Categories

en · official · sha256:c35031670d5359a2a292434585b10d23418dfd0943422d0fdb83627c50d70a6f

\begin{definition} \label{definition-category} A {\it category} $\mathcal{C}$ consists of the following data: \begin{enumerate} \item A set of objects $\Ob(\mathcal{C})$. \item For each pair $x, y \in \Ob(\mathcal{C})$ a set of morphisms $\Mor_\mathcal{C}(x, y)$. \item For each triple $x, y, z\in \Ob(\mathcal{C})$ a composition map $ \Mor_\mathcal{C}(y, z) \times \Mor_\mathcal{C}(x, y) \to \Mor_\mathcal{C}(x, z) $, denoted $(\phi, \psi) \mapsto \phi \circ \psi$. \end{enumerate} These data are to satisfy the following rules: \begin{enumerate} \item For every element $x\in \Ob(\mathcal{C})$ there exists a morphism $\text{id}_x\in \Mor_\mathcal{C}(x, x)$ such that $\text{id}_x \circ \phi = \phi$ and $\psi \circ \text{id}_x = \psi $ whenever these compositions make sense. \item Composition is associative, i.e., $(\phi \circ \psi) \circ \chi = \phi \circ ( \psi \circ \chi)$ whenever these compositions make sense. \end{enumerate} \end{definition}

Referencing norms: SmallCategoryByObjectSet, composes, composition_associative, has_composition_law, has_identity_morphisms, identity_of, is_category, is_category/necessary, is_category/sufficient, morphism_between, object_of

en · official · sha256:c1b316c2a002f39c4168acc283e6fa9a44429cf5d392050a5a496649f644339f

\begin{definition} \label{definition-isomorphism} A morphism $\phi : x \to y$ is an {\it isomorphism} of the category $\mathcal{C}$ if there exists a morphism $\psi : y \to x$ such that $\phi \circ \psi = \text{id}_y$ and $\psi \circ \phi = \text{id}_x$. \end{definition}

Referencing norms: InverseByTwoSidedComposition, IsomorphismByTwoSidedInverse

en · official · sha256:98035b9f693ad4922a00e8ce061ccc605c1e7927ae9d7989609a22f7f53abee3

\begin{definition} \label{definition-functor} A {\it functor} $F : \mathcal{A} \to \mathcal{B}$ between two categories $\mathcal{A}, \mathcal{B}$ is given by the following data: \begin{enumerate} \item A map $F : \Ob(\mathcal{A}) \to \Ob(\mathcal{B})$. \item For every $x, y \in \Ob(\mathcal{A})$ a map $F : \Mor_\mathcal{A}(x, y) \to \Mor_\mathcal{B}(F(x), F(y))$, denoted $\phi \mapsto F(\phi)$. \end{enumerate} These data should be compatible with composition and identity morphisms in the following manner: $F(\phi \circ \psi) = F(\phi) \circ F(\psi)$ for a composable pair $(\phi, \psi)$ of morphisms of $\mathcal{A}$ and $F(\text{id}_x) = \text{id}_{F(x)}$. \end{definition}

Referencing norms: functor_between, is_functor, is_functor/necessary, is_functor/sufficient, preserves_composition, preserves_identities

en · official · sha256:ed7b96856d6a9233e4ed126c39fed76a1571c3043e467aafe9a1c605cf43b477

\begin{definition} \label{definition-faithful} Let $F : \mathcal{A} \to \mathcal{B}$ be a functor. \begin{enumerate} \item We say $F$ is {\it faithful} if for any objects $x, y \in \Ob(\mathcal{A})$ the map $$ F : \Mor_\mathcal{A}(x, y) \to \Mor_\mathcal{B}(F(x), F(y)) $$ is injective. \item If these maps are all bijective then $F$ is called {\it fully faithful}. \item The functor $F$ is called {\it essentially surjective} if for any object $y \in \Ob(\mathcal{B})$ there exists an object $x \in \Ob(\mathcal{A})$ such that $F(x)$ is isomorphic to $y$ in $\mathcal{B}$. \end{enumerate} \end{definition}

Referencing norms: EquivalenceByFullyFaithfulEssentiallySurjective, FaithfulByInjectivity, FullyFaithfulIsFaithful, essentially_surjective, fully_faithful, injective_on_morphisms

en · official · sha256:3e368f55ede1495d627857287d14cba74a860edbaa8145509352af96a24b0065

\begin{definition} \label{definition-transformation-functors} Let $F, G : \mathcal{A} \to \mathcal{B}$ be functors. A {\it natural transformation}, or a {\it morphism of functors} $t : F \to G$, is a collection $\{t_x\}_{x\in \Ob(\mathcal{A})}$ such that \begin{enumerate} \item $t_x : F(x) \to G(x)$ is a morphism in the category $\mathcal{B}$, and \item for every morphism $\phi : x \to y$ of $\mathcal{A}$ the following diagram is commutative $$ \xymatrix{ F(x) \ar[r]^{t_x} \ar[d]_{F(\phi)} & G(x) \ar[d]^{G(\phi)} \\ F(y) \ar[r]^{t_y} & G(y) } $$ \end{enumerate} \end{definition}

en · official · sha256:789d9b2be653dbe7473fc87fdeabbd0b134dd39cadcfe4db34ed16b3f1a35ab9

\begin{definition} \label{definition-equivalence-categories} An {\it equivalence of categories} $F : \mathcal{A} \to \mathcal{B}$ is a functor such that there exists a functor $G : \mathcal{B} \to \mathcal{A}$ such that the compositions $F \circ G$ and $G \circ F$ are isomorphic to the identity functors $\text{id}_\mathcal{B}$, respectively $\text{id}_\mathcal{A}$. In this case we say that $G$ is a {\it quasi-inverse} to $F$. \end{definition}

Referencing norms: EquivalenceByQuasiInverse, quasi_inverse_of

en · official · sha256:8cd0745eb15edeadfa62e9e9bb311c3f24c249ae65467233a0f38d02543bba03

\begin{definition} \label{definition-products} Let $x, y\in \Ob(\mathcal{C})$. A {\it product} of $x$ and $y$ is an object $x \times y \in \Ob(\mathcal{C})$ together with morphisms $p\in \Mor_{\mathcal C}(x \times y, x)$ and $q\in\Mor_{\mathcal C}(x \times y, y)$ such that the following universal property holds: for any $w\in \Ob(\mathcal{C})$ and morphisms $\alpha \in \Mor_{\mathcal C}(w, x)$ and $\beta \in \Mor_\mathcal{C}(w, y)$ there is a unique $\gamma\in \Mor_{\mathcal C}(w, x \times y)$ making the diagram $$ \xymatrix{ w \ar[rrrd]^\beta \ar@{-->}[rrd]_\gamma \ar[rrdd]_\alpha & & \\ & & x \times y \ar[d]_p \ar[r]_q & y \\ & & x & } $$ commute. \end{definition}

Referencing norms: FiniteLimitsByProductsAndEqualizers, has_finite_products

en · official · sha256:82456a3beeaee23c52649970a6a5e816e988b9ccdfa870aaeff3de315f9e5fdb

\begin{definition} \label{definition-initial-final} Let $\mathcal{C}$ be a category. \begin{enumerate} \item An object $x$ of the category $\mathcal{C}$ is called an {\it initial} object if for every object $y$ of $\mathcal{C}$ there is exactly one morphism $x \to y$. \item An object $x$ of the category $\mathcal{C}$ is called a {\it final} object if for every object $y$ of $\mathcal{C}$ there is exactly one morphism $y \to x$. \end{enumerate} \end{definition}

Referencing norms: FinalObjectExists, FiniteLimitsByFinalObjectAndFibreProducts, final_object, initial_object

en · official · sha256:9a725b179f8be18d5729ec5e6b7f4adff6392fede74ca2789d30a490cdd1e555

\begin{definition} \label{definition-limit} A {\it limit} of the $\mathcal{I}$-diagram $M$ in the category $\mathcal{C}$ is given by an object $\lim_\mathcal{I} M$ in $\mathcal{C}$ together with morphisms $p_i : \lim_\mathcal{I} M \to M_i$ such that \begin{enumerate} \item for $\phi : i \to i'$ a morphism in $\mathcal{I}$ we have $p_{i'} = M(\phi) \circ p_i$, and \item for any object $W$ in $\mathcal{C}$ and any family of morphisms $q_i : W \to M_i$ (indexed by $i \in \Ob(\mathcal{I})$) such that for all $\phi : i \to i'$ in $\mathcal{I}$ we have $q_{i'} = M(\phi) \circ q_i$ there exists a unique morphism $q : W \to \lim_\mathcal{I} M$ such that $q_i = p_i \circ q$ for every object $i$ of $\mathcal{I}$. \end{enumerate} \end{definition}

Referencing norms: has_equalizers, has_fibre_products

en · official · sha256:f1458903a141c2863ad4ddfc7139d0b3fb66799aa82fc4b71196724d55f9c821

\begin{definition} \label{definition-colimit} A {\it colimit} of the $\mathcal{I}$-diagram $M$ in the category $\mathcal{C}$ is given by an object $\colim_\mathcal{I} M$ in $\mathcal{C}$ together with morphisms $s_i : M_i \to \colim_\mathcal{I} M$ such that \begin{enumerate} \item for $\phi : i \to i'$ a morphism in $\mathcal{I}$ we have $s_i = s_{i'} \circ M(\phi)$, and \item for any object $W$ in $\mathcal{C}$ and any family of morphisms $t_i : M_i \to W$ (indexed by $i \in \Ob(\mathcal{I})$) such that for all $\phi : i \to i'$ in $\mathcal{I}$ we have $t_i = t_{i'} \circ M(\phi)$ there exists a unique morphism $t : \colim_\mathcal{I} M \to W$ such that $t_i = t \circ s_i$ for every object $i$ of $\mathcal{I}$. \end{enumerate} \end{definition}

Referencing norms: has_finite_colimits

en · official · sha256:7850adcc9f4c3c252de3f473e2eb70895b2a4f50255a93f145d375166f940f06

\begin{lemma} \label{lemma-finite-limits-exist} Let $\mathcal{C}$ be a category. The following are equivalent: \begin{enumerate} \item Finite limits exist in $\mathcal{C}$. \item Finite products and equalizers exist. \item The category has a final object and fibre products exist. \end{enumerate} \end{lemma}

Referencing norms: FiniteLimitsByFinalObjectAndFibreProducts, FiniteLimitsByProductsAndEqualizers

en · official · sha256:7407d5b3e5592473a0541a9117a77025cb05b8ec2052c3e8930d5c4c133661f3

\begin{definition} \label{definition-exact} Let $F : \mathcal{A} \to \mathcal{B}$ be a functor. \begin{enumerate} \item Suppose all finite limits exist in $\mathcal{A}$. We say $F$ is {\it left exact} if it commutes with all finite limits. \item Suppose all finite colimits exist in $\mathcal{A}$. We say $F$ is {\it right exact} if it commutes with all finite colimits. \item We say $F$ is {\it exact} if it is both left and right exact. \end{enumerate} \end{definition}

Referencing norms: ExactIfLeftAndRightExact, LeftExactByFiniteLimits, RightAdjointIsLeftExact, RightExactByFiniteColimits, commutes_with_finite_colimits, commutes_with_finite_limits

en · official · sha256:7880129c4b0eba906cf5a2bd603052723493bdb962086f25871ce89b5d8b41c8

\begin{lemma} \label{lemma-characterize-left-exact} Let $F : \mathcal{A} \to \mathcal{B}$ be a functor. Suppose all finite limits exist in $\mathcal{A}$, see Lemma \ref{lemma-finite-limits-exist}. The following are equivalent: \begin{enumerate} \item $F$ is left exact, \item $F$ commutes with finite products and equalizers, and \item $F$ transforms a final object of $\mathcal{A}$ into a final object of $\mathcal{B}$, and commutes with fibre products. \end{enumerate} \end{lemma}

Referencing norms: LeftExactByProductsAndEqualizers, commutes_with_finite_products_and_equalizers

en · official · sha256:3d4edfd460e9abd34b6ed635af88a1d242927e7fef432173b327a4c0cc3f262f

\begin{definition} \label{definition-adjoint} Let $\mathcal{C}$, $\mathcal{D}$ be categories. Let $u : \mathcal{C} \to \mathcal{D}$ and $v : \mathcal{D} \to \mathcal{C}$ be functors. We say that $u$ is a {\it left adjoint} of $v$, or that $v$ is a {\it right adjoint} to $u$ if there are bijections $$ \Mor_\mathcal{D}(u(X), Y) \longrightarrow \Mor_\mathcal{C}(X, v(Y)) $$ functorial in $X \in \Ob(\mathcal{C})$, and $Y \in \Ob(\mathcal{D})$. \end{definition}

Referencing norms: RightAdjointIsLeftExact, left_adjoint_of

en · official · sha256:e25f3df9f3a8823643b254a954d882312b23b49d765e2a1987ef7d0d81d20fa9

\begin{lemma} \label{lemma-adjoint-exact} Let $u$ be a left adjoint to $v$ as in Definition \ref{definition-adjoint}. \begin{enumerate} \item Suppose that $M : \mathcal{I} \to \mathcal{C}$ is a diagram, and suppose that $\colim_\mathcal{I} M$ exists in $\mathcal{C}$. Then $u(\colim_\mathcal{I} M) = \colim_\mathcal{I} u \circ M$. In other words, $u$ commutes with (representable) colimits. \item Suppose that $M : \mathcal{I} \to \mathcal{D}$ is a diagram, and suppose that $\lim_\mathcal{I} M$ exists in $\mathcal{D}$. Then $v(\lim_\mathcal{I} M) = \lim_\mathcal{I} v \circ M$. In other words $v$ commutes with representable limits. \end{enumerate} \end{lemma}

Referencing norms: RightAdjointIsLeftExact

en · official · sha256:658acca18c1d0501ad036afb269da7c2d49542c030f4f50fac0b69ea821a3d91

\begin{definition} \label{definition-mono-epi} Let $\mathcal{C}$ be a category and let $f : X \to Y$ be a morphism of $\mathcal{C}$. \begin{enumerate} \item We say that $f$ is a {\it monomorphism} if for every object $W$ and every pair of morphisms $a, b : W \to X$ such that $f \circ a = f \circ b$ we have $a = b$. \item We say that $f$ is an {\it epimorphism} if for every object $W$ and every pair of morphisms $a, b : Y \to W$ such that $a \circ f = b \circ f$ we have $a = b$. \end{enumerate} \end{definition}

Referencing norms: epimorphism, epimorphism/necessary, epimorphism/sufficient, left_cancellable, monomorphism, monomorphism/necessary, monomorphism/sufficient, right_cancellable

en · official · sha256:ce1526b3bb4d4fd699176be4f750b85aa0d5a78b01fc818117e9a5880ee54662

\begin{lemma} \label{lemma-equivalence-categories} A functor is an equivalence of categories if and only if it is both fully faithful and essentially surjective. \end{lemma}

Referencing norms: EquivalenceByFullyFaithfulEssentiallySurjective