lectures.tex

\documentclass[a4paper, 11pt]{article}
\newcommand\hmmax{0}
\newcommand\bmmax{0}
%\input{preamble-fancy-very-minimal-newpxtext2}
\input{preamble-fancy-very-minimal-opentype-libertinus}
%\input{preamble}
\usepackage[textwidth=15.5cm,top=2.5cm,bottom=2.5cm]{geometry}
\usepackage{comment}

\title{Lecture Notes}
\author{on assorted topics}
\date{Carl-Fredrik Lidgren}

\defaultfontfeatures[HaranoAjiMincho]{
    Renderer=HarfBuzz,
    Script=Kana,
    Scale=MatchUppercase,
    UprightFont=*-Light,
    BoldFont=*-Medium,
    Extension=.otf
}

\newfontfamily\kanafont{HaranoAjiMincho}
\DeclareRobustCommand\yo{\textup{\kanafont よ}}

\newcommand{\1}{\ensuremath{\mathbb{1}}}
\newcommand{\2}{\ensuremath{\mathbb{2}}}
\newcommand{\3}{\ensuremath{\mathbb{3}}}
\newcommand{\n}{\ensuremath{\mathbb{n}}}
\newcommand{\I}{\ensuremath{\mathbb{I}}}
\newcommand{\Isofib}{\ensuremath{\mathsf{Isofib}}}

\newcommand{\Prob}{\operatorname{Prob}}
\newcommand{\Aff}{\operatorname{Aff}}

\newcommand{\gMod}{\cat{gMod}}
\newcommand{\ugMod}{\underline{\gMod}}
\newcommand{\bgMod}{\cat{bgMod}}
\newcommand{\dgMod}{\cat{dgMod}}
\newcommand{\udgMod}{\underline{\dgMod}}
\newcommand{\dbgMod}{\cat{dbgMod}}

\newcommand{\ladj}{\vdash}
\newcommand{\fib}{\operatorname{fib}}
\newcommand{\cof}{\operatorname{cof}}
\newcommand{\cofib}{\operatorname{cofib}}
\newcommand{\Tot}{\operatorname{Tot}}

\newcommand{\bp}[1]{\prescript{p}{}{#1}}
\newcommand{\bperp}[1]{\prescript{\perp}{}{#1}}

\usepackage[backend=biber, style=alphabetic]{biblatex}
\addbibresource{bibliography.bib}

\includeonly{lectures/graded-algebras,lectures/dg-algebras}

\begin{document}
\maketitle


\tableofcontents

\clearpage
\setcounter{section}{-1}
\section{Welcome}
\subsection{Introduction}
This document contains an assortment of lecture notes prepared initially for use
in lectures of 2--3 hours, focused at a listener with a decent amount of
mathematical maturity and some experience with category theory, algebra,
topology, and so on. There is no overarching goal, other than to cover
interesting topics in and around category theory, homological algebra, geometry,
higher category theory, and higher algebra.

Very often, the notes have been prepared without too much care, and so one should expect mistakes, occasional hand-waving, and sub-optimal approaches. Hopefully, all the essential ideas
remain correct. On a related note, while there has been some effort put into making the exposition clear, we are sometimes rather terse as it makes it easier to use the notes
for lectures, where the exposition may be expanded upon in any case.

References are not always carefully tracked. If in doubt, assume no originality.

\subsection{Foundations}
In this preliminary section, we spell out the general foundational framework we adopt in these notes. Since foundations will never play a \emph{huge} role in what we do, or at least
it won't affect how we do things generally, we will not be too careful. However, the crux of it is as follows: we adopt your favourite common axiomatics for set theory, such as ZFC, along with
Grothendieck's universe axiom. A (Grothendieck) universe is essentially a model for ZF set theory within ZFC set theory itself, so it is a set wherein one can do all basic expected operations (and contains \(\N\)).
The universe axiom postulates that every set is contained in a universe. In particular, we may pick some universe \(\bbU_1\ni\varnothing\), the elements of which we call the \emph{small sets.} Iterating this, the axiom
also implies the existence of a hierarchy of universes
\[ \bbU_1 \in \bbU_2 \in \bbU_3\in \cdots \in \bbU_n\in\cdots \]
and we call elements of \(\bbU_2\) \emph{large,} elements of \(\bbU_3\) \emph{very large,} and so on. If we have picked some universe \(\bbU\), we may also refer to a set \(X\) as \(\bbU\)-small to mean that \(X\in\bbU\), or
that \(X\) is in bijection with a set in \(\bbU\). Note that we will essentially never need to use this nor think about it in any detail. To be noted as well is that in this framework, the notion of a \emph{class} is subsumed
by the notion of a set, in the sense that we may think of elements of \(\bbU_2\backslash\bbU_1\) as analogous to ``proper classes.'' If one wishes, type theory can play a similar role, where one demands a hierarchy of type universes instead.

\subsection{Planned contents}
Some planned contents already have a rough lecture section assigned to them, but some others are a lot looser and so are hard to pin down enough for that to be reasonable, and may span many lectures.
Here is a list to which I aspire.
\begin{enumerate}[label=(\arabic*)]
\item The theory of \((\infty,1)\)-categories, blending the standard approaches (say, \cite{lurie-htt} and \cite{cisinski-book}) with the synthetic approach of \cite{riehl-verity-elements}.
\item Sheaves on sites, probably following \cite{kashiwara-schapira-book}. Relating sheaves between sites will be a focus. Maybe something from \cite[§6.2.2]{lurie-htt} and \cite[§1.3.1]{lurie-sag}
to include \(\infty\)-categorical aspects.
\item Approximable triangulated categories.
\item Perverse sheaves, as an application of the gluing of t-structures.
\item Brown representability.
\item Topics surrounding model categories: Bousfield localizations, stable model categories, algebraic small object argument?
\item Operads, due to their importance in understanding e.g. \(\E_n\)-rings and their modules.
\item Huber's adic spaces, as a ``reasonably elementary'' formalism for rigid analytic geometry.
\item Condensed sets, with a focus on eventually building up to condensed analytic geometry (via Clausen \& Scholze's analytic stacks).
\item \(K\)-theory, particularly of stable \(\infty\)-categories or Waldhausen \(\infty\)-categories. Perhaps recent research extending the \(K\)-theory machine to certain large categories (dualizable ones), à la Efimov.
\item Waldhausen categories, i.e.\ categories with cofibrations and weak equivalences.
\item ``Lower'' higher category theory, so 2-categories and bicategories. (Virtual) double categories, for their applications to \(\infty\)-cosmoi.
\item Topoi and \(\infty\)-topoi.
\item The theorem of Hoshino--Kato--Miyachi on t-structures generated by silting objects. Also requires the computation of \(\Ext^1\) in the heart of a t-structure
in terms of the Hom-sets in the ambient triangulated category.
\item Witt vectors.
\item Tensor triangular geometry.
\end{enumerate}

\include{lectures/abelian-categories}

\include{lectures/basics-of-homological-algebra}

\include{lectures/monadicity}

\include{lectures/more-about-abelian-categories}

\include{lectures/localizations-of-categories}

\include{lectures/homotopical-algebra-through-deformations}

\include{lectures/triangulated-categories}

\include{lectures/localizations-of-triangulated-categories}

\include{lectures/localization-sequences-recollements-triangulated-categories}

\include{lectures/abstract-cohomology-through-t-structures}

\include{lectures/gluing-t-structures}

\include{lectures/ind-categories-and-compactness}

\include{lectures/more-about-ind-categories-and-compact-generation}

\include{lectures/lifting-properties}

\include{lectures/model-categories}

\include{lectures/model-categories-and-homotopy-(co)limits}

\include{lectures/grothendieck-categories-freyd-mitchell}

\include{lectures/graded-algebras}

\include{lectures/dg-algebras}

\begin{comment}
\include{lectures/metric-infinity-categories}

\include{lectures/derived-categories}

\include{lectures/sheaves-on-sites}
\end{comment}


\clearpage
\phantomsection
\addcontentsline{toc}{section}{References}
\printbibliography
\end{document}


% \begin{center}
% \begin{tikzcd}
% 	z\ar[dr,dashed]\ar[drr,bend left]\ar[ddr,bend right] & & \\
% 	& x\ar[r]\ar[d,hook] & * \ar[d,hook] \\
% 	& y\ar[r,"\varphi"] & \Omega
% \end{tikzcd}\(\quad \iff\quad\)
% \begin{tikzcd}
% 	 & z\ar[d,"\psi"]\ar[dl,dashed] & \\
% 	x\ar[r,hook]\ar[dr,hook] & y\times *\ar[d,equal,"\sim" labl] \ar[r,shift left]\ar[r,shift right,"\varphi"'] & \Omega \\
% 	& y &
% \end{tikzcd}
% \end{center}