Properties of functors and their adjoints

Question

I am interested in collecting in this question a list of properties a functor $F$ may have and what those properties imply for left and right adjoints, $F^L$ and $F^R$, assuming they exist. There are different types of functors and different types of categories, but let us begin in complete generality. If this format seems to work, I'd be interested in specializing to abelian and derived categories, and also including higher categories, in separate lists - we'll see (of course, if someone is eager to add and populate the other lists, feel free!).

The motivation here is that, as a "working mathematician", I find it annoying to have to search the literature everytime something like this comes up. In homage to Maclane, I therefore dub this post "Categories for the Lazy Mathematician".

The format of the list is this: give a property of $F:C\to D$, and what it implies for $F^L:D \to C$ or $F^R: D \to C$, as the case may be. Let us try to be concise in each entry of the list. Give any helpful details about each claim as a separate answer, e.g. if you feel a definition is obscure, you can provide it, or if you want to prove an implication or give a reference, do so there. Naturally, there are many more properties of functors than the ones I came up with below. Please add them if they are interesting, even if you do not know the implications.

General categories

• $F$ faithful
  • $\Leftrightarrow$ the unit $\mathrm{id_C} \to F^R \circ F$ is a pointwise monomorphism
  • $\Leftrightarrow$ the counit $F^L\circ F\to \mathrm{id_C}$ is a pointwise epimorphism
• $F$ full
  • $\Leftrightarrow$ the unit $\mathrm{id_C}\to F^R\circ F$ co-splits pointwise in $C$
  • $\Leftrightarrow$ the counit $F^L\circ F\to \mathrm{id_C}$ splits pointwise in $C$
• $F$ is fully faithful
  • $\Leftrightarrow$ the counit $F^L \circ F \to \mathrm{id_C}$ is an isomorphism
  • $\Leftrightarrow$ the unit $\mathrm{id_C} \to F^R \circ F$ is an isomorphism
• (Assuming $C,D$ are essentially small) $F$ essentially surjective $\Rightarrow$ the induced functor on presheaf categories $$\Delta_F\colon[D^{\text{op}},\mathbf{Set}]\to [C^{\text{op}},\mathbf{Set}]$$ is faithful and conservative, so its unit $\mathrm{id}\to\Pi_F\Delta_F$ is pointwise mono and its counit $\Sigma_F\Delta_F\to\mathrm{id}$ is pointwise epi, as above. In this case, $\Delta_F$ is both monadic and comonadic.
• $F$ is a wide inclusion, i.e. $F$ is faithful and essentially surjective $\Rightarrow$
• $F$ dominant $\Rightarrow$ same conclusion as $F$ essentially surjective (since $F$ is essentially surjective on Cauchy completions and the Cauchy completion has the same presheaf category)
• (Assuming $C$ has finite limits) $F$ conservative $\Leftrightarrow$ the counit is a strong epimorphism
• $F$ separable $\Rightarrow$
• $F$ injective on objects $\Rightarrow$
• $F$ preserves limits $\Rightarrow$ $F^L$ exists (assuming $C$ is complete and "small enough")
• $F$ preserves colimits $\Rightarrow$ $F^R$ exists (assuming $C$ is cocomplete and "small enough")
• $F$ preserves $\kappa$-directed colimits $\Rightarrow$ $F^L$ preserves $\kappa$-presentable objects

Abelian categories

• $F$ exact $\Rightarrow$ $F^R$ preserves injectives

• $F^L$ (resp. $F^R$) exists $\Rightarrow$ $F$ and $F^L$ (resp. $F^R$) are additive

Monoidal categories

• $F$ lax monoidal $\Rightarrow$ $F^L$ colax monoidal (doctrinal adjunction, holds in much greater generality). Dually, $F$ colax monoidal $\Rightarrow$ $F^R$ lax monoidal.

Answer 1

THis is a resume from my old notes, the proofs aren't so difficult, but I include proof's if required....

PREMISES

Let $(F, G, \varepsilon , \eta): \mathscr{A} \to \mathscr{B}$ and adjunction.

Let $\Phi:{A, X}: (F(A), X)\cong (A, G(X)$ the natural bijection

give $f: F(A)\to X$ let $f^a:=G(f)\circ \eta_A$ its right adjoint give $g: A\to G(X)$ let ${}^ag:=\epsilon_X\circ F(f)$ its left adjoint

For $f: A\to A'$ da ${}^a(\eta_A'\circ f)=\epsilon_{ F(A')} \circ F(\eta_{ A'})\circ F(f)= F(f)$ follow that

$F_{ A, A'} = \Phi_{ A, FA'}^{-1} \circ \mathscr{A}(A, \eta_{ A'}): \mathscr{A} (A, A') \to \mathscr{A} (A, G(F(A'))) \cong \mathscr{B}(F(A), F(A'))$

THEN WE HAVE THE FOLLOWING PROPERTIES:

a) Give $G: \mathscr{C}\to \mathscr{A}$ let $\mathscr{A'} \subset\mathscr{A}$ the full subcategory with objects the $A\in \mathscr{A}$ such that $h^{A}_{G}: \mathscr{B}\to Set: B\mapsto (A, G(B))$ is representable

This is the maximum sub-category of which is defined a partial left adjoint $F$ of $G$, i.e. exist a bijection $\mathscr{C}(F(A), X)\cong \mathscr{A}(A, G(X))$ natural for $A\in \mathscr{A'}$ and $X\in \mathscr{B}$, then $F$ è unique but isomorphisms. Then $F$ preserves all colimits preserved by $\mathscr{A'} \subset_{fu}\mathscr{A}$ (also large or empty):

give a colimit cocone $(A_i\to A)_{i\in I} A_i$ in $\mathscr{A'}$ and a cocone $e_i: (F(A_i)\to X)_{i\in I}$ from the cocone $(e_i^a : A_i \to G(X))_{i\in I}$ follow unique $g: A\to G(X)$ with $g\circ \epsilon_i=e_i^a$ then ${}^ag: F(A)\to X$ is such that ${}^ag\circ F(\epsilon_i)=e_i$, if $g', g'' : F(A)\to X$ verify the last condition then $g'^a, g''^a : A\to G(X)$ are equal, then $g'={}^a(g'^a)= {}^a(g''^a)=g''$. Is easy proof that $F$ preserving epimorphisms, and dually $G$ preserving monomorphisms, and $F$ preserving strong.epimorphisms and dually $G$ preserving strong-monomorphisms.

b) The following properties are equivalent:

b.1) $F$ is faithful (full, full and faithful)

b.2) $\eta$ is a pointwise-monomorphism (pointwise-Retraction, a Isomorphism)

b.3) $F$ reflect monomorphism

b.4) $\Phi_{ A, B }$ preserving monomorphisms

b.5) For any $X\in\mathscr{C}$ the source $(a:X\to G(A))_{A\in \mathscr{A}, a\in (A, G(A))}$ is a mono-source (is enough considering $A$ belong to cogenerating class).

.

In Particular if $F$ is full from $1_G=G\varepsilon * \eta G$, $1_F= \varepsilon F*F\eta$ follow that $\eta G$, $G\varepsilon $, $F\eta$, $\varepsilon F$ are isomorphisms.

c) Here we call $F$ conservative is reflect isomorphisms, and call a morphisms $m: A\to B$ a co.cover if from $m=f\circ e$ with $e$ epimorphism follow that $e$ is a isomorphism, for straight generalization we have the definition of cocover source.

We have the implication: (1) $F$ is conservative $\Rightarrow $ (2) $F$ reflect co.Cover's $\Rightarrow $ (3) $\eta$ is pointwise-co.cover $\Leftrightarrow$ The source $(a:X\to G(A))_{A\in \mathscr{A}}$ is a co.cover source.

And $(3)\Rightarrow(1)$ if $F$ reflect isomorphisms on epimorphisms (I.e. if $F(e)$ is a isomorphism then $e$ is a epimorphism, in particular this happen if $F$ is faithful).

d) We call $F: \mathscr{B}\to \mathscr{A}$ co.fiathfull if for $H, K: \mathscr{A}\to \mathscr{C}$ and $\phi, \psi: H\to K$ and $\phi\circ F= \psi\circ F$ follow that $\phi=\psi$. ANd call $F$ co.conservative if (on the data above) from $\phi\circ F$ isomorphisms follow that $\phi$ is isomorphism.

We have the following equivalent properties:

d.1) $G$ if full and faithful

d.2) $\epsilon$ is isomorphism

d.3) $F$ is dense

d.4) $F\circ U$ is dense for some (any) $U: \mathcal{C}\to \mathscr{A}$ dense

d.5) the functor $F^*: \mathscr{B}[\Sigma]\to \mathscr{A}[\Sigma]$

where $\Sigma:=F^{-1}(Iso)$ , $F=F^*\circ P$, and $P: \mathscr{B}\to \mathscr{B}[\Sigma]$ canonic, is a equivalence

d.6) $F$ is co.fauthful $\Rightarrow$ $F$ is co.conservative.

e) G riflect strong.epimorphisms $\Leftrightarrow$ $\epsilon$ is pointwise-strong.epimorphisms

f) If $G$ is full and $\eta$ is pointwise-Section then $\eta$ is a Isomorphism.

g) Define a epimorphisms $e: X\to Y$ a (small)source-strong-epimorphism if give $f: X\to A$ and a (small) monosource $(m_i: A\to A_i)_{i\in I}$ and a (small) source $(g_i: Y\to A_i)_{i\in I}$ with $g_i\circ e=m_i\circ f\ i\in I$ exist unique a diagonal $d: Y\to A$ that keep the commutativity of the diagram.

We have te following property:

If for any $A\in \mathscr{A}$ the morphism $\epsilon_A : FG(A)\to A $ is (small)source-strong-epimorphism then $G$ reflect large (small) limits.\

h) Let $F$ such that for $X\in \mathscr{C}$ we have $1_X=s\circ r: X\to F(A)\to X$ for some $s,\ r$. From $\epsilon_X\circ FGF(r)=r\circ \varepsilon _{ FA }$ where $r$ and $\epsilon _{F(A)}$ retractions follow that $\epsilon_X$ is a retraction, then a epimorphisms and $G$ is faithful. If $G_{ A, A'}: \mathscr{B}(F(A), F(A'))\to \mathscr{A}(GF(A), GF(A'))$ is surjective then $G$ is full:

for $u: G(B_1)\to G(B_2)$ with $1=\rho_k\circ \sigma_k: A_k\to F(B_k)\to A_k$ follow $G(\sigma _2)\circ u\circ G(\rho_1): GF(B_1)\to GF(B_2)$ and this is $G(v)$ for some $v: F(B_1)\to F(B_2)$, then $u=G\sigma _2\circ v\circ Q\rho_1$.

i)

Give the adjoint couples $(U_! , U^\ast)$ and $(U^\ast, U_\ast)$ where

$U^\ast: \mathscr{A}\to \mathscr{E}$.

For a category $\mathscr{C}$ let $\mathscr{C}^>:=Fun(\mathscr{C}^{op}, Set)$ the category of presheaves .We have the following equivalents properties:

i1) $U_!$ is faithfull and full (faithful).

i2) The unity $\eta_H: H\to U^\ast U_!(H)$, for $H\in \mathscr{A}^>$ is a isomorphisms (a monomorphism).

i3) $U_\ast$ is faithfull and full (faithful).

i4) The counity $\epsilon_H: U^\ast U_\ast (H)\to H$, for $H\in \mathscr{A}^>$ is a isomorphisms (a epimorphism).

Answer 2

I'm having trouble finding a proof of the following fact written somewhere, so let me record it here.

Fact: Let $F: C \to D$ be a functor with a left adjoint $F^L: D \to C$, and assume that $C$ has finite limits. Then $F$ is conservative if and only if the counit $\varepsilon: F^L F \Rightarrow 1_C$ is a levelwise strong epimorphism.

Proof: For the forward direction, assume that $F$ is conservative; we want to show that $\varepsilon_X : F^L F X \to X$ is a strong epimorphism. Because $C$ has finite limits, it suffices to show that if $\varepsilon_X = gf$ with $g$ a monomorphism, then $g$ is an isomorphism. In this case, we have $F\varepsilon_X = Fg Ff$. Because $F\varepsilon_X$ is a split epimorphism (by one of the triangle equations), so is $Fg$. But $F$ is a right adjoint and so preserves monomorphisms, so $Fg$ is also a monomorphism. It follows that $Fg$ is iso, and by conservativity $g$ is also iso as desired.

For the backward direction, assume that $\varepsilon_X$ is strong epi for each $X \in C$, and suppose that $g: X \to Y$ has the property that $Fg$ is an isomorphism; we want to show that $g$ is an isomorphism. First, we have $g \varepsilon_X = \varepsilon_Y (F^L F g)$. By cancellation properties of strong epimorphisms, $g$ is a strong epimorphism. On the other hand, if $f,f': Z \to X$ have $gf = gf'$, then $(F^L F g)( F^L F f) = (F^L F g)( F^L F f')$. Because $F^L F g$ is an iso, $F^L F f = F^L F f'$. So $f \varepsilon_Z = \varepsilon_X (F^L F f) = \varepsilon_X (F^L F f') = f' \varepsilon_Z$. Because $C$ has finite limits and $\varepsilon_Z$ is strong epi, $\varepsilon_Z$ is in fact epi, so this implies that $f = f'$. Thus $g$ is a monomorphism. Since it is strong epi and mono, $g$ is iso as desired.

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$\infty$-categorical version:

Here’s an $\infty$-categorical analog, which does leave something to be desired I think:

As above, our adjunction is $F^L : D {}^\to_\leftarrow C : F$.

Definition: Say that a morphism $f : c \to c’$ in $C$ is kind of strong monic relative to $F$ if $f$ is right orthogonal to every morphism of the form $F^L(g)$ where $g : d \to d’$ is a split epimorphism in $D$. Say that a morphism $f’ : c \to c’$ in $C$ is kind of strong epic relative to $F$ if $f’$ is left orthogonal to all kind-of-strong-monic-relative-to-$F$ morphisms in $C$. Say that the adjunction $F^L \dashv F$ is admissible if every morphism $f’’ : c \to c’$ in $C$ admits a (necessarily unique) factorization $f’’ = f \circ f’$ where $f$ is kind of strong monic relative to $F$ and $f’$ is kind of strong epic relative to $F$.

Remark: If $C$ is (say) presentable, then admissibility of the adjunction $F^L \dashv F$ is basically a set-theoretic condition on the split epis of $D$, and a reasonable one at that -- but IIRC it is not automatic even if $C,D$ are presentable.

Theorem: Let $F^L : D {}^\to_\leftarrow C : F$ be an admissible adjunction. Then $F$ is conservative if and only if the counits $\varepsilon_c : F^L F c \to c$ are all kind of strong epic relative to $F$.

Proof: Straightforward diagram chase (but I have only run through the argument once without really carefully checking, please don’t take my word for it -- in fact, I invite the industrious reader to edit this CW answer and add the details).

Remark: The least satisfactory part of this theorem to me is that the class of kind-of-strong-epic-relative-to-$F$ morphisms is defined in terms of $F$. Aesthetically, one would prefer to work with a class of morphisms defined independently of $F$. It might be possible to do away with the admissibility hypothesis, I haven’t thought about it.

Answer 3

I have added in Proposition 2.5 of https://web.archive.org/web/20210506152517/http://homepages.vub.ac.be/~scaenepe/Full19.pdf, which outlines the descriptions of full and faithful in terms of the adjunction data.

Ardizzoni, A.; Menini, C.; Caenepeel, S.; Militaru, G., Naturally full functors in nature., Acta Math. Sin., Engl. Ser. 22, No. 1, 233-250 (2006). ZBL1110.16039.