diff --git a/R/geom-predicates.R b/R/geom-predicates.R
index 38c20980f..4ff90f983 100644
--- a/R/geom-predicates.R
+++ b/R/geom-predicates.R
@@ -81,7 +81,7 @@ st_geos_binop = function(op, x, y, par = 0.0, pattern = NA_character_,
 #' @param y object of class \code{sf}, \code{sfc} or \code{sfg}
 #' @param pattern character; define the pattern to match to, see details.
 #' @param sparse logical; should a sparse matrix be returned (`TRUE`) or a dense matrix?
-#' @return In case \code{pattern} is not given, \code{st_relate} returns a dense \code{character} matrix; element `[i,j]` has nine characters, referring to the DE9-IM relationship between `x[i]` and `y[j]`, encoded as IxIy,IxBy,IxEy,BxIy,BxBy,BxEy,ExIy,ExBy,ExEy where I refers to interior, B to boundary, and E to exterior, and e.g. BxIy the dimensionality of the intersection of the the boundary of `x[i]` and the interior of `y[j]`, which is one of: 0, 1, 2, or F; digits denoting dimensionality of intersection, F denoting no intersection. When \code{pattern} is given, a dense logical matrix or sparse index list returned with matches to the given pattern; see \link{st_intersection} for a description of the returned matrix or list. See also \url{https://en.wikipedia.org/wiki/DE-9IM} for further explanation.
+#' @return In case \code{pattern} is not given, \code{st_relate} returns a dense \code{character} matrix; element `[i,j]` has nine characters, referring to the DE9-IM relationship between `x[i]` and `y[j]`, encoded as IxIy,IxBy,IxEy,BxIy,BxBy,BxEy,ExIy,ExBy,ExEy where I refers to interior, B to boundary, and E to exterior, and e.g. BxIy the dimensionality of the intersection of the the boundary of `x[i]` and the interior of `y[j]`, which is one of: 0, 1, 2, or F; digits denoting dimensionality of intersection, F denoting no intersection. When \code{pattern} is given, a dense logical matrix or sparse index list returned with matches to the given pattern; see \link{st_intersects} for a description of the returned matrix or list. See also \url{https://en.wikipedia.org/wiki/DE-9IM} for further explanation.
 #' @export
 #' @examples
 #' p1 = st_point(c(0,0))
@@ -117,7 +117,7 @@ st_relate	= function(x, y, pattern = NA_character_, sparse = !is.na(pattern)) {
 #' @inheritDotParams s2::s2_options
 #' @param prepared logical; prepare geometry for `x`, before looping over `y`? See Details.
 #' @details If \code{prepared} is \code{TRUE}, and \code{x} contains POINT geometries and \code{y} contains polygons, then the polygon geometries are prepared, rather than the points.
-#' @return If \code{sparse=FALSE}, \code{st_predicate} (with \code{predicate} e.g. "intersects") returns a dense logical matrix with element \code{i,j} \code{TRUE} when \code{predicate(x[i], y[j])} (e.g., when geometry of feature i and j intersect); if \code{sparse=TRUE}, an object of class \code{\link{sgbp}} with a sparse list representation of the same matrix, with list element \code{i} an integer vector with all indices j for which \code{predicate(x[i],y[j])} is \code{TRUE} (and hence a zero-length integer vector if none of them is \code{TRUE}). From the dense matrix, one can find out if one or more elements intersect by \code{apply(mat, 1, any)}, and from the sparse list by \code{lengths(lst) > 0}, see examples below.
+#' @return If \code{sparse=FALSE}, \code{st_predicate} (with \code{predicate} e.g. "intersects") returns a dense logical matrix with element \code{i,j} equal to \code{TRUE} when \code{predicate(x[i], y[j])} (e.g., when geometry of feature i and j intersect); if \code{sparse=TRUE}, an object of class \code{\link{sgbp}} is returned, which is a sparse list representation of the same matrix, with list element \code{i} an integer vector with all indices \code{j} for which \code{predicate(x[i],y[j])} is \code{TRUE} (and hence a zero-length integer vector if none of them is \code{TRUE}). From the dense matrix, one can find out if one or more elements intersect by \code{apply(mat, 1, any)}, and from the sparse list by \code{lengths(lst) > 0}, see examples below.
 #' @details For most predicates, a spatial index is built on argument \code{x}; see \url{https://r-spatial.org/r/2017/06/22/spatial-index.html}.
 #' Specifically, \code{st_intersects}, \code{st_disjoint}, \code{st_touches} \code{st_crosses}, \code{st_within}, \code{st_contains}, \code{st_contains_properly}, \code{st_overlaps}, \code{st_equals}, \code{st_covers} and \code{st_covered_by} all build spatial indexes for more efficient geometry calculations. \code{st_relate}, \code{st_equals_exact}, and do not; \code{st_is_within_distance} uses a spatial index for geographic coordinates when \code{sf_use_s2()} is true.
 #'
diff --git a/man/geos_binary_pred.Rd b/man/geos_binary_pred.Rd
index f41859698..71a2b2fc3 100644
--- a/man/geos_binary_pred.Rd
+++ b/man/geos_binary_pred.Rd
@@ -102,7 +102,7 @@ boolean operation.}
 \item{dist}{distance threshold; geometry indexes with distances smaller or equal to this value are returned; numeric value or units value having distance units.}
 }
 \value{
-If \code{sparse=FALSE}, \code{st_predicate} (with \code{predicate} e.g. "intersects") returns a dense logical matrix with element \code{i,j} \code{TRUE} when \code{predicate(x[i], y[j])} (e.g., when geometry of feature i and j intersect); if \code{sparse=TRUE}, an object of class \code{\link{sgbp}} with a sparse list representation of the same matrix, with list element \code{i} an integer vector with all indices j for which \code{predicate(x[i],y[j])} is \code{TRUE} (and hence a zero-length integer vector if none of them is \code{TRUE}). From the dense matrix, one can find out if one or more elements intersect by \code{apply(mat, 1, any)}, and from the sparse list by \code{lengths(lst) > 0}, see examples below.
+If \code{sparse=FALSE}, \code{st_predicate} (with \code{predicate} e.g. "intersects") returns a dense logical matrix with element \code{i,j} equal to \code{TRUE} when \code{predicate(x[i], y[j])} (e.g., when geometry of feature i and j intersect); if \code{sparse=TRUE}, an object of class \code{\link{sgbp}} is returned, which is a sparse list representation of the same matrix, with list element \code{i} an integer vector with all indices \code{j} for which \code{predicate(x[i],y[j])} is \code{TRUE} (and hence a zero-length integer vector if none of them is \code{TRUE}). From the dense matrix, one can find out if one or more elements intersect by \code{apply(mat, 1, any)}, and from the sparse list by \code{lengths(lst) > 0}, see examples below.
 }
 \description{
 Geometric binary predicates on pairs of simple feature geometry sets
diff --git a/man/st_relate.Rd b/man/st_relate.Rd
index ddb70f550..05b93dee5 100644
--- a/man/st_relate.Rd
+++ b/man/st_relate.Rd
@@ -16,7 +16,7 @@ st_relate(x, y, pattern = NA_character_, sparse = !is.na(pattern))
 \item{sparse}{logical; should a sparse matrix be returned (\code{TRUE}) or a dense matrix?}
 }
 \value{
-In case \code{pattern} is not given, \code{st_relate} returns a dense \code{character} matrix; element \verb{[i,j]} has nine characters, referring to the DE9-IM relationship between \code{x[i]} and \code{y[j]}, encoded as IxIy,IxBy,IxEy,BxIy,BxBy,BxEy,ExIy,ExBy,ExEy where I refers to interior, B to boundary, and E to exterior, and e.g. BxIy the dimensionality of the intersection of the the boundary of \code{x[i]} and the interior of \code{y[j]}, which is one of: 0, 1, 2, or F; digits denoting dimensionality of intersection, F denoting no intersection. When \code{pattern} is given, a dense logical matrix or sparse index list returned with matches to the given pattern; see \link{st_intersection} for a description of the returned matrix or list. See also \url{https://en.wikipedia.org/wiki/DE-9IM} for further explanation.
+In case \code{pattern} is not given, \code{st_relate} returns a dense \code{character} matrix; element \verb{[i,j]} has nine characters, referring to the DE9-IM relationship between \code{x[i]} and \code{y[j]}, encoded as IxIy,IxBy,IxEy,BxIy,BxBy,BxEy,ExIy,ExBy,ExEy where I refers to interior, B to boundary, and E to exterior, and e.g. BxIy the dimensionality of the intersection of the the boundary of \code{x[i]} and the interior of \code{y[j]}, which is one of: 0, 1, 2, or F; digits denoting dimensionality of intersection, F denoting no intersection. When \code{pattern} is given, a dense logical matrix or sparse index list returned with matches to the given pattern; see \link{st_intersects} for a description of the returned matrix or list. See also \url{https://en.wikipedia.org/wiki/DE-9IM} for further explanation.
 }
 \description{
 Compute DE9-IM relation between pairs of geometries, or match it to a given pattern