This course develops the categorical language needed to describe algebraic structures through universal properties, then pushes that language into the setting of limits, exactness, and abelian categories. It begins with adjunctions as the organizing principle behind many familiar constructions, such as free objects, forgetful functors, and algebraic examples from groups, rings, modules, and related structures. From there, it studies how adjunctions connect to equivalences and monads, and how universal properties reappear in the theory of limits and colimits.

The later chapters build the machinery needed for modern homological algebra. After limits and colimits, the course turns to completeness, cocontinuity, creation of limits, and Kan extensions, culminating in a preview of adjoint functor theorems. It then shifts to additive and preadditive categories, introduces kernels, cokernels, and abelian categories, and uses exact sequences and diagram lemmas to formalize the behavior of algebraic objects under functors. The final chapters treat projectives, injectives, exact functors, and a preview of derived functors, so the course moves from structural category theory to the tools that underpin much of algebra and homological algebra.

# 1. Adjunctions From Universal Properties

Adjunctions organise universal constructions by recording not just an object with a property, but a coherent way of transporting maps through that property. The chapter moves from external data, namely natural bijections of hom-sets, to internal data, namely units and counits, and then to the universal-arrow formulation that appears in concrete constructions. The main obstruction throughout is coherence: pointwise bijections or isolated universal objects are not enough unless they vary correctly with morphisms.

## Hom-Set Adjunctions

Suppose a construction assigns to each object $c$ of a category $\mathcal C$ an object $Fc$ of a category $\mathcal D$. When should $Fc$ deserve to be called the object freely generated by $c$, or the best $\mathcal D$-object associated to $c$? The answer is that maps out of $Fc$ should be controlled exactly by maps out of $c$ after applying a comparison functor back to $\mathcal C$.

A pointwise bijection of hom-sets is not enough. If the chosen bijection changes arbitrarily when $c$ or $d$ is replaced by an isomorphic object, then it does not describe a construction in the categories; it describes unrelated set-theoretic coincidences. Naturality is the condition that rules out this failure.

[definition: Hom-Set Adjunction] Let $F: \mathcal C \to \mathcal D$ and $G: \mathcal D \to \mathcal C$ be functors. An adjunction $F \dashv G$ is a family of bijections \begin{align*} \Phi_{c,d}: \mathcal D(Fc,d) \longrightarrow \mathcal C(c,Gd) \end{align*} for all objects $c \in \mathcal C$ and $d \in \mathcal D$, natural in both $c$ and $d$. [/definition]

The functor $F$ is called the left adjoint and $G$ is called the right adjoint. Naturality means that the bijection is compatible with changing $c$ by a morphism in $\mathcal C$ and changing $d$ by a morphism in $\mathcal D$. Thus an adjunction is a coherent identification of two bifunctors

\begin{align*} \mathcal D(F-, -), \mathcal C(-,G-) : \mathcal C^{\mathrm{op}} \times \mathcal D \to \operatorname{Set}. \end{align*}

To use this coherence in calculations, we need the explicit two-variable naturality equation. It is the rule that lets one move morphisms in the source and target through the adjunction bijection without changing the represented map.

[definition: Natural In Both Variables] For a family $\Phi_{c,d}: \mathcal D(Fc,d) \to \mathcal C(c,Gd)$, naturality in both variables means that for every morphism $u:c' \to c$ in $\mathcal C$, every morphism $v:d \to d'$ in $\mathcal D$, and every morphism $f:Fc \to d$, one has \begin{align*} \Phi_{c',d'}(v \circ f \circ Fu) = Gv \circ \Phi_{c,d}(f) \circ u. \end{align*} [/definition]

This formula is often the most efficient way to use an adjunction. It says that the adjunct of a composite is the corresponding composite of adjuncts after applying the functors on the correct sides.

[example: Free Group And Underlying Set] Let $U:\operatorname{Grp}\to\operatorname{Set}$ be the forgetful functor, and let $F(S)$ be the free group on a set $S$. Its elements are reduced words \begin{align*} s_1^{\epsilon_1}s_2^{\epsilon_2}\cdots s_n^{\epsilon_n}, \end{align*} where $s_i\in S$, $\epsilon_i\in\{1,-1\}$, and adjacent inverse pairs have been cancelled. We compute the adjunction bijection \begin{align*} \operatorname{Grp}(F(S),H)\cong \operatorname{Set}(S,UH). \end{align*} Given a group homomorphism $\varphi:F(S)\to H$, define its underlying function on generators by \begin{align*} \Phi(\varphi):S&\longrightarrow U(H),\\ s&\longmapsto \varphi(s). \end{align*} Conversely, given a function $a:S\to U(H)$, define $\widetilde a:F(S)\to H$ on a reduced word by \begin{align*} \widetilde a(s_1^{\epsilon_1}s_2^{\epsilon_2}\cdots s_n^{\epsilon_n}) = a(s_1)^{\epsilon_1}a(s_2)^{\epsilon_2}\cdots a(s_n)^{\epsilon_n}, \end{align*} where $a(s)^1=a(s)$ and $a(s)^{-1}$ is the inverse of $a(s)$ in $H$. This respects cancellation because if a word contains an adjacent pair $ss^{-1}$, its image contains \begin{align*} a(s)a(s)^{-1}=e_H, \end{align*} and if it contains $s^{-1}s$, its image contains \begin{align*} a(s)^{-1}a(s)=e_H. \end{align*} Thus deleting either adjacent inverse pair does not change the product in $H$. The two constructions are inverse. Starting with $a:S\to U(H)$ and restricting $\widetilde a$ to generators gives \begin{align*} \Phi(\widetilde a)(s)=\widetilde a(s)=a(s) \end{align*} for every $s\in S$. Starting with a homomorphism $\varphi:F(S)\to H$, the extension of its restriction to $S$ sends a reduced word to \begin{align*} \widetilde{\Phi(\varphi)}(s_1^{\epsilon_1}\cdots s_n^{\epsilon_n}) &= \Phi(\varphi)(s_1)^{\epsilon_1}\cdots \Phi(\varphi)(s_n)^{\epsilon_n}\\ &= \varphi(s_1)^{\epsilon_1}\cdots \varphi(s_n)^{\epsilon_n}\\ &= \varphi(s_1^{\epsilon_1}\cdots s_n^{\epsilon_n}), \end{align*} where the last equality uses preservation of multiplication and inverses by the group homomorphism $\varphi$. Hence a homomorphism out of $F(S)$ is exactly a choice of elements of $H$ indexed by $S$, which is the universal property encoded by $F\dashv U$. [/example]

The example shows the slogan: left adjoints build freely, right adjoints forget structure or record a space of observations. The word "free" is not part of the definition, but many familiar free constructions are left adjoints because their universal property is a hom-set bijection.

[example: Tensor Hom For Modules] Let $R$ be a commutative ring and let $M$ be an $R$-module. We compute the adjunction \begin{align*} \operatorname{Hom}_R(M \otimes_R N, P) \cong \operatorname{Hom}_R(N,\operatorname{Hom}_R(M,P)) \end{align*} for $R$-modules $N$ and $P$. Given an $R$-linear map $b:M\otimes_R N\to P$, define \begin{align*} \Phi(b):N&\longrightarrow \operatorname{Hom}_R(M,P),\\ n&\longmapsto \bigl(m\longmapsto b(m\otimes n)\bigr). \end{align*} For each fixed $n\in N$, the map $m\mapsto b(m\otimes n)$ is $R$-linear because \begin{align*} b((m_1+m_2)\otimes n) &=b(m_1\otimes n+m_2\otimes n)\\ &=b(m_1\otimes n)+b(m_2\otimes n), \end{align*} and \begin{align*} b((r m)\otimes n) &=b(r(m\otimes n))\\ &=r\,b(m\otimes n). \end{align*} The assignment $n\mapsto \Phi(b)(n)$ is also $R$-linear, since for $m\in M$, \begin{align*} \Phi(b)(n_1+n_2)(m) &=b(m\otimes(n_1+n_2))\\ &=b(m\otimes n_1+m\otimes n_2)\\ &=b(m\otimes n_1)+b(m\otimes n_2)\\ &=\bigl(\Phi(b)(n_1)+\Phi(b)(n_2)\bigr)(m), \end{align*} and \begin{align*} \Phi(b)(r n)(m) &=b(m\otimes r n)\\ &=b(r(m\otimes n))\\ &=r\,b(m\otimes n)\\ &=\bigl(r\,\Phi(b)(n)\bigr)(m). \end{align*} Conversely, given an $R$-linear map $a:N\to \operatorname{Hom}_R(M,P)$, define \begin{align*} \beta_a:M\times N&\longrightarrow P,\\ (m,n)&\longmapsto a(n)(m). \end{align*} This rule is $R$-bilinear. In the first variable, \begin{align*} \beta_a(m_1+m_2,n) &=a(n)(m_1+m_2)\\ &=a(n)(m_1)+a(n)(m_2), \end{align*} and \begin{align*} \beta_a(rm,n) &=a(n)(rm)\\ &=r\,a(n)(m). \end{align*} In the second variable, \begin{align*} \beta_a(m,n_1+n_2) &=a(n_1+n_2)(m)\\ &=(a(n_1)+a(n_2))(m)\\ &=a(n_1)(m)+a(n_2)(m), \end{align*} and \begin{align*} \beta_a(m,rn) &=a(rn)(m)\\ &=(r\,a(n))(m)\\ &=r\,a(n)(m). \end{align*} Hence $\beta_a$ is balanced, so by the universal property of the tensor product there is a unique $R$-linear map \begin{align*} \widetilde a:M\otimes_R N\longrightarrow P \end{align*} such that \begin{align*} \widetilde a(m\otimes n)=a(n)(m) \end{align*} for all $m\in M$ and $n\in N$. The two constructions are inverse. Starting with $a:N\to \operatorname{Hom}_R(M,P)$, for every $n\in N$ and $m\in M$, \begin{align*} \Phi(\widetilde a)(n)(m) &=\widetilde a(m\otimes n)\\ &=a(n)(m), \end{align*} so $\Phi(\widetilde a)=a$. Starting with $b:M\otimes_R N\to P$, the map $\widetilde{\Phi(b)}$ satisfies \begin{align*} \widetilde{\Phi(b)}(m\otimes n) &=\Phi(b)(n)(m)\\ &=b(m\otimes n) \end{align*} on every pure tensor $m\otimes n$; by the uniqueness part of the tensor product universal property, $\widetilde{\Phi(b)}=b$. Thus maps out of $M\otimes_R N$ are exactly $R$-linear families of maps out of $M$ indexed by $N$, which is the adjunction $M\otimes_R -\dashv \operatorname{Hom}_R(M,-)$. [/example]

This second example is central because it turns a bilinear operation into an ordinary hom-set. Many later exactness arguments in homological algebra come from knowing which side of this adjunction preserves limits or colimits.

[example: Product And Mapping Space] Let $\mathcal E$ be a cartesian closed category, fix an object $X$, and write $Z^X$ for the exponential object. By definition of the exponential, there is an evaluation morphism \begin{align*} \operatorname{ev}_{X,Z}:Z^X\times X\longrightarrow Z \end{align*} such that for every object $Y$ and every morphism $f:Y\times X\to Z$, there is a unique morphism $\lambda f:Y\to Z^X$ satisfying \begin{align*} \operatorname{ev}_{X,Z}\circ(\lambda f\times \operatorname{id}_X)=f. \end{align*} Thus the transposition map is \begin{align*} \Phi_{Y,Z}:\mathcal E(Y\times X,Z)&\longrightarrow \mathcal E(Y,Z^X),\\ f&\longmapsto \lambda f. \end{align*} Conversely, from a morphism $g:Y\to Z^X$, recover a morphism $Y\times X\to Z$ by \begin{align*} \Psi_{Y,Z}(g) &=\operatorname{ev}_{X,Z}\circ(g\times \operatorname{id}_X). \end{align*} These two operations are inverse. Starting with $f:Y\times X\to Z$, the defining equation for $\lambda f$ gives \begin{align*} \Psi_{Y,Z}(\Phi_{Y,Z}(f)) &=\Psi_{Y,Z}(\lambda f)\\ &=\operatorname{ev}_{X,Z}\circ(\lambda f\times \operatorname{id}_X)\\ &=f. \end{align*} Starting with $g:Y\to Z^X$, the morphism $g$ itself satisfies \begin{align*} \operatorname{ev}_{X,Z}\circ(g\times \operatorname{id}_X)=\Psi_{Y,Z}(g), \end{align*} so by the uniqueness clause in the exponential universal property, \begin{align*} \Phi_{Y,Z}(\Psi_{Y,Z}(g))=g. \end{align*} Therefore \begin{align*} \mathcal E(Y\times X,Z)\cong \mathcal E(Y,Z^X), \end{align*} naturally in $Y$ and $Z$. In compactly generated weak Hausdorff spaces, this says that a continuous map $Y\times X\to Z$ is equivalently a continuous map $Y\to Z^X$, whose value at $y$ is the function $x\mapsto f(y,x)$. [/example]

This topological example explains why adjunctions are not confined to algebraic freeness. They also encode exponential objects, currying, and internal function spaces whenever the ambient category supports them.

## Units And Counits