensionality. One option is to postulate
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these as axioms. This approach, however, has other shortcomings, e.g.; you lose
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\nomenindex{canonicity} (\TODO{Pageno!} \cite{huber-2016}). Canonicity means that any
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well typed term evaluates to a \emph{canonical} form. For example for a closed
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term $e \tp \bN$ it will be the case that $e$ reduces to $n$ applications of
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$\mathit{suc}$ to $0$ for some $n$; $e = \mathit{suc}^n\ 0$. Without canonicity
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terms in the language can get ``stuck'' -- meaning that they do not reduce to a
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canonical form.
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There are alternative approaches to working in a cubical setting where
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one can still have univalence and functional extensionality. One
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option is to postulate these as axioms. This approach, however, has
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other shortcomings, e.g. you lose \nomenindex{canonicity}
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(\TODO{Pageno!} \cite{huber-2016}). Canonicity means that any well
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typed term evaluates to a \emph{canonical} form. For example for a
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closed term $e \tp \bN$ it will be the case that $e$ reduces to $n$
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applications of $\mathit{suc}$ to $0$ for some $n$; $e =
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\mathit{suc}^n\ 0$. Without canonicity terms in the language can get
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``stuck'' -- meaning that they do not reduce to a canonical form.
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Another approach is to use the \emph{setoid interpretation} of type
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theory (\cite{hofmann-1995,huber-2016}). With this approach one works
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@ -0,0 +1,236 @@
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\documentclass[a4paper,handout]{beamer}
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\input{packages.tex}
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\input{macros.tex}
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\title{Univalent Categories}
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\author{Frederik Hangh{\o}j Iversen}
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\institute{Chalmers University of Technology}
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\begin{document}
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\frame{\titlepage}
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\begin{frame}
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\frametitle{Motivating example}
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\framesubtitle{Functional extensionality}
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Consider the functions
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\begin{align*}
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\var{zeroLeft} & \defeq (n \tp \bN) \mto (0 + n \tp \bN) \\
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\var{zeroRight} & \defeq (n \tp \bN) \mto (n + 0 \tp \bN)
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\end{align*}
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\pause
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We have
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%
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$$
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\prod_{n \tp \bN} n + 0 \equiv 0 + n
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$$
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%
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\pause
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But not
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%
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$$
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\var{zeroLeft} \equiv \var{zeroRight}
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$$
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%
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\pause
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We need
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%
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$$
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\funExt \tp \prod_{a \tp A} f\ a \equiv g\ a \to f \equiv g
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$$
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\end{frame}
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\begin{frame}
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\frametitle{Motivating example}
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\framesubtitle{Univalence}
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Consider the set
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$\{x \mid \phi\ x \land \psi\ x\}$
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\pause
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If we show $\forall x . \psi\ x \equiv \top$
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then we want to conclude
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$\{x \mid \phi\ x \land \psi\ x\} \equiv \{x \mid \phi\ x\}$
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\pause
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We need univalence:
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$$(A \simeq B) \simeq (A \equiv B)$$
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\pause
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%
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We will return to $\simeq$, but for not, think of it as an
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isomorphism, so it induces maps:
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\begin{align*}
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\var{toPath} & \tp (A \simeq B) \to (A \equiv B) \\
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\var{toEquiv} & \tp (A \equiv B) \to (A \simeq B)
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\end{align*}
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\end{frame}
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\begin{frame}
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\frametitle{Paths}
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\framesubtitle{Definition}
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Heterogeneous paths
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\begin{equation*}
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\Path \tp (P \tp I → \MCU) → P\ 0 → P\ 1 → \MCU
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\end{equation*}
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\pause
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For $P \tp I \to \MCU$, $A \tp \MCU$ and $a_0, a_1 \tp A$
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inhabitants of $\Path\ P\ a_0\ a_1$ are like functions
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%
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$$
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p \tp \prod_{i \tp I} P\ i
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$$
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%
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Which satisfy $p\ 0 & = a_0$ and $p\ 1 & = a_1$
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\pause
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Homogenous paths
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$$
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a_0 \equiv a_1 \defeq \Path\ (\var{const}\ A)\ a_0\ a_1
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$$
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\end{frame}
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\begin{frame}
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\frametitle{Paths}
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\framesubtitle{Functional extenstionality}
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$$
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\funExt & \tp \prod_{a \tp A} f\ a \equiv g\ a \to f \equiv g
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$$
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\pause
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$$
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\funExt\ p \defeq λ i\ a → p\ a\ i
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$$
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\pause
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$$
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\funExt\ (\var{const}\ \refl)
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\tp
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\var{zeroLeft} \equiv \var{zeroRight}
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$$
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\end{frame}
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\begin{frame}
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\frametitle{Paths}
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\framesubtitle{Homotopy levels}
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\begin{align*}
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& \isContr && \tp \MCU \to \MCU \\
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& \isContr\ A && \defeq \sum_{c \tp A} \prod_{a \tp A} a \equiv c
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\end{align*}
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\pause
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\begin{align*}
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& \isProp && \tp \MCU \to \MCU \\
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& \isProp\ A && \defeq \prod_{a_0, a_1 \tp A} a_0 \equiv a_1
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\end{align*}
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\pause
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\begin{align*}
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& \isSet && \tp \MCU \to \MCU \\
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& \isSet\ A && \defeq \prod_{a_0, a_1 \tp A} \isProp\ (a_0 \equiv a_1)
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\end{align*}
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\pause
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\end{frame}
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\begin{frame}
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\frametitle{Paths}
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\framesubtitle{A few lemmas}
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Let $D$ be a type-family:
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$$
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D \tp \prod_{b \tp A} \prod_{p \tp a ≡ b} \MCU
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$$
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%
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\pause
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And $d$ and in inhabitant of $D$ at $\refl$:
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%
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$$
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d \tp D\ a\ \refl
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$$
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%
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\pause
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We then have the function:
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%
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\begin{equation}
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\pathJ\ D\ d \tp \prod_{b \tp A} \prod_{p \tp a ≡ b} D\ a\ p
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\end{equation}
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\end{frame}
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\begin{frame}
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\frametitle{Paths}
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\framesubtitle{A few lemmas}
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Given
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\begin{align*}
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A & \tp \MCU \\
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P & \tp A \to \MCU \\
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\var{propP} & \tp \prod_{x \tp A} \isProp\ (P\ x) \\
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p & \tp a_0 \equiv a_1 \\
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p_0 & \tp P\ a_0 \\
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p_1 & \tp P\ a_1
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\end{align*}
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%
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We have
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$$
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\lemPropF\ \var{propP}\ p
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\tp
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\Path\ (\lambda\; i \mto P\ (p\ i))\ p_0\ p_1
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$$
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%
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\end{frame}
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\begin{frame}
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\frametitle{Paths}
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\framesubtitle{A few lemmas}
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$\prod$ preserves $\isProp$:
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$$
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\mathit{propPi}
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\tp
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\left(\prod_{a \tp A} \isProp\ (P\ a)\right)
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\to \isProp\ \left(\prod_{a \tp A} P\ a\right)
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$$
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\pause
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$\sum$ preserves $\isProp$:
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$$
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\mathit{propSig} \tp \isProp\ A \to \left(\prod_{a \tp A} \isProp\ (P\ a)\right) \to \isProp\ \left(\sum_{a \tp A} P\ a\right)
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$$
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\end{frame}
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\begin{frame}
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\frametitle{Categories}
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\framesubtitle{Definition}
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Data:
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\begin{align*}
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\Object & \tp \Type \\
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\Arrow & \tp \Object \to \Object \to \Type \\
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\identity & \tp \Arrow\ A\ A \\
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\lll & \tp \Arrow\ B\ C \to \Arrow\ A\ B \to \Arrow\ A\ C
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\end{align*}
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%
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Laws:
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%
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$$
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h \lll (g \lll f) ≡ (h \lll g) \lll f
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$$
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$$
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\identity \lll f ≡ f \x
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f \lll \identity ≡ f
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$$
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\pause
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1-categories:
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$$
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\isSet\ (\Arrow\ A\ B)
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$$
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\pause
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Univalent categories:
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$$
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\isEquiv\ (A \equiv B)\ (A \approxeq B)\ \idToIso
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$$
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\end{frame}
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\begin{frame}
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\frametitle{Categories}
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\framesubtitle{Univalence}
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\begin{align*}
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\var{IsIdentity} & \defeq
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\prod_{A\ B \tp \Object} \prod_{f \tp \Arrow\ A\ B} \phi\ f
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%% \\
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%% & \mathrel{\ } \identity \lll f \equiv f \x f \lll \identity \equiv f
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\end{align*}
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where
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$$
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\phi\ f \defeq \identity \lll f \equiv f \x f \lll \identity \equiv f
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$$
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Let $\approxeq$ denote ismorphism of objects. We can then construct
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the identity isomorphism in any category:
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$$
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\identity , \identity , \var{isIdentity} \tp A \approxeq A
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$$
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Likewise since paths are substitutive we can promote a path to an isomorphism:
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$$
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\idToIso \tp A ≡ B → A ≊ B
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$$
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For a category to be univalent we require this to be an equivalence:
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\end{frame}
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\end{document}
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@ -0,0 +1,95 @@
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%% FRONTMATTER
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\frontmatter
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%% \newgeometry{top=3cm, bottom=3cm,left=2.25 cm, right=2.25cm}
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\begingroup
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\thispagestyle{empty}
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{\Huge\thetitle}\\[.5cm]
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{\Large A formalization of category theory in Cubical Agda}\\[6cm]
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\begin{center}
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\includegraphics[width=\linewidth,keepaspectratio]{isomorphism.png}
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\end{center}
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% Cover text
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\vfill
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%% \renewcommand{\familydefault}{\sfdefault} \normalfont % Set cover page font
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{\Large\theauthor}\\[.5cm]
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Master's thesis in Computer Science
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\endgroup
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%% \end{titlepage}
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% BACK OF COVER PAGE (BLANK PAGE)
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\newpage
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%% \newgeometry{a4paper} % Temporarily change margins
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%% \restoregeometry
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\thispagestyle{empty}
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\null
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%% \begin{titlepage}
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% TITLE PAGE
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\newpage
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\thispagestyle{empty}
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\begin{center}
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\textsc{\LARGE Master's thesis \the\year}\\[4cm] % Report number is currently not in use
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\textbf{\LARGE \thetitle} \\[1cm]
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{\large \subtitle}\\[1cm]
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{\large \theauthor}
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\vfill
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\centering
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\includegraphics[width=0.2\pdfpagewidth]{logo_eng.pdf}
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\vspace{5mm}
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\textsc{Department of Computer Science and Engineering}\\
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\textsc{{\researchgroup}}\\
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%Name of research group (if applicable)\\
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\textsc{\institution} \\
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\textsc{Gothenburg, Sweden \the\year}\\
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\end{center}
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% IMPRINT PAGE (BACK OF TITLE PAGE)
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\newpage
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\thispagestyle{plain}
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\textit{\thetitle}\\
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\subtitle\\
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\copyright\ \the\year ~ \textsc{\theauthor}
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\vspace{4.5cm}
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\setlength{\parskip}{0.5cm}
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\textbf{Author:}\\
|
||||
\theauthor\\
|
||||
\href{mailto:\authoremail>}{\texttt{<\authoremail>}}
|
||||
|
||||
\textbf{Supervisor:}\\
|
||||
\supervisor\\
|
||||
\href{mailto:\supervisoremail>}{\texttt{<\supervisoremail>}}\\
|
||||
\supervisordepartment
|
||||
|
||||
\textbf{Co-supervisor:}\\
|
||||
\cosupervisor\\
|
||||
\href{mailto:\cosupervisoremail>}{\texttt{<\cosupervisoremail>}}\\
|
||||
\cosupervisordepartment
|
||||
|
||||
\textbf{Examiner:}\\
|
||||
\examiner\\
|
||||
\href{mailto:\examineremail>}{\texttt{<\examineremail>}}\\
|
||||
\examinerdepartment
|
||||
|
||||
\vfill
|
||||
Master's Thesis \the\year\\ % Report number currently not in use
|
||||
\department\\
|
||||
%Division of Division name\\
|
||||
%Name of research group (if applicable)\\
|
||||
\institution\\
|
||||
SE-412 96 Gothenburg\\
|
||||
Telephone +46 31 772 1000 \setlength{\parskip}{0.5cm}\\
|
||||
% Caption for cover page figure if used, possibly with reference to further information in the report
|
||||
%% Cover: Wind visualization constructed in Matlab showing a surface of constant wind speed along with streamlines of the flow. \setlength{\parskip}{0.5cm}
|
||||
%Printed by [Name of printing company]\\
|
||||
Gothenburg, Sweden \the\year
|
||||
|
||||
%% \restoregeometry
|
||||
%% \end{titlepage}
|
||||
|
||||
\tableofcontents
|
||||
Loading…
Reference in New Issue