- the map Δ is
called the
comultiplication (or coproduct) of C and ε is the
counit of C. Take an
arbitrary set S and form the K-vector
space C = K(S) with...
- in
counit here is not
consistent with the
terminology of
limits and colimits,
because a
colimit satisfies an
initial property whereas the
counit morphisms...
-
compatible with a few more axioms. Specifically, the
comultiplication and the
counit are both
unital algebra homomorphisms, or equivalently, the multiplication...
-
particularly nice,
since the
existence of
compatible comultiplication,
counit, and
antipode allows for the
construction of
tensor products of representations...
-
adjunction can be
described by its
counit and unit
natural transformations.
Using the
notation from the
previous section, the
counit ε : F G → 1 C {\displaystyle...
- {\displaystyle \Delta :T^{m}V\to \bigoplus _{k=0}^{m}T^{k}V\boxtimes T^{(m-k)}V} The
counit ϵ : T V → K {\displaystyle \epsilon :TV\to K} is
given by the projection...
- → T) → W T
extend and
counit must also
satisfy duals of the
monad laws:
counit ∘ ( (wa =>> f) → wb ) ↔ f(wa) → b wa =>>
counit ↔ wa wa ( (=>> f(wx = wa))...
- K {\displaystyle \epsilon :A\to K} (corresponding to trace,
called the
counit). The
composition ϵ ∘ η : K → K {\displaystyle \epsilon \circ \eta :K\to...
- \rho =\mathrm {id} } ,
where Δ is the
comultiplication for C, and ε is the
counit. Note that in the
second rule we have
identified M ⊗ K {\displaystyle M\otimes...
-
multiplication map is
constructed using the
counit map of the adjunction: T 2 = G ∘ F ∘ G ∘ F → G ∘
counit ∘ F G ∘ F = T . {\displaystyle T^{2}=G\circ...
