CatGloss

\quad \begin{enumerate} \item There is an endofunctor P ⁣:catcatP \colon \textup{\textsf{cat}} \to \textup{\textsf{cat}} that sends a set AA to its power set PA=AAPA = { A’ \subset A} and a function f ⁣:ABf \colon A \to B to the direct-image function f ⁣:PAPBf_* \colon PA \to PB that sends AAA’ \subset A to f(A)Bf(A’) \subset B. \item Each of the categories listed in Example \ref{exs:concrete-categories} has a forgetful functor, a general term that is used for any functor that forgets structure, whose codomain is the category of sets. For example, U ⁣:catcatU \colon \textup{\textsf{cat}} \to \textup{\textsf{cat}} sends a group to its underlying set and a group homomorphism to its underlying function. The functor U ⁣:catcatU \colon \textup{\textsf{cat}} \to \textup{\textsf{cat}} sends a space to its set of points. There are two natural forgetful functors V,E ⁣:catcatV,E \colon \textup{\textsf{cat}} \rightrightarrows \textup{\textsf{cat}} that send a graph to its vertex or edge sets, respectively; if desired, these can be combined to define a single functor VE ⁣:catcatV \sqcup E \colon \textup{\textsf{cat}} \to \textup{\textsf{cat}} that carries a graph to the disjoint union of its vertex and edge sets. These mappings are functorial because in each instance a morphism in the domain category has an underlying function. \item There are intermediate forgetful functors $\textup{\textsf{cat}}R \to \textup{\textsf{cat}}and and \textup{\textsf{cat}} \to \textup{\textsf{cat}}thatforgetsomebutnotallofthealgebraicstructure.Theinclusionfunctors that forget some but not all of the algebraic structure. The inclusion functors \textup{\textsf{cat}} \hookrightarrow \textup{\textsf{cat}}and and \textup{\textsf{cat}} \hookrightarrow \textup{\textsf{cat}}$ may also be regarded as forgetful.'' Note that the latter two, but neither of the former, are injective on objects: a group is either abelian or not, but an abelian group might admit the structure of a ring in multiple ways. \item Similarly, there are forgetful functors $\textup{\textsf{cat}} \to \textup{\textsf{cat}}_*$ and $\textup{\textsf{cat}} \to \textup{\textsf{cat}}_*$ that take the basepoint to be the identity and zero elements, respectively. These assignments are functorial because group and ring homomorphisms necessarily preserve these elements. \item There are functors $\textup{\textsf{cat}} \to \textup{\textsf{cat}}$ and $\textup{\textsf{cat}}_* \to \textup{\textsf{cat}}_*$ that act as the identity on objects and send a (based) continuous function to its homotopy class. \item The \emph{fundamental group} defines a functor $\pi_1 \colon \textup{\textsf{cat}}_* \to \textup{\textsf{cat}}$; a continuous function $f\colon (X,x) \to (Y,y)$ of based spaces induces a group homomorphism $f_* \colon \pi_1(X,x) \to \pi_1(Y,y)$ and this assignment is functorial, satisfying the two functoriality axioms described above. A precise expression of the statement thatthe fundamental group is a homotopy invariant’’ is that this functor factors through the functor $\textup{\textsf{cat}}* \to \textup{\textsf{cat}}*todefineafunctor to define a functor \pi_1 \colon \textup{\textsf{cat}}* \to \textup{\textsf{cat}}$. \item A related functor $\Pi_1 \colon \textup{\textsf{cat}} \to \textup{\textsf{cat}}$ assigns an unbased topological space its fundamental groupoid, the category defined in Example \ref{exs:groupoid}\eqref{ex:fundamental}. A continuous function $f \colon X \to Yinducesafunctor induces a functor f_* \colon \Pi_1(X) \to \Pi_1(Y)thatcarriesapoint that carries a point x \in Xtothepoint to the point f(x) \in Y.Thismappingextendstomorphismsin. This mapping extends to morphisms in \Pi_1(X)$ because continuous functions preserve paths and path homotopy classes. \item For each $n \in \mathbb{Z},therearefunctors, there are functors Z_n, B_n, H_n \colon \textup{\textsf{cat}}R \to \textup{\textsf{cat}}_R.Thefunctor. The functor Z_ncomputesthe computes the n$-cycles defined by $Z_nC\bullet = \textup{fun}(d \colon C_n \to C_{n-1}).Thefunctor. The functor B_ncomputesthe computes the n$-boundary defined by BnC=fun(d ⁣:Cn+1Cn)B_nC_\bullet = \textup{fun}(d \colon C_{n+1} \to C_n). The functor HnH_n computes the nnth homology HnCZnC/BnCH_nC_\bullet \coloneqq Z_nC_\bullet / B_nC_\bullet. We leave it to the reader to verify that each of these three constructions is functorial. Considering all degrees simultaneously, the cycle, boundary, and homology functors assemble into functors $Z_,B_, H_* \colon \textup{\textsf{cat}}R \to \textup{\textsf{cat}}_Rfromthecategoryofchaincomplexestothecategoryofgraded from the category of chain complexes to the category of graded Rmodules.Thesingularhomologyofatopologicalspaceisdefinedbyprecomposing-modules. The singular homology of a topological space is defined by precomposing Hwithasuitablefunctor with a suitable functor \textup{\textsf{cat}} \to \textup{\textsf{cat}}_R$. \item There is a functor $F \colon \textup{\textsf{cat}} \to \textup{\textsf{cat}}thatsendsaset that sends a set X$ to the free group on XX. This is the group whose elements are finite words'' whose letters are elements $x \in X$ or their formal inverses $x^{-1}$, modulo an equivalence relation that equates the words $xx^{-1}$ and $x^{-1}x$ with the empty word. Multiplication is by concatenation, with the empty word serving as the identity. This is one instance of a large family offree’’ functors studied in Chapter \ref{ch:adjunction}. \item The chain rule expresses the functoriality of the derivative. Let $\textup{\textsf{cat}}_denotethecategorywhoseobjectsarepointedfinitedimensionalEuclideanspaces denote the category whose objects are pointed finite-dimensional Euclidean spaces (\mathbb{R}^n,a)$—or, better, open subsets thereof—and whose morphisms are pointed differentiable functions. The total derivative of f ⁣:RnRmf \colon \mathbb{R}^n \to \mathbb{R}^m, evaluated at the designated basepoint aRna \in \mathbb{R}^n, gives rise to a matrix called the Jacobian matrix defining the directional derivatives of ff at the point aa. If ff is given by component functions f1,,fm ⁣:RnRf_1,\ldots, f_m \colon \mathbb{R}^n \to \mathbb{R}, the (i,j)(i,j)-entry of this matrix is xjfi(a)\frac{\partial}{\partial x_j} f_i(a). This defines the action on morphisms of a functor $D \colon \textup{\textsf{cat}}* \to \textup{\textsf{cat}}\mathbb{R};onobjects,; on objects, DassignsapointedEuclideanspaceitsdimension.Given assigns a pointed Euclidean space its dimension. Given g \colon \mathbb{R}^m \to \mathbb{R}^kcarryingthedesignatedbasepoint carrying the designated basepoint f(a) \in \mathbb{R}^mto to gf(a) \in \mathbb{R}^k,functorialityof, functoriality of DassertsthattheproductoftheJacobianof asserts that the product of the Jacobian of fat at awiththeJacobianof with the Jacobian of gat at f(a)equalstheJacobianof equals the Jacobian of gfat at a.Thisisthechainrulefrommultivariablecalculus.. This is the chain rule from multivariable calculus.\textup{\textsf{cat}}*$ to the category of real vector spaces that sends a pointed manifold to its tangent space.} \item Any commutative monoid $Mcanbeusedtodefineafunctor can be used to define a functor M^{-} \colon \textup{\textsf{cat}}* \to \textup{\textsf{cat}}.Writing. Writing n_+ \in \textup{\textsf{cat}}*forthesetwith for the set with nnonbasepointelements,define non-basepoint elements, define M^{n+}tobe to be M^n,the, the nfoldcartesianproductoftheset-fold cartesian product of the set Mwithitself.Byconvention, with itself. By convention, M^{0+}isasingletonset.Foranybasedmap is a singleton set. For any based map f \colon m+ \to n_+,definethe, define the ithcomponentofthecorrespondingfunctionth component of the corresponding function M^f \colon M^m \to M^nbyprojectingfrom by projecting from M^mtothecoordinatesindexedbyelementsinthefiber to the coordinates indexed by elements in the fiber f^{-1}(i)andthenmultiplyingtheseusingthecommutativemonoidstructure;ifthefiberisempty,thefunction and then multiplying these using the commutative monoid structure; if the fiber is empty, the function M^finsertstheunitelementinthe inserts the unit element in the ithcoordinate.Noteeachofthesetsth coordinate. Note each of the sets M^nitselfhasabasepoint,the itself has a basepoint, the ntupleofunitelements,andeachofthemapsintheimageofthefunctorarebased.Itfollowsthatthefunctor-tuple of unit elements, and each of the maps in the image of the functor are based. It follows that the functor M^{-}liftsalongtheforgetfulfunctor lifts along the forgetful functor U \colon \textup{\textsf{cat}}_* \to \textup{\textsf{cat}}$.