\[\newcommand{\rv}[1]{\mathbf{#1}} \newcommand{\x}{\rv x} \newcommand{\y}{\rv y} \newcommand{\bar}[1]{\overline{#1}} \newcommand{\wtil}[1]{\widetilde{#1}} \newcommand{\what}[1]{\widehat{#1}} \newcommand{\ep}{\varepsilon} \newcommand{\ph}{\varphi} \newcommand{\maps}{\colon} \newcommand{\to}{\rightarrow} \newcommand{\xraw}{\xrightarrow} \newcommand{\darr}{\downarrow} \newcommand{\To}{\Rightarrow} \renewcommand{\dot}{\centerdot} \renewcommand{\tensor}{\otimes} \newcommand{\pr}{^\prime} \newcommand{\op}[1]{#1^{\mathrm{op}}} \newcommand{\hom}{\textit{Hom}} \newcommand{\oo}{\circ} \newcommand{\lang}{\langle} \newcommand{\rang}{\rangle} \newcommand{\str}[1]{#1^{*}} \newcommand{\intvl}{\mathbb{I}} \newcommand{\yon}{\mathcal{Y}} \newcommand{\topos}{\mathfrak{Top}} \newcommand{\bTopos}{\mathfrak{BTop}} \newcommand{\BTop}{\mathfrak{BTop}} \newcommand{\con}{\mathfrak{Con}} \newcommand{\pos}{\mathfrak{Poset}} \newcommand{\topl}{\mathbb{Top}} \newcommand{\grpd}{\mathbb{Grpd}} \newcommand{\ab}{\mathbf{Ab}} \newcommand{\thT}{\mathbb{T}} \newcommand{\mod}{\mathbf{Mod}} \newcommand{\th}{\mathbf{Th}} \newcommand{\cl}{\mathbf{Cl}} \newcommand{\ob}{\mathbf{Ob}} \newcommand{\aut}{\mathbf{Aut}} \newcommand{\bun}{\mathbf{Bun}} \newcommand{\geom}{\mathbf{Geom}} \def\catg{\mathop{\mathcal{C}\! {\it at}}\nolimits} \def\Cat{\mathop{\mathfrak{Cat}}} \def\Con{\mathop{\mathfrak{Con}}} \def\CAT{\mathop{\mathbf{2} \mathfrak{Cat}}} \def\Cat{\mathop{\mathfrak{Cat}}} \newcommand{\Topos}{\mathfrak{Top}} \newcommand{\ETopos}{\mathcal{E}\mathfrak{Top}} \newcommand{\BTopos}{\mathcal{B}\mathfrak{Top}} \newcommand{\GTopos}{\mathcal{G}\mathfrak{Top}} \newcommand{\Psh}{\textit{Psh}} \newcommand{\Sh}{\textit{Sh}} \newcommand{\psh}[1]{\textit{Psh}(\cat{#1})} \newcommand{\sh}[1]{\textit{Sh}(\cat{#1})} \newcommand{\Id}{\operatorname{Id}} \newcommand{\Ho}{\operatorname{Ho}} \newcommand{\ad}{\operatorname{ad}} \newcommand{\Adj}{\operatorname{Adj}} \newcommand{\Sym}{\operatorname{Sym}} \newcommand{\Set}{\operatorname{Set}} \newcommand{\Pull}{\operatorname{Pull}} \newcommand{\Push}{\operatorname{Push}} \newcommand{\dom}{\operatorname{dom}} \newcommand{\cod}{\operatorname{cod}} \newcommand{\fun}{\operatorname{Fun}} \newcommand{\colim}{\operatorname{Colim}} \newcommand{\cat}[1]{\mathcal{#1}} \newcommand{\scr}[1]{\mathscr{#1}} \newcommand{\frk}[1]{\mathfrak{#1}} \newcommand{\bb}[1]{\mathbb{#1}} \newcommand{\CS}{\mathcal{S}} \newcommand{\CE}{\mathcal{E}} \newcommand{\CF}{\mathcal{F}}\]Theories to learn
Logical Relations: In particular the classic Plotkin’s paper Lambda-definability and logical relations
Homological Algebra: I never had a proper course in homological algebra (I am thinking for instance Hilton-Stambach’s level or Manin’s). I learnt basic stuff from sparse lecture notes certain constructions in homological algebra from nLab.
Connection between logic and topological/geometric invariants such as cohomology: There is a very nice paper by A.Blass which shows EAC (EAC= external axiom of choice which states every epi splits. (or equivalently every object is projective.)) holds iff all first non-abelian cohomologies (for all coefficient groups, not necessarily abelian) of discrete spaces are trivial. If you wonder what cohomology has to do with choice, roughly consider choice as a local to global statement, and first cohomology detects the obstructions of local to global. If there is no such obstruction then you can transition from local to global. If you would like to see the proof read his paper. Be warned that his proofs use excluded middle, so it is not constructive. If you want the idea of proof, and indeed a constructive proof read a post from 3 years ago by M.Shulman.
Here is a quote from that post:
“…I think it’s quite cute that we can connect geometric ideas like characteristic classes of fiber bundles in cohomology with logical ideas like the axiom of choice. In some sense, this is what topos theory is supposed to be all about.”
Books and papers to read
Dan Kan’s original paper on adjoint functors
Emily Riehl and Dominic Verity’s Infinity category theory (and infinity cosmos) from scratch
Carlos Simpson’s Homotopy Theory of Higher Categories: From Segal Categories to N-Categories
Charles Rezk’s Toposes and homotopy toposes
Peter Johnstone’s older topos theory book, chapter 8 on Cohomology in toposes
