diff --git a/bowtie.cabal b/bowtie.cabal
--- a/bowtie.cabal
+++ b/bowtie.cabal
@@ -1,11 +1,11 @@
 cabal-version: 1.12
 
--- This file has been generated from package.yaml by hpack version 0.37.0.
+-- This file has been generated from package.yaml by hpack version 0.38.0.
 --
 -- see: https://github.com/sol/hpack
 
 name:           bowtie
-version:        0.6.0
+version:        0.7.0
 synopsis:       Tying knots in polynomial functors
 description:    Please see the README on GitHub at <https://github.com/ejconlon/bowtie#readme>
 homepage:       https://github.com/ejconlon/bowtie#readme
diff --git a/src/Bowtie/Free.hs b/src/Bowtie/Free.hs
--- a/src/Bowtie/Free.hs
+++ b/src/Bowtie/Free.hs
@@ -5,9 +5,7 @@
 -- See https://hackage.haskell.org/package/free-5.1.7/docs/Control-Monad-Trans-Free.html
 module Bowtie.Free
   ( FreeF (..)
-  , Free (..)
-  , pattern FreeEmbed
-  , pattern FreePure
+  , Free (.., FreeEmbed, FreePure)
   , substFree
   , liftFree
   , iterFree
diff --git a/src/Bowtie/Jot.hs b/src/Bowtie/Jot.hs
--- a/src/Bowtie/Jot.hs
+++ b/src/Bowtie/Jot.hs
@@ -1,12 +1,10 @@
 {-# LANGUAGE UndecidableInstances #-}
 
 module Bowtie.Jot
-  ( JotF (..)
-  , pattern JotFP
+  ( JotF (.., JotFP)
   , jotFKey
   , jotFVal
-  , Jot (..)
-  , pattern JotP
+  , Jot (.., JotP)
   , mkJot
   , unMkJot
   , annoJot
diff --git a/src/Bowtie/Memo.hs b/src/Bowtie/Memo.hs
--- a/src/Bowtie/Memo.hs
+++ b/src/Bowtie/Memo.hs
@@ -1,12 +1,10 @@
 {-# LANGUAGE UndecidableInstances #-}
 
 module Bowtie.Memo
-  ( MemoF (..)
-  , pattern MemoFP
+  ( MemoF (.., MemoFP)
   , memoFKey
   , memoFVal
-  , Memo (..)
-  , pattern MemoP
+  , Memo (.., MemoP)
   , mkMemo
   , mkMemoM
   , reMkMemo
@@ -37,7 +35,7 @@
 
 -- | The base functor for a 'Memo'
 newtype MemoF f k r = MemoF {unMemoF :: Anno k (f r)}
-  deriving stock (Show, Functor)
+  deriving stock (Show, Functor, Foldable, Traversable)
   deriving newtype (Eq, Ord)
 
 pattern MemoFP :: k -> f r -> MemoF f k r
diff --git a/src/Bowtie/SMap.hs b/src/Bowtie/SMap.hs
--- a/src/Bowtie/SMap.hs
+++ b/src/Bowtie/SMap.hs
@@ -15,17 +15,22 @@
   , Member
   , NonMember
   , Inserted
-  , Without
   , Deleted
   , Reordered
+  , Unioned
+  , Merged
   , SMap
   , emptySMap
+  , unionSMap
+  , MergeFn
+  , mergeSMap
   , singletonSMap
   , indexSMap
   , updateSMap
   , insertSMap
   , deleteSMap
   , reorderSMap
+  , keysSMap
   )
 where
 
@@ -36,9 +41,10 @@
 import Data.GADT.Compare (GCompare (..), GEq (..), GOrdering (..), defaultCompare, defaultEq)
 import Data.Kind (Type)
 import Data.Proxy (Proxy (..))
+import Data.Some (Some (..))
 import Data.Type.Bool (type (||))
 import Data.Type.Equality ((:~:) (..), type (==))
-import GHC.TypeLits (KnownSymbol, OrderingI (..), Symbol, cmpSymbol, sameSymbol)
+import GHC.TypeLits (KnownSymbol, OrderingI (..), Symbol, cmpSymbol, sameSymbol, symbolVal)
 
 type family Val (d :: Type) (s :: Symbol) :: Type
 
@@ -69,6 +75,9 @@
 key :: (KnownSymbol s, v ~ Val d s) => Proxy s -> Key d v
 key = flip Key Proxy
 
+keyString :: Key d v -> String
+keyString (Key p _) = symbolVal p
+
 type DM (d :: Type) = DMap (Key d) Identity
 
 emptyDM :: DM d
@@ -115,17 +124,6 @@
 
 instance (zs ~ InsertedF x (y : ys), Member x zs) => Inserted x (y : ys) zs
 
-type family WithoutF (x :: Symbol) (xs :: [Symbol]) :: [Symbol] where
-  WithoutF x '[] = '[]
-  WithoutF x (x : zs) = zs
-  WithoutF x (y : zs) = y : WithoutF x zs
-
-class (NonMember x zs) => Without (x :: Symbol) (xs :: [Symbol]) (zs :: [Symbol]) | x xs -> zs
-
-instance (zs ~ WithoutF x '[]) => Without x '[] zs
-
-instance (zs ~ WithoutF x (y : ys), NonMember x zs) => Without x (y : ys) zs
-
 type family DeletedF (x :: Symbol) (xs :: [Symbol]) :: [Symbol] where
   DeletedF x (x : zs) = zs
   DeletedF x (y : zs) = y : DeletedF x zs
@@ -140,11 +138,39 @@
 
 instance (Reordered xs (DeletedF x zs)) => Reordered (x : xs) zs
 
+type family UnionedF (xs :: [Symbol]) (ys :: [Symbol]) :: [Symbol] where
+  UnionedF '[] ys = ys
+  UnionedF (x : xs) ys = UnionedF xs (InsertedF x ys)
+
+class Unioned (xs :: [Symbol]) (ys :: [Symbol]) (zs :: [Symbol]) | xs ys -> zs
+
+instance Unioned '[] ys ys
+
+instance (NonMember x ys, UnionedF xs ys ~ zs) => Unioned (x : xs) ys (x : zs)
+
+type family MergedF (xs :: [Symbol]) (ys :: [Symbol]) :: [Symbol] where
+  MergedF '[] ys = ys
+  MergedF (x : xs) ys = MergedF xs (InsertedF x ys)
+
+class Merged (xs :: [Symbol]) (ys :: [Symbol]) (zs :: [Symbol]) | xs ys -> zs
+
+instance Merged '[] ys ys
+
+instance (MergedF xs ys ~ ws, InsertedF x ws ~ zs) => Merged (x : xs) ys zs
+
 newtype SMap (d :: Type) (xs :: [Symbol]) = SMap (DM d)
 
 emptySMap :: SMap d '[]
 emptySMap = SMap emptyDM
 
+unionSMap :: (Unioned xs ys zs) => SMap d xs -> SMap d ys -> SMap d zs
+unionSMap (SMap dmx) (SMap dmy) = SMap (DMap.union dmx dmy)
+
+type MergeFn (d :: Type) = forall (s :: Symbol). (KnownSymbol s) => Proxy s -> Val d s -> Val d s -> Val d s
+
+mergeSMap :: (Merged xs ys zs) => MergeFn d -> SMap d xs -> SMap d ys -> SMap d zs
+mergeSMap f (SMap dmx) (SMap dmy) = SMap (DMap.unionWithKey (\(Key p _) (Identity vx) (Identity vy) -> Identity (f p vx vy)) dmx dmy)
+
 singletonSMap :: (KnownSymbol s) => Proxy s -> Val d s -> SMap d '[s]
 singletonSMap ps v = SMap (singletonDM ps v)
 
@@ -162,3 +188,6 @@
 
 reorderSMap :: (Reordered xs zs) => SMap d xs -> SMap d zs
 reorderSMap = coerce
+
+keysSMap :: SMap d xs -> [String]
+keysSMap (SMap m) = fmap (\(Some k) -> keyString k) (DMap.keys m)