# Properties of functors and their adjoints

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.
– user19475
Jun 27, 2012 at 21:28
• I didn't know the notion of a dominant functor, which can be found here: en.wikipedia.org/wiki/Dominant_functor . What is a separated functor? I only know separated presheaves (global data injects into gluing data). Jun 28, 2012 at 7:37
• I've added some trivial equivalences and the implication about preservation of presentable objects. Does it work that way or should we always explain it in some answer? Jun 28, 2012 at 7:55
• Hi Martin, your additions are great, thanks! I think if the assertion should be an easy exercise, then it's not necessary to spell it out (but no harm if someone wants to). My hope is for a concise and easy reference for what is true, not necessarily a complete compendium. Jun 28, 2012 at 17:32
• Hi Martin, by separated, I meant separable, as here: ncatlab.org/nlab/show/separable+functor It's an analog of separability for an algebra A over a commutative ring R. Jun 28, 2012 at 17:36

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).

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.

• Do you happen to know what's the analogous statement for $\infty$-categories? Dec 9, 2019 at 13:00
• @SaalHardali I'm actually pretty confused about what the $\infty$-categorical statement would be. Dec 9, 2019 at 14:21
• Yeah, me too. That's why I asked. Dec 9, 2019 at 14:24
• @SaalHardali We do get one direction $\infty$-categorically -- if $F$ is conservative and has a left adjoint $F^L$, then the counit $\varepsilon_X: F^L F X \to X$ is extremal in the sense that it doesn't factor through any non-iso mono. I'm not sure if the reverse implication holds. Dec 9, 2019 at 15:08

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

• This is also a reference for the properties I have mentioned. It also contains Rafael's theorem characterizing separable functors (as well as the dual version). Jul 2, 2012 at 8:19