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Contents

1. Idea

The left part of a pair of adjoint functors is one of two best approximations to a weak inverse of the other functor of the pair. (The other best approximation is the functor’s right adjoint, if it exists.) Note that a weak inverse itself, if it exists, must be a left adjoint, forming an adjoint equivalence.

A left adjoint to a forgetful functor is called a free functor. Many left adjoints can be constructed as quotients of free functors.

The concept generalises immediately to enriched categories and in 2-categories.

2. Definitions

\subsection{For categories}

\begin{defn} \label{DefinitionLeftAdjointForCategories} Given categories $\mathcal{C}$ and $\mathcal{D}$ and a functor $R: \mathcal{D} \to \mathcal{C}$, a left adjoint of $R$ is a functor $L: \mathcal{C} \to \mathcal{D}$ together with natural transformations $\iota: id_\mathcal{C} \to R \circ L$ and $\epsilon: L \circ R \to id_\mathcal{D}$ such that the following diagrams (known as the triangle identities) commute, where $\cdot$ denotes thewhiskering of a functor with a natural transformations. transformation.

\begin{centre} \begin{tikzcd} L \ar[r, id(L) L \cdot \iota] \ar[dr, swap, id] & L \circ R \circ L \ar[d, \epsilon \cdot id(R)] L] \ & L \end{tikzcd} \end{centre}

\begin{centre} \begin{tikzcd} R \ar[r, \iota \cdot id(R)] R] \ar[dr, swap, id] & R \circ L \circ R \ar[d, id(R) R \cdot \epsilon] \ & R \end{tikzcd} \end{centre}

\end{defn}

\begin{rmk} \label{RemarkEquivalentDefinitionLeftAdjointForCategories} Requiring the commutativity of the two diagrams in Definition \ref{DefinitionLeftAdjointForCategories} is equivalent to requiring that there is a natural isomorphism between the Hom functors

Depending upon one’s interpretation of $\mathsf{Set}$, the category of sets, one may strictly speaking need to restrict to locally small categories for this equivalence to parse. \end{rmk}

\subsection{For enriched categories}

The equivalent formulation of Definition \ref{DefinitionLeftAdjointForCategories} given in Remark \ref{RemarkEquivalentDefinitionLeftAdjointForCategories} generalises immediately to the setting of enriched categories.

\begin{defn} Given $\mathbb{V}$-enriched categories $\mathcal{C}$ and $\mathcal{D}$ and a $\mathbb{V}$-enriched functor $R: \mathcal{D} \to \mathcal{C}$, a left adjoint of $R$ is a $\mathbb{V}$-enriched functor $L: \mathcal{C} \to \mathcal{D}$ together with a $\mathbb{V}$-enriched natural isomorphism between the Hom functors

\end{defn}

\subsection{In a 2-category}

Definition \ref{DefinitionLeftAdjointInACategory} generalises immediately from Cat, the 2-category of categories, to any 2-category.

\begin{defn} \label{DefinitionLeftAdjointInA2Category} Let $\mathcal{A}$ be a 2-category. Given objects $\mathcal{C}$ and $\mathcal{D}$, and a 1-arrow $R: \mathcal{D} \to \mathcal{C}$ of $\mathcal{A}$, a left adjoint of $R$ is a 1-arrow $L: \mathcal{C} \to \mathcal{D}$ together with 2-arrows $\iota: id_\mathcal{C} \to R \circ L$ and $\epsilon: L \circ R \to id_\mathcal{D}$ such that the following diagrams commute, where $\cdot$ denotes whiskering in $\mathcal{A}$.

\begin{centre} \begin{tikzcd} L \ar[r, id(L) L \cdot \iota] \ar[dr, swap, id] & L \circ R \circ L \ar[d, \epsilon \cdot id(R)] L] \ & L \end{tikzcd} \end{centre}

\begin{centre} \begin{tikzcd} R \ar[r, \iota \cdot id(R)] R] \ar[dr, swap, id] & R \circ L \circ R \ar[d, id(R) R \cdot \epsilon] \ & R \end{tikzcd} \end{centre}

\end{defn}

\begin{rmk} If one assumes that one’s ambient 2-category has more structure, bringing it closer to being a 2-topos, for example a Yoneda structure, one should be able to give an equivalent formulation of Definition \ref{DefinitionLeftAdjointInA2Category} akin to that of Remark \ref{RemarkEquivalentDefinitionLeftAdjointForCategories}. \end{rmk}

\subsection{For preorders and posets}

Restricted to preorders or posets, Definition \ref{DefinitionLeftAdjointForCategories} in its equivalent formulation of Remark \ref{RemarkEquivalentDefinitionLeftAdjointForCategories} can be expressed in the following terminology.

\begin{defn} Given posets or preorders $\mathcal{C}$ and $\mathcal{D}$ and a monotone function $R: \mathcal{D} \to \mathcal{C}$, a left adjoint of $R$ is a monotone function $L: \mathcal{C} \to \mathcal{D}$ such that, for all $x$ in $\mathcal{D}$ and $y$ in $\mathcal{C}$, we have that $L(x) \leq y$ holds if and only if $x \leq R(y)$ holds. \end{defn}

3. Properties

• left adjoints preserve colimits

• left adjoints preserve epimorphisms.

\section{Examples}

• The left adjoint of the nerve functor $N: \mathsf{Grpd} \to \mathsf{Set}^{\Delta^{op}}$ from the category of groupoids to the category of simplicial sets is the fundamental groupoid functor.

4. Related entries

• Galois connection for more on left adjoints of monotone functions.
• adjoint functor for more on left adjoints of functors.
• adjunction for more on left adjoints in $2$-categories.
• examples of adjoint functors for examples.
• pro-left adjoint
• 2-adjunction for a categorified notion of adjunction.