nLab
comparison functor
Contents
Context
Category theory
category theory
Concepts
Universal constructions
Theorems
Extensions
Applications
Categorical algebra
Contents
Idea
While every functor might be understood as constituting a “comparison” between its domain and codomain -categories , the term “comparison functor” is often understood by default as referring, in categorical algebra ., to the unique functor that relates any adjunction to its monadic adjunction – this is the case we discuss here.
But other instances of functorial “comparison” are bound to be relevant. For instance, for the “comparison lemma ” in topos theory see there.
Statement
Proposition
(the comparison functor from any adjunction to its monadic adjunction ) Every pair of adjoint functors ( F ⊣ U ) : C ′ ↔ C (F \dashv U) \;\colon\; \mathbf{C}' \leftrightarrow \mathbf{C} (with unit η U F \eta^{U F} and counit ϵ F U \epsilon^{F U} ) between categories C \mathbf{C} , C ′ \mathbf{C}' fits into a commuting diagram in Cat of the following form:
Here:
ℰ ≔ U F ≔ U ∘ F \mathcal{E} \;\coloneqq\; U F \;\coloneqq\; U \circ F denotes the induced endofunctor on C \mathbf{C}
which carries the structure of a monad with
unit given by
η ℰ : id C → η U F U F = ℰ
\eta^{\mathcal{E}}
\;\;
\colon
\;\;
id_{\mathbf{C}}
\xrightarrow{\;\;
\eta^{U F}
\;\;}
U F
\;=\;
\mathcal{E}
product given by
μ ℰ : ℰ ℰ = U F U F → U ( ϵ F U ) F U F = ℰ
\mu^{\mathcal{E}}
\;\;
\colon
\;\;
\mathcal{E} \mathcal{E}
\;=\;
U F U F
\xrightarrow{\;\;
U ( \epsilon^{F U} ) F
\;\;}
U F
\;=\;
\mathcal{E}
C ℰ \mathbf{C}^{\mathcal{E}} denotes the (“Eilenberg-Moore ”-)category of ℰ \mathcal{E} -algebras in C \mathbf{C}
A = ( ℰ U ℰ ( A ) → ρ A U ℰ ( A ) )
A
\;=\;
\Big(
\mathcal{E}
U^{\mathcal{E}}(A)
\xrightarrow{\;\; \rho_A \;\;}
U^{\mathcal{E}}(A)
\Big)
in C \mathbf{C} with homomorphisms between them,
K U F K^{U F} denotes the “comparison functor ” given by
C ′ → K U F C ℰ C ′ ↦ ( ℰ U ( C ′ ) = U F U ( C ′ ) → U ϵ C ′ F U U ( C ′ ) )
\array{
\mathbf{C}'
&
\xrightarrow{\;\;\;\;\;\;\;\;\;\;
K^{U F}
\;\;\;\;\;\;\;\;\;\;}
&
\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\mathbf{C}^{\mathcal{E}}
\\
C'
&\mapsto&
{
\Big(
\mathcal{E}
U(C')
\,=\,
U F U (C')
\xrightarrow{\;
U \epsilon^{F U}_{C'}
\;}
U (C')
\Big)
}
}
C ℰ \mathbf{C}_{\mathcal{E}} denotes the (“Kleisli ”-)category of free ℰ \mathcal{E} -algebras
F ℰ ( C ) = ( ℰ ℰ ( C ) → μ C ℰ ℰ ( C ) ) = K U F F ( C ) = ( ℰ U F ( C ) → U ϵ F ( C ) F U U F ( C ) ) .
\begin{array}{rl}
F^{\mathcal{E}}(C)
&
\;=\;
\Big(
\mathcal{E}
\mathcal{E}
(C)
\xrightarrow{\;\;
\mu^{\mathcal{E}}_C
\;\;}
\mathcal{E}(C)
\Big)
\\
\;=\;
K^{U F} F(C)
&
\;=\;
\Big(
\mathcal{E}
U
F
(C)
\xrightarrow{\;\;
U \epsilon^{ F U }_{F(C)}
\;\;}
U
F
(C)
\Big)
\,.
\end{array}
Here the last line makes explicit that
U ℰ F ℰ = U F = ℰ .
U^{\mathcal{E}} F^{\mathcal{E}}
\;=\;
U F
\;=\;
\mathcal{E}
\,.
In fact, F ℰ ⊣ U ℰ F^{\mathcal{E}} \dashv U^{\mathcal{E}} and this realizes the monadic adjunction whose induced monad is ℰ \mathcal{E} .
Hence the original F ⊣ U F \dashv U is monadic iff the comparison functor K U F K^{U F} is an equivalence of categories .
(See for instance MacLane (1971), §VI.3 )
References
Last revised on November 8, 2022 at 15:48:10.
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