# 18-02-14: Adjoint and Classifying Spaces In general, we define the classifying space $K(G, n)$ (also known as an Eilenberg-MacLane space) to be a space $X$ such that $\pi_n(X) = G$ and for $k\neq n,~\pi_k(X) = 0$. *Note: in my notation, I will simply write this as $\pi_*(X) = G\delta_n$* It is worth mentioning here that there are nice Serre spectral sequences for this family of fibrations: $$K(\ZZ, n-1) \to \pt \to K(\ZZ, n)$$ By examining an appropriate spectral sequence, we were able to find that $H_*(\RP^\infty) = \ZZ_2\delta_1$, which makes $\RP^\infty$ an geometric model of the classifying space $K(\ZZ_2, 1)$. Recall that $\CP^\infty$ is defined as the limit of the sequence of inclusions $$\CP^1 \subset \CP^2 \subset \CP^3 \subset \ldots$$ together with the weak limit topology. There are a handful of easily recognizable geometric models for a few other types of classifying spaces. $G$ \\ $n$ | 1 | 2 | 3 -- |--- |--- |-- $\ZZ$ |$S^1$ |$\CP^\infty$ | No good model! $\ZZ_2$ |$\RP^\infty$ |$\cdot$ |$\cdot$ $\ZZ_p$ |$L(\infty, p)$ |$\cdot$ |$\cdot$ $\ast_n \ZZ$ |$\bigvee_n S^1$ |$\cdot$ |$\cdot$ *Note: $\ast_n \ZZ$ is the free group on $n$ generators. Also, these spaces can all be constructed as a CW complex for any given $G$ - just start with some $\bigvee S^1$ and add cells to kill off all higher homotopy.* Using spectral sequences, we also found that $K(\ZZ, 3)$ was a space that, although simple from the point of view of homotopy, had a more complicated structure in homology. It was a number of odd properties- it has torsion in infinitely many dimensions, doesn't satisfy Poincare duality (even in a truncated sense). --- Consider the fibration $$S^1 \to S^{2\infty + 1} \to \CP^\infty$$ where these infinite-dimensional spaces are defined using the weak topology. There is a perfectly good filtration arising from the inclusions in this diagram: $$\begin{CD} S^3 @>\subseteq>> S^5 @>\subseteq>> S^7 @>\subseteq>> \cdots\\ @VVV @VVV @VVV \\ \CP^1 @>\subseteq>> \CP^3 @>\subseteq>> \CP^5 @>\subseteq>> \cdots \end{CD}$$ So we can apply the usual spectral sequence to this filtration. We know that $E_\infty$ can only contain $\ZZ$ in dimension zero, and we obtain the following $E_2$ page: latex {cmd=true, hide=true, run_on_save=true} \documentclass{standalone} \usepackage{tikz} \usepackage{dsfont} \usepackage{amsmath, amsthm, amssymb} \usetikzlibrary{cd} \begin{document} \begin{tikzcd} \mathbb{Z} \arrow[rrd, "d_2 \cong"] & 0 & 0 \arrow[rrd, "d_2 \cong"] & 0 & 0 \\ 0 & 0 & \mathbb{Z} & 0 & \mathbb{Z} \end{tikzcd} \end{document}  Since $d_2$ is an isomorphism, it must take generators to generators, and so we can deduce the following facts: - $d_2(\alpha \tensor 1) = 1\tensor\beta$ - $d_2(1\tensor\beta) = 0$ We can now compute \begin{align} d_2(\alpha\tensor\beta) &= d_2(\alpha\tensor 1) \cup (1\tensor\beta) + 0\\ &= 1\tensor\beta^2 \end{align} And using the cup product structure on cohomology, we can fill out the following diagram that summarizes these results: latex {cmd=true, hide=true, run_on_save=true} \documentclass{standalone} \usepackage{tikz} \usepackage{dsfont} \usepackage{amsmath, amsthm, amssymb} \usetikzlibrary{cd} \newcommand*\ZZ{\mathds{Z}} \begin{document} \begin{tikzcd} \alpha\otimes 1 \arrow[rrd, "d_2 \cong"] & \cdot & \alpha\otimes\beta \arrow[rrd, "d_2 \cong"] & \cdot & \cdot \\ \cdot & \cdot & 1\otimes \beta \arrow[u, "\cup"] & \cdot & 1\otimes\beta^2 \end{tikzcd} \end{document}  Thus, just from knowing that $d_2$ is an isomorphism, we can conclude that $H^4(\CP^\infty) = \ZZ<\beta^2>$. Alternatively, we'll write this as $H^4(\CP^\infty) = \ZZ.\beta^2$ By a repeated application of this argument, we find that $H^{2n}(\CP^\infty) = \ZZ.\beta^n$, allowing us to conclude that $$H^*(\CP^\infty) = \ZZ[\beta_{(2)}]. \qed$$ ---- If we know $H^*(\CP^\infty)$, which is the easiest case, we can then use the inclusion $\CP^n \mapsvia{i} \CP^\infty$ (as a cellular map) to induce $$H^*(\CP^n) \mapsvia{i^*} H^*(\CP^\infty)\\ \beta \mapsto \beta$$ which is a actually a *ring* homomorphism instead of just a group homomorphism. This presents a good argument for the use of cohomology, due to its extra ring structure. This is an isomorphism on low-dimensional (co)homology, which reflects the idea encapsulated in the weak limit that these should be approximately equal for large enough $n$. This is indicative of a general principle: if $X$ is a CW complex and $X^n$ is its $n$-skeleton, then the inclusion $X^n \mapsvia{i} X$ induces an isomorphism $H_k(X^n) \cong H_k(X)$ for $k < n$. (Note that this may or may not be an isomorphism for $k=n$.) In particular, it is again a ring homomorphism, and so carries true relations/equations to true relations/equations. Dually, homology does have *some* type of ring structure, however it is slightly unnatural and onerous to define and use. There is a natural coproduct on $H_*(X)$ for any space $X$, which has a "one in, two out" type and takes this form: $$H_*(X) \mapsvia{\Delta} H_*(X) \cross H_*(X) \\ a \mapsto \sum a' \tensor a''$$ This coproduct satisfies a form of coassociativity, i.e. if we have $$\Delta(a) = \sum b_i \otimes c_i \\ (\Delta \tensor 1) \Delta (a) = \sum_{i,j} (d_j^i \tensor e_j^i)\tensor c_i \\ (1 \tensor \Delta ) \Delta (a) = \sum_{i,j} b_i \tensor (f_k^i \tensor g_k^i)$$ then the "structure coefficients" agree, i.e. we have $b_i = \sum_j (d_j^i \tensor e_j^i)$ and $c_i = \sum_k (f_k^i \tensor g_k^i)$. In other words, just note that each element on the right hand side of these equations is an element of $H_*^{\otimes 3}$, and so coassociativity simply requires that they are the same element of this space. ---- We can specialize by looking at the case where $V$ is a vector space, with a coproduct $V \mapsvia{\Delta} V\tensor V$. Then pick a basis $\{e_i\}_{i\in I}$, and write $$\Delta(e_i) = \sum_{j,k} \Delta_i^{j,k}(e_j\tensor e_k)$$ where $\Delta_i^{j,k} \in k$, the ground field of $V$. Then coassociativity requires that we have $$\sum_{j,k,l,m} \Delta_i^{j,k} \Delta_j^{l,m} (e_l \tensor e_m \tensor e_k) = \sum_{j,k}\Delta_i^{j,k} e_j \tensor \Delta_k^{p,q}(e_p \tensor e_q)$$ or in other words, that $$\sum_j \Delta_i^{j,k} \Delta_j^{l,m} = \sum_i \Delta_i^{l,r} \Delta_r^{m,k}\tag*{\forall k,l,m}$$ It is worth noting that there is also a version of the universal coefficient theorem for homology, which comes in the form $$0 \to \ext(H_{n-1}(X, \ZZ), \ZZ) \to H_n(X, \ZZ) \to \hom(H_n(X,\ZZ),\ZZ) \to 0$$ ---- One question that comes up here is whether or not there is a sense in which Ext and Hom are "duals" of each other. In some way, this is case, using the "Frobenius duality" of $\wait \tensor R$ and $\hom(\wait, S)$. *Aside: Frobenius duality occurs in algebras $A$ over some field $k$ possessing a nondegenerate bilinear form $A\cross A \mapsvia{\sigma} k$ satisfying $\sigma(ab, c) = \sigma(a, bc)$. Such a \sigma$is called a Frobenius norm. A simple example is the trace of a matrix, another example is any Hopf algebra.* This kind of duality comes in the form of something like $$\hom(M\tensor N, P) = \hom(M, \hom_{\text{in}}(N, P))$$ where$\hom_\text{in}$is an "internal hom", which is actually an object in the category whose underlying set is the usual$\hom$. One might also call this "map", and denote it$[N, P]$, then the above statement translates to the condition that if$N, P \in \mathcal{C}$for some category, then$\hom_\text{in}(N, P) = [N, P] \in \mathcal{C}$is also an object in the same category. (This might also be denoted$\mathcal{Hom}$.) For an analogy, let$\mathcal{C} = \mathbf{Top}$, and$\hom_\mathbf{Top}(X,Y)$be the set of continuous maps from$X$to$Y$. Then notice that we can put a topology on this space, say$\mathcal{T}$, so define$\text{Map}(X, Y) = (\hom_\mathbf{Top}(X,Y), \mathcal{T})$, which is in fact an **object** in$\mathbf{Top}$. This becomes the aforementioned "internal hom". Then, the previous adjunction becomes $$\hom_\textbf{Top}(X\cross Y, Z) = \hom_\textbf{Top}(X, \text{Map}(X, Y)) \tag*{(\in \mathbf{Set})}$$ ---- More generally, consider what happens in categories of$R$modules, where$R$is generally non-commutative. We can then take objects like$M_R \in \mathbf{mod\dash R}$and${}_{R}N_{S} \in \mathbf{R\dash mod\dash S}$. We can then form the tensor product$M_R \tensor_R {}_{R}N_{S}$, and the adjunction becomes $$\hom_\mathbf{mod\dash S} (M_R \tensor_R {}_{R}N_{S}, P_S) = \hom_\mathbf{mod\dash R} (M_R, \hom_\mathbf{mod\dash S} ({}_{R}N_{S}, P_S)) \tag*{(\in \mathbf{Ab})}$$ Again, in the second argument of the right-hand side, we identify this as an internal hom - this works because the object$\hom_\mathbf{mod\dash S} ({}_{R}N_{S}, P_S)$actually becomes a right$R$-module by precomposition. ---- In some ways, this resembles the kind of adjunction that occurs in an inner product space - for example, given a matrix$A$, it may have an "adjoint" matrix$A^*$that satisfies $$\inner{Av}{w} = \inner{w}{A^*v}$$ and so we can think of$\hom$like a Hermitian inner product of this form, which is contravariant (re: conjugate) in the first argument. Note that the choice of which argument is contrvariant varies! In Physics, the second argument is often conjugate-linear, while the first is linear. We can also look at this as an almost-commuting of the following diagram latex {cmd=true, hide=true, run_on_save=true} \documentclass{standalone} \usepackage[utf8]{inputenc} \usepackage{tikz-cd} \newcommand*\wait{\,\cdot\,} \newcommand*\tensor{\otimes} \newcommand*\dash{\hbox{-}} \newcommand*\cross{\times} \newcommand\inner[2]{\langle #1, #2 \rangle} \begin{document} \begin{tikzcd} \mathbf{mod\dash R} \arrow[ddd, "\wait \tensor_R N_s"', dashed, maps to, bend right=49] \arrow[ddd] & \cross & \mathbf{mod\dash R} \arrow[rr, "\inner{\wait}{\wait}"] & & \mathbf{Ab} \arrow[ddd] \\ & & & & \\ & & & & \\ \mathbf{mod\dash S} & \cross & \mathbf{mod\dash S} \arrow[uuu, "{\hom_R(N_{S}, \wait)}"', dashed, bend right=49] \arrow[rr, "\inner{\wait}{\wait}"] \arrow[uuu] & & \mathbf{Ab} \arrow[uuu, "\cong"'] \end{tikzcd} \end{document}  where we can simplify by choosing elements, yielding latex {cmd=true, hide=true, run_on_save=true} \documentclass{standalone} \usepackage[utf8]{inputenc} \usepackage{tikz-cd} \newcommand*\wait{\,\cdot\,} \newcommand*\tensor{\otimes} \newcommand*\dash{\hbox{-}} \newcommand*\cross{\times} \newcommand\inner[2]{\langle #1, #2 \rangle} \begin{document} \begin{tikzcd} M \arrow[ddd] & \cross & \hom_R(N, P) \arrow[rr, "\hom_R"] & & \mathbf{Ab} \arrow[ddd] \\ & & & & \\ & & & & \\ M \tensor_R N & \cross & P \arrow[rr, "\hom_S"] \arrow[uuu] & & \mathbf{Ab} \arrow[uuu, "\cong"'] \end{tikzcd} \end{document}  In this framework, we can now talk about pairs of adjoint functors$\mathcal{C}\underset{L}{\overset{R}\leftrightarrow} \mathcal{D}$between categories, which satisfy $$\hom_\mathcal{C}(LA, X) = \hom_\mathcal{D}(A, RX)$$ for every$A\in\mathcal{D}, X\in\mathcal{C}$, plus a few more properties concerning how these act under natural transformations. Then$L$is said to be left adjoint to$R$, and$R$is right adjoint to$L$, which is sometimes denoted$L \vdash R$. **Example**: Free and forgetful functors. Work in$\mathbf{Grp}$and$\mathbf{Set}$, then let$F$be the free group functor and$U$by the forgetful functor. Then we have $$\hom_\mathbf{Grp}(F(S), G) \cong \hom_\mathbf{Set}(S, U(G))$$ **Example**: The classifying space functor. Define the classifying space functor$\mathbf{Cat} \mapsvia{B} \mathbf{Set}$, denoted$\abs{\wait}$. As an input, it takes a category$\mathcal{C}$, then define a simplicial complex where the - The vertices (0-simplices) are the objects, - The edges (1-simplices) are the morphisms, - The 2-simplices are triangles latex {cmd=true, hide=true, run_on_save=true} \documentclass{standalone} \usepackage[utf8]{inputenc} \usepackage{tikz-cd} \begin{document} \begin{tikzcd} & & B \arrow[rrdd, "g"] & & \\ & & & & \\ A \arrow[rruu, "f"] \arrow[rrrr, "fg"] & & & & C \end{tikzcd} \end{document}  where the inside is considered "filled in" to denote the equivalence between the bottom$fg$and the top "$f$then$g$" path. - The 3-simplices are the tetrahedra latex {cmd=true, hide=true, run_on_save=true} \documentclass{standalone} \usepackage[utf8]{inputenc} \usepackage{tikz-cd} \begin{document} \begin{tikzcd} & B \arrow[rrrdd] \arrow[rr, "g"] & & D \arrow[rdd, "h"] & \\ & & & & \\ A \arrow[ruu, "f"] \arrow[rrrr, "fgh"] \arrow[rrruu] & & & & C \end{tikzcd} \end{document}  with the interior space filled in similarly. (Note that we only label the outer morphisms, because the rest can be named as concatenations of others.) - And so on, etc. This produces a CW complex, and hence a topological space, from the input category. **Example**: Let$G$be a discrete group of order$n$-- it is equivalently a category with one object and$n$morphisms. latex {cmd=true, hide=true, run_on_save=true} \documentclass{standalone} \usepackage[utf8]{inputenc} \usepackage{tikz} \usetikzlibrary{arrows,positioning,shadows,fit,shapes} \begin{document} \begin{tikzpicture}[] \node[](0){G}; \path[->](0)edge[loop above]node{$g_1$}(); \foreach \looseness/\label [count=\n] in {10/$g_2$,15/$g_3$,20/$g_4$,27/$\cdots$} \path [->] (0) edge [ loop above, every loop/.append style={ looseness=\looseness, in=60-0.8*\looseness, out=120+0.8*\looseness }] node {\label} (); \end{tikzpicture} \end{document}  Then$BG$is called _the classifying space of$G$_.$H_*(BG, \ZZ)$is denoted the homology of the group, and we have -$\pi_0(BG) = \pt$-$\pi_1(BG) = G$-$\pi_k(BG) = 0$for$k \geq 2$. Some concrete examples of these are: -$B\ZZ_2 = \RP^\infty$-$B\ZZ = S^1$-$BS_3 = ?$This construction can be carried out for _topological_ groups as well, with the following sequence of gluings: - A point -$G\cross I$-$G\cross G \cross \Delta^2$-$G\cross G \cross G \cross \Delta^3$-$\cdots$etc A concrete example of this is$BS^1 = \CP^\infty = K(\ZZ, 2)$. This is related to the homogeneous space fibration $$H \to G \mapsvia{g\mapsto g.p} G/H$$ for a chosen basepoint$p\in G/H$such that$H$stabilizes$p$. ---- On a different note, it is worth mentioning some of the fibrations to which a spectral sequence might apply. One that comes up is $$U_{n-k} \cross U_k \to U_n \to Gr_\CC(n, k)$$ where$Gr_\CC(n, k)$is the set of$k$-planes in$\CC^n$. Here, it is worth noting that$U_n \homotopic GL_n(\CC)$and$O_n \homotopic GL_n(\RR)$. From this, it can be concluded that$G_\CC(k, n) = \frac{U_n}{U_{n-k} \cross U_k}$, and further that if we take$\lim_{n\to \infty}$we obtain$Gr_\CC(k, \infty) = \frac{X}{U_k}$, where$X$is some contractible space, and we thus find that$Gr_\CC(k, \infty) = BU_k$, the classifying space for$U_k$. It can further be shown that there is another fibration $$U_k \to EU_k \to BU_k$$ where$EU_k$is a contractible space on which$U_k$acts and$BU_k$is the above quotient. We can then find interesting structure here arising from the fact that$H^*(Gr_\CC(k, \infty)) = H^*(BU_k)\$.