diff --git a/doc/Makefile b/doc/Makefile
index 4d43828..c942a7e 100644
--- a/doc/Makefile
+++ b/doc/Makefile
@@ -4,16 +4,20 @@
 #
 # Change only the variable below to the name of the main tex file.
 PROJNAME=univalent-categories
+MAIN=main.tex
 
 # You want latexmk to *always* run, because make does not have all the info.
 # Also, include non-file targets in .PHONY so they are run regardless of any
 # file of the given name existing.
-.PHONY: $(PROJNAME).pdf all clean
+.PHONY: $(PROJNAME).pdf all clean preview
 
 # The first rule in a Makefile is the one executed by default ("make"). It
 # should always be the "all" rule, so that "make" and "make all" are identical.
 all: $(PROJNAME).pdf
 
+preview: $(MAIN)
+	latexmk -pvc -jobname=$(PROJNAME) -pdf -pdflatex="pdflatex -interactive=nonstopmode" $<
+
 # CUSTOM BUILD RULES
 
 # In case you didn't know, '$@' is a variable holding the name of the target,
@@ -36,7 +40,7 @@ all: $(PROJNAME).pdf
 # -interactive=nonstopmode keeps the pdflatex backend from stopping at a
 # missing file reference and interactively asking you for an alternative.
 
-$(PROJNAME).pdf: main.tex
+$(PROJNAME).pdf: $(MAIN)
 	latexmk -jobname=$(PROJNAME) -pdf -pdflatex="pdflatex -interactive=nonstopmode" -use-make $<
 
 cleanall:
diff --git a/doc/halftime.tex b/doc/halftime.tex
index 96f6094..2f3d2fa 100644
--- a/doc/halftime.tex
+++ b/doc/halftime.tex
@@ -236,7 +236,7 @@ however; the first one is that the standard library's notion of a functor is
 indexed by the object map:
 %
 $$
-\Functor   & \tp (\Type \to \Type) \to \Type
+\Functor \tp (\Type \to \Type) \to \Type
 $$
 %
 Where $\Functor\ F$ has the member:
@@ -248,7 +248,7 @@ $$
 Whereas the object map in my definition is existentially quantified:
 %
 $$
-\Functor   & \tp \Type
+\Functor \tp \Type
 $$
 %
 And $\Functor$ has these members: