diff --git a/.claude/skills/proofread/SKILL.md b/.claude/skills/proofread/SKILL.md index ca4e05683..b66c45005 100644 --- a/.claude/skills/proofread/SKILL.md +++ b/.claude/skills/proofread/SKILL.md @@ -24,6 +24,7 @@ Name the proofread file or folder in the response. Always proofread whole files, - Correct genuine spelling, grammar, punctuation, and typographical mistakes in prose, comments, and human-readable string values. - Support common source formats such as YAML, Markdown, Svelte, and TypeScript. In structured or code files, edit only human-language text and obvious text typos; preserve keys, identifiers, APIs, program behavior, markup, links, interpolation, and syntax. +- Do fix violations of the project's writing and format conventions (see CLAUDE.md and CONTRIBUTING.md), even in markup and links: for example citation link texts (authors' last names, not titles, with the location after the link), `target="_blank"` on links, and notation conventions such as `\varnothing` and `\coloneqq`. Keep link targets unchanged. - In mathematical notation, correct only an obvious local typo, such as a variable name that inconsistently changes from `$a$` to `$x$` where the surrounding text makes the intended symbol unambiguous. Preserve formulas and claims otherwise. - Do not assess or correct the mathematical validity of proofs. That is the role of `/check-proofs`. Language mistakes inside proof text may still be corrected without changing the mathematical argument. - Make stylistic changes sparingly. Only adjust wording when it is clearly awkward or ambiguous and a small change improves readability while preserving the author's meaning and voice. Prefer leaving acceptable personal style alone. diff --git a/.cspell.json b/.cspell.json index 0a2477ec4..383586790 100644 --- a/.cspell.json +++ b/.cspell.json @@ -41,6 +41,7 @@ "Baer", "Baire", "Barnea", + "Baumslag", "Beke", "bijection", "bijections", @@ -62,13 +63,6 @@ "catdat", "Catégories", "cauchy", - "Dieudonné", - "Krause", - "Kurz", - "Paré", - "Shelah", - "Strecker", - "Séminaire", "Čech", "characterisation", "clopen", @@ -131,6 +125,7 @@ "concretizability", "concretizable", "conormal", + "coordinatewise", "copower", "copowers", "copresentability", @@ -169,6 +164,7 @@ "deloopings", "Demazure", "Diers", + "Dieudonné", "diffeomorphism", "diffeomorphisms", "disjointness", @@ -177,6 +173,7 @@ "Duskin", "Easton", "Eilenberg", + "elementwise", "endofunctor", "endofunctors", "Engelking", @@ -252,8 +249,10 @@ "Kawase", "Kerodon", "Kolmogorov", + "Krause", "Kunen", "Kuratowski", + "Kurz", "Lawvere", "libsql", "Lindelöf", @@ -283,6 +282,7 @@ "Multialgébriques", "multigraphs", "naturality", + "Neumann", "Neves", "Niefield", "nilradical", @@ -292,6 +292,7 @@ "Noncommutative", "objectwise", "opfibration", + "Paré", "Perrone", "pointwise", "Pontryagin", @@ -333,15 +334,19 @@ "Schapira", "Schepler", "Schreier", + "semidirect", "semigroup", "semigroups", + "Séminaire", "semisimple", "Serre", "setoid", "Sheafifiable", + "Shelah", "Sierpiński", "simplicial", "Specker", + "Strecker", "subalgebra", "subalgebras", "subbasic", @@ -387,6 +392,7 @@ "tripleability", "Turso", "Tychonoff", + "Tyrer", "Ulmer", "ultrafilters", "uncountably", diff --git a/CLAUDE.md b/CLAUDE.md index 490dcf06a..101d7bfe4 100644 --- a/CLAUDE.md +++ b/CLAUDE.md @@ -125,7 +125,7 @@ Format: - Write in full sentences, using "we". Introduce notation with `\coloneqq`, and use `$$...$$` for displayed formulas. In long YAML proofs, use a `>-` block and separate paragraphs with a blank line. - Internal links in YAML: structures as `$\Set$` (with the notation as link text), properties as `...`, and content pages as `here` or `this lemma`. -- External links in YAML point to a specific result: `See Prop. 4.2 at the nLab.`, `MSE/601463` (`MO/...` for MathOverflow; the link text always names the question ID, while the URL may target a specific answer as `/a/` without a user-ID suffix), and books and papers with the author's last name as link text (not the title; multiple authors joined by an en-dash, e.g. `Carboni–Lack–Walters`), followed by the location, e.g. `Mac Lane, Ch. V, Theorem 5.1`. See "Citations" in CONTRIBUTING.md. Do not add `target="_blank"`: external links get it automatically when rendered (`render_nested_text` in `src/lib/server/text.ts`). +- External links, in YAML and in content pages (as Markdown links there), point to a specific result: `See Prop. 4.2 at the nLab.`, `MSE/601463` (`MO/...` for MathOverflow; the link text always names the question ID, while the URL may target a specific answer as `/a/` without a user-ID suffix), and books and papers with the author's last name as link text (not the title; multiple authors joined by an en-dash, e.g. `Carboni–Lack–Walters`), followed by the location, e.g. `Mac Lane, Ch. V, Theorem 5.1`. See "Citations" in CONTRIBUTING.md. Do not add `target="_blank"`: external links get it automatically when rendered (`render_nested_text` in `src/lib/server/text.ts`). - Content pages are Markdown with `title` and `description` in the front matter and use Markdown links. Statements go in blocks such as `::: Lemma 1` (also `Proposition`, `Corollary`, `Claim`), closed by `:::`, followed by a `::: Proof` block. Do not state the dual version of a result unless it is used often (as for Lemma 2 in [content/subcategories.md](content/subcategories.md)). ## Workflow after changing data diff --git a/content/fg-groups-aleph1-cofiltered-limits.md b/content/fg-groups-aleph1-cofiltered-limits.md new file mode 100644 index 000000000..3e8e9952a --- /dev/null +++ b/content/fg-groups-aleph1-cofiltered-limits.md @@ -0,0 +1,129 @@ +--- +title: Finitely generated groups are not closed under ℵ₁-cofiltered limits +description: We construct an ℵ₁-cofiltered diagram of finitely generated groups whose limit in the category of groups is uncountable. +--- + +# Finitely generated groups are not closed under $\aleph_1$-cofiltered limits + +We construct a diagram $\Delta : \omega_1^{\op} \to \Grp$ in which every group is isomorphic to a fixed finitely generated group $G$, but whose limit is uncountable, and hence not finitely generated. Here, $\omega_1$ denotes the poset of countable ordinals. + +Throughout, we fix a non-trivial finitely generated group $G$ with $G \cong G \times G$. Such a group has been constructed by [Tyrer Jones](https://doi.org/10.1017/S144678870001675X) (1974), Theorem B. + +**Definition.** A _good pair_ $(p,s) : A \rightleftarrows B$ consists of two homomorphisms $p : A \to B$ and $s : B \to A$ such that + +- $B$ and $\ker(p)$ are isomorphic to $G$, +- $p \circ s = \id_B$, +- the subgroups $s(B)$ and $\ker(p)$ of $A$ commute elementwise. + +In this case, $A$ is the internal direct product of $s(B)$ and $\ker(p)$, which we indicate by writing $A = s(B) \times \ker(p)$. In particular, $A \cong B \times \ker(p) \cong G \times G \cong G$. + +::: Lemma 1 +Let $B$ be a group with $B \cong G$. Let $p : B \times G \to B$ be the projection defined by $p(b,g) \coloneqq b$ and $s : B \to B \times G$ be the inclusion defined by $s(b) \coloneqq (b,1)$. Then $(p,s) : B \times G \rightleftarrows B$ is a good pair. +::: + +::: Proof +We have $\ker(p) = 1 \times G \cong G$. Clearly, $p \circ s = \id_B$, and $s(B) = B \times 1$ commutes with $1 \times G$. +::: + +::: Lemma 2 +Good pairs compose: If $(p,s) : A \rightleftarrows B$ and $(q,t) : B \rightleftarrows C$ are good pairs, then $(q \circ p, s \circ t) : A \rightleftarrows C$ is a good pair. +::: + +::: Proof +We have $q \circ p \circ s \circ t = q \circ t = \id_C$. + +Next, we compute the kernel of $q \circ p$. By the internal direct product decomposition $A = s(B) \times \ker(p)$, every $a \in A$ can be written uniquely as $a = s(b) k$ with $b \in B$ and $k \in \ker(p)$, namely $b = p(a)$. Then $q(p(a)) = q(b)$, so $a \in \ker(q \circ p)$ if and only if $b \in \ker(q)$. This shows $\ker(q \circ p) = s(\ker(q)) \cdot \ker(p)$. Since $s(\ker(q)) \subseteq s(B)$, the two factors commute elementwise and intersect trivially. Therefore, +$$\ker(q \circ p) \cong s(\ker(q)) \times \ker(p) \cong \ker(q) \times \ker(p) \cong G \times G \cong G.$$ + +Finally, we show that $s(t(C))$ commutes elementwise with $\ker(q \circ p) = s(\ker(q)) \cdot \ker(p)$. Since $t(C)$ commutes elementwise with $\ker(q)$, applying $s$ shows that $s(t(C))$ commutes elementwise with $s(\ker(q))$. And since $s(t(C)) \subseteq s(B)$, it commutes elementwise with $\ker(p)$. +::: + +::: Lemma 3 +Good pairs lift: Let $(\pi,\iota) : X \rightleftarrows B$ and $(p,s) : A \rightleftarrows B$ be good pairs. Then there is a good pair $(\pi',\iota') : X \rightleftarrows A$ with +$$p \circ \pi' = \pi, \qquad \iota' \circ s = \iota.$$ +::: + +::: Proof +Let $C \coloneqq \ker(\pi)$ and $K \coloneqq \ker(p)$. Since $A \cong G$ and $C \cong G$, we may choose an isomorphism $c : A \to C$. We have the internal direct product decompositions $A = s(B) \times K$ and $X = \iota(B) \times C$. Applying $c$ to the former, we get $C = c(K) \times c(s(B))$, and therefore +$$X = \iota(B) \times c(K) \times c(s(B))$$ +as an internal direct product of three subgroups. That is, every element of $X$ can be written uniquely as $\iota(b) \, c(k) \, c(s(b'))$ with $b,b' \in B$ and $k \in K$. We define the homomorphisms $\iota' : A \to X$ and $\pi' : X \to A$ by +$$\iota'(s(b) k) \coloneqq \iota(b) \, c(k), \qquad \pi'\bigl(\iota(b) \, c(k) \, c(s(b'))\bigr) \coloneqq s(b) k.$$ +They are well-defined homomorphisms, since they are defined factor by factor on internal direct products. We now verify the required properties. + +- We have $\pi' \circ \iota' = \id_A$ by construction. +- We have $p\bigl(\pi'(\iota(b) \, c(k) \, c(s(b')))\bigr) = p(s(b) k) = b$. This equals $\pi\bigl(\iota(b) \, c(k) \, c(s(b'))\bigr)$, since $\pi \circ \iota = \id_B$ and $c(A) = C = \ker(\pi)$. Hence, $p \circ \pi' = \pi$. +- We have $\iota'(s(b)) = \iota(b)$, i.e. $\iota' \circ s = \iota$. +- The kernel of $\pi'$ is $c(s(B)) \cong B \cong G$. +- The image $\iota'(A) = \iota(B) \cdot c(K)$ commutes elementwise with $\ker(\pi') = c(s(B))$. Indeed, $\iota(B)$ commutes elementwise with $C$, and $c(K)$ commutes elementwise with $c(s(B))$ because $K$ commutes elementwise with $s(B)$. + +::: + +Next, we construct the diagram $\Delta : \omega_1^{\op} \to \Grp$. + +::: Proposition 4 +There are groups $\Delta_\beta \cong G$ for $\beta < \omega_1$ and homomorphisms $\pi_{\alpha\beta} : \Delta_\beta \to \Delta_\alpha$ and $\iota_{\alpha\beta} : \Delta_\alpha \to \Delta_\beta$ for $\alpha \leq \beta < \omega_1$ with the following properties: + +1. We have $\pi_{\alpha\alpha} = \iota_{\alpha\alpha} = \id$, and for $\alpha \leq \beta \leq \gamma$ we have $\pi_{\alpha\beta} \circ \pi_{\beta\gamma} = \pi_{\alpha\gamma}$ and $\iota_{\beta\gamma} \circ \iota_{\alpha\beta} = \iota_{\alpha\gamma}$. +2. For $\alpha < \beta$, the pair $(\pi_{\alpha\beta},\iota_{\alpha\beta}) : \Delta_\beta \rightleftarrows \Delta_\alpha$ is good. + +::: + +::: Proof +We construct $\Delta_\beta$ and the homomorphisms $\pi_{\alpha\beta},\iota_{\alpha\beta}$ for $\alpha \leq \beta$ by transfinite recursion on $\beta$. + +We start with $\Delta_0 \coloneqq G$ and $\pi_{00} \coloneqq \iota_{00} \coloneqq \id_G$. + +For the successor step, let $\Delta_{\beta+1} \coloneqq \Delta_\beta \times G$, and let $(\pi_{\beta,\beta+1},\iota_{\beta,\beta+1})$ be the good pair from Lemma 1. For $\alpha < \beta$ we define +$$\pi_{\alpha,\beta+1} \coloneqq \pi_{\alpha\beta} \circ \pi_{\beta,\beta+1}, \qquad \iota_{\alpha,\beta+1} \coloneqq \iota_{\beta,\beta+1} \circ \iota_{\alpha\beta}.$$ +Then (1) clearly holds, and (2) follows from Lemma 2. + +For the limit step, let $\lambda < \omega_1$ be a limit ordinal. Since $\lambda$ is countable, we may choose a cofinal sequence $\alpha_0 < \alpha_1 < \cdots$ in $\lambda$. First, we define good pairs $(\pi_{\alpha_n\lambda},\iota_{\alpha_n\lambda}) : \Delta_\lambda \rightleftarrows \Delta_{\alpha_n}$ by recursion on $n$. Let $\Delta_\lambda \coloneqq \Delta_{\alpha_0} \times G$, and let $(\pi_{\alpha_0\lambda},\iota_{\alpha_0\lambda})$ be the good pair from Lemma 1. If $(\pi_{\alpha_n\lambda},\iota_{\alpha_n\lambda})$ has been defined, we apply Lemma 3 to this good pair and to the good pair $(\pi_{\alpha_n\alpha_{n+1}},\iota_{\alpha_n\alpha_{n+1}}) : \Delta_{\alpha_{n+1}} \rightleftarrows \Delta_{\alpha_n}$. This gives a good pair $(\pi_{\alpha_{n+1}\lambda},\iota_{\alpha_{n+1}\lambda}) : \Delta_\lambda \rightleftarrows \Delta_{\alpha_{n+1}}$ with +$$\pi_{\alpha_n\alpha_{n+1}} \circ \pi_{\alpha_{n+1}\lambda} = \pi_{\alpha_n\lambda}, \qquad \iota_{\alpha_{n+1}\lambda} \circ \iota_{\alpha_n\alpha_{n+1}} = \iota_{\alpha_n\lambda}.$$ +By induction, using (1) below $\lambda$, we get +$$\pi_{\alpha_n\alpha_m} \circ \pi_{\alpha_m\lambda} = \pi_{\alpha_n\lambda}, \qquad \iota_{\alpha_m\lambda} \circ \iota_{\alpha_n\alpha_m} = \iota_{\alpha_n\lambda}$$ +for all $n \leq m$. + +Now, for an arbitrary $\alpha < \lambda$, choose some $n$ with $\alpha < \alpha_n$ and define +$$\pi_{\alpha\lambda} \coloneqq \pi_{\alpha\alpha_n} \circ \pi_{\alpha_n\lambda}, \qquad \iota_{\alpha\lambda} \coloneqq \iota_{\alpha_n\lambda} \circ \iota_{\alpha\alpha_n}.$$ +Also, let $\pi_{\lambda\lambda} \coloneqq \iota_{\lambda\lambda} \coloneqq \id$. By the previous equations and (1) below $\lambda$, this definition does not depend on the choice of $n$, and it agrees with the previous definition when $\alpha = \alpha_m$. For (1), let $\alpha \leq \beta < \lambda$ and choose $n$ with $\beta < \alpha_n$. Then +$$\pi_{\alpha\beta} \circ \pi_{\beta\lambda} = \pi_{\alpha\beta} \circ \pi_{\beta\alpha_n} \circ \pi_{\alpha_n\lambda} = \pi_{\alpha\alpha_n} \circ \pi_{\alpha_n\lambda} = \pi_{\alpha\lambda},$$ +and similarly $\iota_{\beta\lambda} \circ \iota_{\alpha\beta} = \iota_{\alpha\lambda}$. For (2), the pair $(\pi_{\alpha\lambda},\iota_{\alpha\lambda})$ is the composite of the good pairs $(\pi_{\alpha_n\lambda},\iota_{\alpha_n\lambda})$ and $(\pi_{\alpha\alpha_n},\iota_{\alpha\alpha_n})$ in the sense of Lemma 2, hence it is good. +::: + +By (1), we obtain a diagram $\Delta : \omega_1^{\op} \to \Grp$ with $\alpha \mapsto \Delta_\alpha$ and $(\alpha \leq \beta) \mapsto \pi_{\alpha\beta}$. The homomorphisms $\iota_{\alpha\beta}$ are not part of the diagram; they are only used to produce elements of its limit. + +::: Proposition 5 +The limit of $\Delta$ in $\Grp$ is uncountable. +::: + +::: Proof +The limit is the subgroup +$$\textstyle L \coloneqq \bigl\{x \in \prod_{\beta < \omega_1} \Delta_\beta : \pi_{\beta\gamma}(x_\gamma) = x_\beta \text{ for all } \beta \leq \gamma\bigr\}.$$ +For $\alpha < \omega_1$ we define a homomorphism $j_\alpha : \Delta_\alpha \to \prod_{\beta < \omega_1} \Delta_\beta$ by +$$j_\alpha(x)_\beta \coloneqq \begin{cases} \iota_{\alpha\beta}(x) & \beta \geq \alpha, \\ \pi_{\beta\alpha}(x) & \beta \leq \alpha. \end{cases}$$ +For $\beta = \alpha$, both cases give $x$, so this is well-defined. We claim that $j_\alpha$ maps into $L$, i.e. that for all $x \in \Delta_\alpha$ and $\beta \leq \gamma$ we have +$$\pi_{\beta\gamma}(j_\alpha(x)_\gamma) = j_\alpha(x)_\beta.$$ +This follows from (1) and $\pi_{\alpha\beta} \circ \iota_{\alpha\beta} = \id$, distinguishing three cases: + +$$ +\begin{align*} +\pi_{\beta\gamma}(j_\alpha(x)_\gamma) & = \pi_{\beta\gamma}(\iota_{\alpha\gamma}(x)) = \pi_{\beta\gamma}(\iota_{\beta\gamma}(\iota_{\alpha\beta}(x))) = \iota_{\alpha\beta}(x) = j_\alpha(x)_\beta && \text{if } \alpha \leq \beta, \\ +\pi_{\beta\gamma}(j_\alpha(x)_\gamma) & = \pi_{\beta\gamma}(\iota_{\alpha\gamma}(x)) = \pi_{\beta\alpha}(\pi_{\alpha\gamma}(\iota_{\alpha\gamma}(x))) = \pi_{\beta\alpha}(x) = j_\alpha(x)_\beta && \text{if } \beta \leq \alpha \leq \gamma, \\ +\pi_{\beta\gamma}(j_\alpha(x)_\gamma) & = \pi_{\beta\gamma}(\pi_{\gamma\alpha}(x)) = \pi_{\beta\alpha}(x) = j_\alpha(x)_\beta && \text{if } \gamma \leq \alpha. +\end{align*} +$$ + +So we get a homomorphism $j_\alpha : \Delta_\alpha \to L$. A coordinatewise check of the same kind shows $j_\beta \circ \iota_{\alpha\beta} = j_\alpha$ for $\alpha \leq \beta$. Hence, the subgroups $L_\alpha \coloneqq j_\alpha(\Delta_\alpha)$ of $L$ satisfy +$$L_\alpha \subseteq L_\beta$$ +for $\alpha \leq \beta$. Next, we show that $L_\alpha \neq L_{\alpha+1}$. Since $G$ is non-trivial and $\ker(\pi_{\alpha,\alpha+1}) \cong G$ by (2), there is some element $1 \neq k \in \ker(\pi_{\alpha,\alpha+1})$. Assume that $j_{\alpha+1}(k) = j_\alpha(x)$ for some $x \in \Delta_\alpha$. Comparing the $(\alpha+1)$-coordinates gives $k = \iota_{\alpha,\alpha+1}(x)$. Applying $\pi_{\alpha,\alpha+1}$ gives $1 = x$, and hence $k = 1$, a contradiction. Thus, $j_{\alpha+1}(k) \in L_{\alpha+1} \setminus L_\alpha$. + +Now choose an element $y_\alpha \in L_{\alpha+1} \setminus L_\alpha$ for every $\alpha < \omega_1$. These elements are pairwise distinct: if $\alpha < \beta$, then $y_\alpha \in L_{\alpha+1} \subseteq L_\beta$, but $y_\beta \notin L_\beta$. Therefore, $L$ has at least $\aleph_1$ elements. +::: + +::: Corollary 6 +The full subcategory of finitely generated groups $\Grp_\fg \subseteq \Grp$ is not closed under $\aleph_1$-cofiltered limits. +::: + +::: Proof +The diagram $\Delta : \omega_1^{\op} \to \Grp$ is $\aleph_1$-cofiltered, and each $\Delta_\alpha \cong G$ is finitely generated. But its limit is uncountable by Proposition 5, whereas every finitely generated group is countable. +::: diff --git a/content/fg-groups-aleph1-filtered-colimits.md b/content/fg-groups-aleph1-filtered-colimits.md new file mode 100644 index 000000000..e297bee5f --- /dev/null +++ b/content/fg-groups-aleph1-filtered-colimits.md @@ -0,0 +1,108 @@ +--- +title: Finitely generated groups do not have ℵ₁-filtered colimits +description: We construct an uncountable group in which every countable subset is contained in a finitely generated subgroup, and deduce that the category of finitely generated groups does not have ℵ₁-filtered colimits. +--- + +# Finitely generated groups do not have $\aleph_1$-filtered colimits + +We show that the category $\Grp_\fg$ of finitely generated groups does not have $\aleph_1$-filtered colimits. + +::: Lemma 1 +Assume that an uncountable group $H$ exists with the following two properties: + +1. Every countable subset of $H$ is contained in a finitely generated subgroup of $H$. +2. $H$ is a subgroup of a product $\prod_{n \in \IN} H_n$ of finitely generated groups $H_n$. + +Then $\Grp_\fg$ does not have $\aleph_1$-filtered colimits. +::: + +::: Proof +Let $I$ be the set of finitely generated subgroups of $H$, partially ordered by inclusion, and let $D : I \to \Grp_\fg$ be the diagram which maps each $U \in I$ to $U$ and each inclusion $U \subseteq U'$ to the inclusion homomorphism $U \hookrightarrow U'$. The poset $I$ is $\aleph_1$-filtered by (1): every countable subset of $I$ has an upper bound in $I$, since the union of countably many finitely generated subgroups is countable. + +Assume that $D$ has a colimit $C$ in $\Grp_\fg$ with colimit cocone $\lambda_U : U \to C$ for $U \in I$. Since $H$ is the directed union of all $U \in I$, these homomorphisms glue to a homomorphism $f : H \to C$ with $f|_U = \lambda_U$. For each $n$, the restrictions $\pi_n|_U : U \to H_n$ of the projection $\pi_n : H \to H_n$ form a cocone of $D$ with tip $H_n$, so there is a homomorphism $h_n : C \to H_n$ with $h_n \circ \lambda_U = \pi_n|_U$ for all $U \in I$, i.e. $h_n \circ f = \pi_n$. By (2), the $\pi_n$ are jointly injective, so $f$ is injective. But $H$ is uncountable, whereas $C$ is countable as a finitely generated group. +::: + +It remains to find such a group $H$. + +## Idea + +For (2), it is natural to look for $H$ of the form $V^{\IN} \rtimes G$, where $V$ is a countable vector space and $G \subseteq \Aut(V)$ is a finitely generated group acting on $V^{\IN}$ coordinatewise. Such a group is uncountable and embeds into $(V \rtimes G)^{\IN}$. The real task is (1): given countably many $w_0,w_1,\dotsc \in V^{\IN}$, we want to find them in a finitely generated subgroup. Our plan is to store all of them in a single element $c \in V^{\IN}$ and to compute them from $c$ using $G$. In $H$, the elements of $G$ act on $V^{\IN}$ by conjugation, so everything we compute from $c$ in this way lies in the subgroup generated by $G$ and $c$. This is the purpose of the semidirect product. Let us see which operations $G$ needs for this. + +First, $c$ needs room for infinitely many vectors. So we let $V$ contain infinitely many independent copies $V_0,V_1,V_2,\dotsc$ of one subspace, together with an element $u \in G$ mapping $V_i$ to $V_{i+1}$. We will put the data of $w_{k,n}$ into the coordinate $c_n$. Since vectors in $V$ have finite support, $c_n$ can only use finitely many copies, so we only put $w_{k,n}$ for $k \leq n$ into $c_n$. + +Second, we need to read off a single copy. Moving $V_i$ back to $V_0$ with $u^{-i}$, it remains to discard everything outside of $V_0$, that is, to apply the projection $P$ onto $V_0$ along a complement $W$. Since $P$ is not invertible, it cannot be an element of $G$. But if $s \in G$ acts as $-1$ on $V_0$ and as $1$ on $W$, then $s \cdot v - v = -2 P(v)$, which is $P(v)$ when we work over $\IF_3$. + +Third, $w_{k,n}$ is an arbitrary vector of $V$ and does not fit into the single copy $V_0$. So we split it into two vectors of $V_0$ with an involution $\varphi \in G$ which interchanges $V_0$ and $W$: both $P(v)$ and $P(\varphi \cdot v)$ lie in $V_0$, and $v = P(v) + \varphi \cdot P(\varphi \cdot v)$. Thus, $w_{k,n}$ takes up two copies, and we use $V_{2k+1}$ and $V_{2k+2}$. + +Finally, since $c$ only contains $w_{k,n}$ for $n \geq k$, we only recover $w_k$ up to finitely many coordinates. To correct these, we add an element $b \in V^{\IN}$ which, together with $G$, generates all sequences with only finitely many non-zero coordinates. For this, it is convenient if a single basis vector of $V$ can be moved to all others by $G$, which also makes $V \rtimes G$ finitely generated. This is the reason for a second shift $r \in G$. + +## Construction + +We now make this precise. Let $\Omega \coloneqq \IZ \times \IZ$, and let $V$ be the $\IF_3$-vector space of finitely supported functions $\Omega \to \IF_3$, with basis $(e_p)_{p \in \Omega}$. Every bijection $\sigma : \Omega \to \Omega$ induces an automorphism of $V$ by $\sigma \cdot e_p \coloneqq e_{\sigma(p)}$. For $i \in \IZ$ let $V_i \subseteq V$ be the subspace of functions supported on $\{i\} \times \IZ$, and let $W \subseteq V$ be the subspace of functions vanishing on $\{0\} \times \IZ$. Then $V = V_0 \oplus W$, and we let $P : V \to V$ be the projection onto $V_0$. + +Let $u,r$ be the automorphisms of $V$ induced by the bijections $u(i,j) \coloneqq (i+1,j)$ and $r(i,j) \coloneqq (i,j+1)$ of $\Omega$, and let $\varphi$ be the automorphism induced by an involution of $\Omega$ which maps $\{0\} \times \IZ$ onto its complement (both sets are countably infinite). Thus, we have +$$u \cdot V_i = V_{i+1}, \qquad \varphi \cdot V_0 = W, \qquad \varphi \cdot W = V_0, \qquad \varphi^2 = \id_V.$$ +Let $s$ be the automorphism of $V$ with $s \cdot x = -x$ for $x \in V_0$ and $s \cdot y = y$ for $y \in W$. Let $G \subseteq \Aut(V)$ be the subgroup generated by $u,r,\varphi,s$, and define the semidirect product +$$H \coloneqq V^{\IN} \rtimes G,$$ +where $G$ acts on $V^{\IN}$ coordinatewise. + +::: Lemma 2 +The group $H$ is uncountable and satisfies (2) in Lemma 1. +::: + +::: Proof +Since $V^{\IN}$ is uncountable, so is $H$. The map $(z,g) \mapsto ((z_n,g))_{n \in \IN}$ embeds $H$ into $(V \rtimes G)^{\IN}$, and $V \rtimes G$ is generated by $u,r,\varphi,s,e_{(0,0)}$, since $u^i r^j \cdot e_{(0,0)} = e_{(i,j)}$. +::: + +For the proof of (1), we regard $V^{\IN}$ and $G$ as subgroups of $H$ via $z \mapsto (z,1)$ and $g \mapsto (0,g)$, and we keep writing the group $V^{\IN}$ additively. We extend $P$ coordinatewise to $V^{\IN}$. + +::: Lemma 3 +Let $x \in V_0$ and $d \in \IZ$. Then $P(u^d \cdot x) = x$ if $d = 0$, and $P(u^d \cdot x) = 0$ otherwise. +::: + +::: Proof +For $d = 0$ this holds since $P$ is the identity on $V_0$. For $d \neq 0$, we have $u^d \cdot x \in V_d \subseteq W$, and $P$ vanishes on $W$. +::: + +::: Lemma 4 +For $v \in V$ we have +$$v = P(v) + \varphi \cdot P(\varphi \cdot v).$$ +::: + +::: Proof +Write $v = x + y$ with $x \in V_0$ and $y \in W$, so that $P(v) = x$. Since $\varphi \cdot x \in W$ and $\varphi \cdot y \in V_0$, we have $P(\varphi \cdot v) = \varphi \cdot y$, and hence $\varphi \cdot P(\varphi \cdot v) = \varphi^2 \cdot y = y$. Therefore, $P(v) + \varphi \cdot P(\varphi \cdot v) = x + y = v$. +::: + +::: Lemma 5 +Let $\Gamma \subseteq H$ be a subgroup containing $G$. Then $\Gamma \cap V^{\IN}$ is a subgroup of $V^{\IN}$ which is closed under the action of $G$ and under $P$. +::: + +::: Proof +By definition of the semidirect product, conjugation with $g \in G$ in $H$ acts on $V^{\IN}$ by $z \mapsto g \cdot z$. Hence, $\Gamma \cap V^{\IN}$ is closed under the action of $G$. For $v = x + y$ with $x \in V_0$ and $y \in W$, we have +$$s \cdot v - v = (-x + y) - (x + y) = -2x = x = P(v),$$ +since $-2 = 1$ in $\IF_3$. Hence, $P(z) = s \cdot z - z$ for $z \in V^{\IN}$, so $\Gamma \cap V^{\IN}$ is also closed under $P$. +::: + +::: Proposition 6 +The group $H$ satisfies (1) in Lemma 1. +::: + +::: Proof +Let $X \subseteq H$ be countable. Since every element of $H$ is a product of an element of $V^{\IN}$ and an element of $G$, there are $w_0,w_1,\dotsc \in V^{\IN}$ such that $X$ is contained in the subgroup generated by $G$ and $w_0,w_1,\dotsc$. Write $w_k = (w_{k,n})_{n \in \IN}$ and define $b,c \in V^{\IN}$ by +$$\textstyle b_n \coloneqq u^n \cdot e_{(0,0)}, \qquad c_n \coloneqq \sum_{k=0}^{n} \bigl(u^{2k+1} \cdot P(w_{k,n}) + u^{2k+2} \cdot P(\varphi \cdot w_{k,n})\bigr).$$ +Let $\Gamma \subseteq H$ be the finitely generated subgroup generated by $u,r,\varphi,s,b,c$, i.e. by $G$, $b$ and $c$. By Lemma 5, $\Gamma \cap V^{\IN}$ is a subgroup of $V^{\IN}$ containing $b$ and $c$ which is closed under the action of $G$ and under $P$. We show that $w_k \in \Gamma$ for all $k$, which implies $X \subseteq \Gamma$. + +First, $\Gamma$ contains every $z \in V^{\IN}$ with $z_n = 0$ for almost all $n$. Indeed, let $n \in \IN$. The coordinate of $P(u^{-n} \cdot b) \in \Gamma$ at $n'$ is $P(u^{n'-n} \cdot e_{(0,0)})$, which is $e_{(0,0)}$ for $n' = n$ and $0$ otherwise. Acting on this element with $u^i r^j$ replaces $e_{(0,0)}$ by $e_{(i,j)}$, and these elements generate all $z \in V^{\IN}$ with $z_{n'} = 0$ for $n' \neq n$. + +Second, let $k \in \IN$ and $n \geq k$. All vectors $P(w_{k',n})$ and $P(\varphi \cdot w_{k',n})$ lie in $V_0$. If we apply the linear map $P \circ u^{-(2k+1)}$ to $c_n$, Lemma 3 shows that only the summand with $u^{2k+1}$ survives, so that $P(u^{-(2k+1)} \cdot c_n) = P(w_{k,n})$. Similarly, $P(u^{-(2k+2)} \cdot c_n) = P(\varphi \cdot w_{k,n})$. Therefore, by Lemma 4, the element +$$y_k \coloneqq P(u^{-(2k+1)} \cdot c) + \varphi \cdot P(u^{-(2k+2)} \cdot c) \in \Gamma$$ +has coordinate $P(w_{k,n}) + \varphi \cdot P(\varphi \cdot w_{k,n}) = w_{k,n}$ for all $n \geq k$. Hence, $w_k - y_k$ has only finitely many non-zero coordinates, so it lies in $\Gamma$ by the first step, and therefore $w_k \in \Gamma$. +::: + +::: Corollary 7 +The category $\Grp_\fg$ of finitely generated groups does not have $\aleph_1$-filtered colimits. +::: + +::: Proof +Apply Lemma 1 to the group $H$, using Lemma 2 and Proposition 6. +::: diff --git a/content/regular-epis-kernel-pairs.md b/content/regular-epis-kernel-pairs.md new file mode 100644 index 000000000..7b3bb99a9 --- /dev/null +++ b/content/regular-epis-kernel-pairs.md @@ -0,0 +1,20 @@ +--- +title: Regular epimorphisms are coequalizers of their kernel pairs +description: A regular epimorphism that has a kernel pair is the coequalizer of that kernel pair. +--- + +# Regular epimorphisms are coequalizers of their kernel pairs + +A regular epimorphism is the coequalizer of _some_ pair of morphisms. It turns out that, in many cases, a canonical pair is available. + +::: Lemma +Let $f : X \to Y$ be a regular epimorphism in a category $\C$ that has a kernel pair $p_1,p_2 : X \times_Y X \rightrightarrows X$. Then $f$ is the coequalizer of $p_1,p_2$, i.e. $f$ is an [effective epimorphism](/morphism-property/effective_epimorphism). +::: + +::: Proof +Say $f$ is the coequalizer of $a,b : A \rightrightarrows X$. Since $f \circ a = f \circ b$, there is a morphism $c : A \to X \times_Y X$ with $p_1 \circ c = a$ and $p_2 \circ c = b$. Now let $g : X \to T$ be a morphism with $g \circ p_1 = g \circ p_2$. Then +$$g \circ a = g \circ p_1 \circ c = g \circ p_2 \circ c = g \circ b,$$ +so $g$ factors uniquely through $f$. Since also $f \circ p_1 = f \circ p_2$, this shows that $f$ is the coequalizer of $p_1,p_2$. +::: + +**Remark.** The dual statement is that any regular monomorphism that has a cokernel pair is the equalizer of this cokernel pair, and hence an effective monomorphism. diff --git a/content/subcategories.md b/content/subcategories.md index 6173513b7..43182e86f 100644 --- a/content/subcategories.md +++ b/content/subcategories.md @@ -147,3 +147,13 @@ Let $U : \C \to \D$ be a faithful conservative functor (for example, a fully fai ::: Proof This is straightforward. We need to prove that finite coproducts are disjoint and stable under pullbacks in $\C$. If $A,B \in \C$, the coproduct inclusion $A \to A + B$ is a monomorphism: Since $U$ is faithful, it suffices to prove that its image under $U$ is a monomorphism. Since $U$ preserves finite coproducts, the image identifies with the coproduct inclusion $U(A) \to U(A) + U(B)$, which is a monomorphism since $\D$ is extensive. Moreover, the unique morphism $0 \to A \times_{A + B} B$ is an isomorphism: Since $U$ is conservative, it suffices to prove that its image under $U$ is an isomorphism. Since $U$ preserves finite coproducts and pullbacks along coproduct inclusions, the image identifies with the unique morphism $0 \to U(A) \times_{U(A) + U(B)} U(B)$, which is an isomorphism since $\D$ is extensive. This proves that finite coproducts are disjoint in $\C$. To prove that they are stable under pullbacks, let $T \to A + B$ be any morphism in $\C$, and consider the pullbacks $T_A \coloneqq T \times_{A + B} A$ and $T_B \coloneqq T \times_{A + B} B$. We need to show that the canonical morphism $T_A + T_B \to T$ is an isomorphism. Since $U$ is conservative, it suffices to prove that its image under $U$ is an isomorphism. Since $U$ preserves finite coproducts and pullbacks along coproduct inclusions, the image identifies with the canonical morphism $U(T)_{U(A)} + U(T)_{U(B)} \to U(T)$ induced by the morphism $U(T) \to U(A) + U(B)$ in $\D$, which is an isomorphism since $\D$ is extensive. ::: + +::: Lemma 12 +Let $U : \C \to \D$ be a fully faithful functor that preserves kernel pairs. Let $f : X \to Y$ be a morphism in $\C$ that has a kernel pair in $\C$. If $U(f)$ is a regular epimorphism in $\D$, then $f$ is a regular epimorphism in $\C$. In particular, if $U$ maps epimorphisms to regular epimorphisms and $\C$ is not epi-regular, then $\C$ does not have kernel pairs. +::: + +::: Proof +Let $p_1,p_2 : E \rightrightarrows X$ be the kernel pair of $f$ in $\C$. Then $U(p_1),U(p_2)$ is the kernel pair of $U(f)$ in $\D$. Since $U(f)$ is a regular epimorphism, it is the coequalizer of $U(p_1),U(p_2)$ by [this lemma](/content/regular-epis-kernel-pairs). Since $U$ is fully faithful, it reflects colimits, so $f$ is the coequalizer of $p_1,p_2$ in $\C$, i.e. a regular epimorphism. + +For the second statement, assume that $\C$ has kernel pairs and that $U$ maps epimorphisms to regular epimorphisms. By the first statement, every epimorphism of $\C$ is regular, i.e. $\C$ is epi-regular. +::: diff --git a/database/data/categories/Ab_fg.yaml b/database/data/categories/Ab_fg.yaml index 847077ab4..9d623122a 100644 --- a/database/data/categories/Ab_fg.yaml +++ b/database/data/categories/Ab_fg.yaml @@ -12,6 +12,7 @@ tags: related: - Ab - FinAb + - Grp_fg - FinVect_c - FreeAb_fg diff --git a/database/data/categories/Bin_irr.yaml b/database/data/categories/Bin_irr.yaml index 4551cd860..b361d2aca 100644 --- a/database/data/categories/Bin_irr.yaml +++ b/database/data/categories/Bin_irr.yaml @@ -133,7 +133,7 @@ special_morphisms: proof: 'For the non-trivial direction, the construction of cokernel pairs shows that the forgetful functor $\Bin_{\irr} \to \Bin$ preserves cokernel pairs, and hence preserves epimorphisms (using the general fact that $f : X \to Y$ is an epimorphism if and only if $\nabla : Y \sqcup_X Y \to Y$ is an isomorphism). Moreover, the epimorphisms in $\Bin$ have already been identified as being surjective.' regular monomorphisms: description: injective maps that reflect and preserve the relation - proof: 'First of all, we know that the regular monomorphisms in $\Bin$ have this description. The forgetful functor $\Bin_{\irr} \to \Bin$ preserves equalizers, in fact all non-empty limits (by their construction), and hence preserves regular monomorphisms. This proves one direction. Now assume that $X \to Y$ is a morphism in $\Bin_{\irr}$ that is injective and reflects the relation, i.e. is an embedding. Then it is a regular monomorphism in $\Bin$. Since $\Bin$ has cokernel pairs, it follows that $X \to Y$ is the equalizer of its cokernel pair $Y \rightrightarrows Y \sqcup_X Y$ (see here). Since $Y \sqcup_X Y \in \Bin_{\irr}$ (see our proof that cokernel pairs exist), we conclude that $X \to Y$ is also the equalizer of its cokernel pair in $\Bin_{\irr}$.' + proof: 'First of all, we know that the regular monomorphisms in $\Bin$ have this description. The forgetful functor $\Bin_{\irr} \to \Bin$ preserves equalizers, in fact all non-empty limits (by their construction), and hence preserves regular monomorphisms. This proves one direction. Now assume that $X \to Y$ is a morphism in $\Bin_{\irr}$ that is injective and reflects the relation, i.e. is an embedding. Then it is a regular monomorphism in $\Bin$. Since $\Bin$ has cokernel pairs, it follows that $X \to Y$ is the equalizer of its cokernel pair $Y \rightrightarrows Y \sqcup_X Y$ by this lemma. Since $Y \sqcup_X Y \in \Bin_{\irr}$ (see our proof that cokernel pairs exist), we conclude that $X \to Y$ is also the equalizer of its cokernel pair in $\Bin_{\irr}$.' regular epimorphisms: description: >- morphisms $f : (X,R) \to (Y,S)$ such that $f : X \to Y$ is surjective and $S = \{(f(x),f(x')) : (x,x') \in R\}$ diff --git a/database/data/categories/Grp.yaml b/database/data/categories/Grp.yaml index 159ca0f32..269e2e5cf 100644 --- a/database/data/categories/Grp.yaml +++ b/database/data/categories/Grp.yaml @@ -11,6 +11,7 @@ tags: related: - FinGrp + - Grp_fg - Grp_c - Ab - Mon diff --git a/database/data/categories/Grp_c.yaml b/database/data/categories/Grp_c.yaml index 3240fa854..33b27cc9c 100644 --- a/database/data/categories/Grp_c.yaml +++ b/database/data/categories/Grp_c.yaml @@ -11,6 +11,7 @@ tags: related: - Grp + - Grp_fg - FinGrp - Set_c - Vect_c @@ -48,6 +49,7 @@ satisfied_properties: - property: mono-regular proof: >- This can be deduced from the corresponding property of $\Grp$ as follows: Let $i : K \hookrightarrow G$ be a monomorphism in $\Grp_\c$, i.e. an injective homomorphism of countable groups. Since $\Grp$ is mono-regular, there is a group $H$ and two homomorphisms $f,g : G \rightrightarrows H$ with $i = \eq(f,g)$. Let $H' \subseteq H$ be the subgroup generated by $\im(f) \cup \im(g)$. Since $G$ is countable, $H'$ is countable as well, and $f,g$ corestrict to homomorphisms $f', g' : G \rightrightarrows H'$. Hence, $i = \eq(f',g')$. + label: grpc_mono-regular - property: conormal proof: 'If $f : G \to H$ is an epimorphism in $\Grp_\c$, i.e. a surjective homomorphism of countable groups, then $f$ is the cokernel of $K \hookrightarrow G$ in $\Grp$, where $K$ is the kernel of $f$. Since $K$ is countable, it is also the cokernel in $\Grp_\c$.' @@ -82,6 +84,7 @@ unsatisfied_properties: - property: countable powers proof: Since the forgetful functor $\Grp_\c \to \Set$ is representable, it preserves products. Therefore, if the power $\IZ^{\IN}$ exists in $\Grp_\c$, its underlying set must be the ordinary cartesian product, which however is uncountable. + label: grpc_no_countable_powers - property: ℵ₂-small copowers proof: 'Assume that the copower $G \coloneqq I \otimes \IZ$ exists in $\Grp_\c$. We will prove that $I$ is countable; in particular, $\aleph_1 \otimes \IZ$ does not exist. The copower is a countable group $G$ with a universal family of elements $(x_i)_{i \in I}$. For every $i \in I$ there is a homomorphism $G \to \IZ$ defined by $x_i \mapsto 1$ and $x_j \mapsto 0$ for $j \neq i$. In particular, $x_i \neq x_j$ whenever $i \neq j$. Thus, $x : I \to G$ is an injective map, which proves that $I$ is countable.' @@ -96,6 +99,7 @@ unsatisfied_properties: - property: cogenerator proof: 'Assume that a cogenerator $Q$ exists in $\Grp_\c$. There are only countably many finitely generated subgroups of $Q$. But there are continuum many finitely generated simple groups; this follows from Hyde–Lodha, Corollary 1.5. Hence, there is a finitely generated (and hence countable) simple group $H$ which does not embed into $Q$. Since $H$ is simple, any homomorphism $H \to Q$ must be trivial then. But then $\id_H, 1 : H \rightrightarrows H$ are not separated by a homomorphism $H \to Q$.' + label: grpc_no_cogenerator - property: ℵ₁-cofiltered limits proof: >- diff --git a/database/data/categories/Grp_fg.yaml b/database/data/categories/Grp_fg.yaml new file mode 100644 index 000000000..3630bd6ed --- /dev/null +++ b/database/data/categories/Grp_fg.yaml @@ -0,0 +1,173 @@ +id: Grp_fg +name: category of finitely generated groups +notation: $\Grp_{\fg}$ +objects: finitely generated groups +morphisms: group homomorphisms +description: This is a full subcategory of $\Grp$ with $\FinGrp \subseteq \Grp_{\fg} \subseteq \Grp_{\c}$. +nlab_link: null + +tags: + - algebra + +related: + - Grp + - Ab_fg + - Grp_c + - FinGrp + +satisfied_properties: + - property: locally small + proof: It is a full subcategory of $\Grp$, which is locally small. + + - property: essentially small + proof: Every finitely generated group is isomorphic to a quotient $F_n / N$ of a free group $F_n$ of finite rank $n \in \IN$ by some normal subgroup $N \subseteq F_n$, and these quotients form a set. + + - property: pointed + proof: The trivial group is finitely generated and is a zero object. + + - property: finite products + proof: This is because $\Grp$ has finite products, and $\Grp_\fg \hookrightarrow \Grp$ is closed under finite products. Indeed, if $G$ is generated by $g_1,\dotsc,g_n$ and $H$ is generated by $h_1,\dotsc,h_m$, then $G \times H$ is generated by $(g_1,1),\dotsc,(g_n,1),(1,h_1),\dotsc,(1,h_m)$. + + - property: finite coproducts + proof: This is because $\Grp$ has finite coproducts (free products), and $\Grp_\fg \hookrightarrow \Grp$ is closed under finite coproducts, since $G \sqcup H$ is generated by the images of generating sets of $G$ and $H$. + + - property: coequalizers + proof: 'This is because $\Grp$ has coequalizers, and $\Grp_\fg \hookrightarrow \Grp$ is closed under coequalizers: the coequalizer of $f,g : G \rightrightarrows H$ in $\Grp$ is a quotient of $H$, and quotients of finitely generated groups are finitely generated.' + label: grp_fg_coequalizers + + - property: extremal generator + proof: The finitely generated group $\IZ$ is an extremal generator even in $\Grp$. Now use Lemma 10 here. + references: + - Grp_extremal_generator + + - property: mono-regular + proof: >- + This is the same argument as for $\Grp_\c$ and can be deduced from the corresponding property of $\Grp$ as follows: Let $i : K \hookrightarrow G$ be a monomorphism in $\Grp_\fg$, i.e. an injective homomorphism of finitely generated groups (see below). Since $\Grp$ is mono-regular, there is a group $H$ and two homomorphisms $f,g : G \rightrightarrows H$ with $i = \eq(f,g)$ in $\Grp$. Let $H' \subseteq H$ be the subgroup generated by $\im(f) \cup \im(g)$. Since $G$ is finitely generated, also $H'$ is finitely generated. The homomorphisms $f,g$ corestrict to homomorphisms $f', g' : G \rightrightarrows H'$, and we still have $i = \eq(f',g')$ in $\Grp$. Since $K$ is finitely generated and $\Grp_\fg \hookrightarrow \Grp$ is fully faithful, $i = \eq(f',g')$ also holds in $\Grp_\fg$. + references: + - grpc_mono-regular + + - property: effective congruences + proof: >- + Let $p_1,p_2 : E \rightrightarrows G$ be a congruence in $\Grp_\fg$. Since the pair is jointly monomorphic, the induced homomorphism $(p_1,p_2) : E \to G \times G$ is a monomorphism, hence injective (see below). Thus, $E$ is isomorphic to a subgroup $R \subseteq G \times G$, which contains the diagonal of $G$. We know from the proof that $\Grp$ is Malcev that $R$ is the congruence induced by the normal subgroup $N \coloneqq \{x \in G : (x,1) \in R\}$, i.e. $R$ is the kernel pair of the projection $h : G \to G/N$ in $\Grp$. Since $G/N$ and $R \cong E$ are finitely generated and $\Grp_\fg \hookrightarrow \Grp$ is fully faithful, $R$ is also the kernel pair of $h$ in $\Grp_\fg$. + references: + - Grp_malcev + + - property: equalizers of cokernel pairs + proof: >- + Let $f : A \to B$ be a homomorphism of finitely generated groups and $C \coloneqq f(A)$. We already know that finite colimits in $\Grp_\fg$ are computed as in $\Grp$. Hence, the cokernel pair $i_1,i_2 : B \rightrightarrows B \sqcup_A B$ of $f$ is computed as in $\Grp$. Since $A \to C$ is surjective, two homomorphisms $B \rightrightarrows T$ agree after composing with $f$ if and only if they agree on $C$. Therefore, $i_1,i_2$ is also the cokernel pair of the inclusion $C \hookrightarrow B$ in $\Grp$. Since $\Grp$ is mono-regular, $C \hookrightarrow B$ is a regular monomorphism in $\Grp$, so it is the equalizer of its cokernel pair $i_1,i_2$ in $\Grp$ by this lemma. Since $C$ is finitely generated as a quotient of $A$, and $\Grp_\fg \hookrightarrow \Grp$ is fully faithful, $C \hookrightarrow B$ is also the equalizer of $i_1,i_2$ in $\Grp_\fg$. + +unsatisfied_properties: + - property: skeletal + proof: This is trivial. + + - property: small + proof: Even the collection of all trivial groups is not a set. + + - property: locally finite + proof: The set $\Hom(\IZ,\IZ) \cong \IZ$ is infinite. + + - property: essentially countable + proof: There are uncountably many pairwise non-isomorphic finitely generated groups. This is a classical result by Neumann (1937), who even constructed continuum many pairwise non-isomorphic $2$-generated groups. It also follows from Hyde–Lodha, Corollary 1.5, which provides continuum many finitely generated simple groups. + label: grp_fg_uncountably_many_objects + + - property: kernels + proof: >- + First of all, the forgetful functor $\Grp_\fg \hookrightarrow \Grp$ is continuous because the forgetful functor $\Grp \to \Set$ is continuous and conservative, and the composite functor $\Grp_\fg \to \Set$ is continuous (in fact, representable). Therefore, it suffices to show that $\Grp_\fg$ is not closed under kernels in $\Grp$. + + Consider the free group $F_2 = \langle x,y \rangle$ and the homomorphism $\varphi : F_2 \to \IZ$ with $\varphi(x) = 1$ and $\varphi(y) = 0$. Its kernel $L \subseteq F_2$ in $\Grp$ is not finitely generated. To see this, consider the restricted wreath product + $$\textstyle \IZ \wr \IZ = (\bigoplus_{n \in \IZ} \IZ) \rtimes \IZ = \langle (e_n)_{n \in \IZ}, t : t e_n t^{-1} = e_{n+1}, \, [e_n,e_m]=1 \rangle.$$ + Define the homomorphism $\psi : F_2 \to \IZ \wr \IZ$ by $\psi(x)=t$ and $\psi(y)=e_0$. Since $\varphi$ is the composition of $\psi$ with the projection onto the right factor, $\psi$ maps $L$ into $\bigoplus_{n \in \IZ} \IZ$. In fact, $\psi(L) = \bigoplus_{n \in \IZ} \IZ$, since $L$ contains $x^n y x^{-n}$ and + $$\psi(x^n y x^{-n}) = t^n e_0 t^{-n} = e_n.$$ + But $\bigoplus_{n \in \IZ} \IZ$ is not finitely generated. Hence, $L$ is not finitely generated either. + label: grp_fg_no_kernels + + - property: normal + proof: 'Every non-normal finitely generated subgroup of a finitely generated group (such as $C_2 \hookrightarrow S_3$) provides a counterexample. Indeed, if an injective homomorphism $i : K \to G$ is the kernel of some $f : G \to H$ in $\Grp_\fg$, then, using again that $\Grp_\fg \hookrightarrow \Grp$ is continuous, it would also be the kernel in $\Grp$, so that $i$ is the inclusion of a normal subgroup.' + references: + - grp_fg_no_kernels + + - property: countable powers + proof: This is the same argument as for $\Grp_\c$. Since the forgetful functor $\Grp_\fg \to \Set$ is representable (by $\IZ$), it preserves products. Therefore, if the power $\IZ^{\IN}$ exists in $\Grp_\fg$, its underlying set must be the ordinary cartesian product, which however is uncountable. But every finitely generated group is countable. + references: + - grpc_no_countable_powers + + - property: countable copowers + proof: Assume that the copower $C \coloneqq \IN \otimes \IZ$ exists in $\Grp_\fg$. It is a finitely generated group $C$ with a universal family of elements $(x_n)_{n \in \IN}$. Let us say that $C$ is generated by $d$ elements. Then every finitely generated group $G$ is generated by $d$ elements, since for a generating set $\{g_0,\dotsc,g_m\}$ of $G$ we find a homomorphism $C \to G$ mapping $x_n \mapsto g_n$ for $n \leq m$ and $x_n \mapsto 1$ for $n > m$, and it is clearly surjective. This is absurd, since for example $\IZ^{d+1}$ is not generated by $d$ elements. + + - property: ℵ₁-cofiltered limits + proof: Since the inclusion $\Grp_\fg \hookrightarrow \Grp$ is continuous (see the proof that $\Grp_\fg$ has no kernels), it suffices to show that $\Grp_\fg$ is not closed under $\aleph_1$-cofiltered limits in $\Grp$. This is done here, using a non-trivial finitely generated group $G$ satisfying $G \cong G \times G$ to build an $\aleph_1$-cofiltered diagram of finitely generated groups whose limit in $\Grp$ is uncountable. Remark that $\Ab_\fg$ is closed under $\aleph_1$-cofiltered limits. + references: + - grp_fg_no_kernels + + - property: ℵ₁-filtered colimits + proof: This is shown here, using an uncountable group in which every countable subset is contained in a finitely generated subgroup, and which is a subgroup of a countable product of finitely generated groups. + + - property: epi-regular + proof: >- + Consider the homomorphism $\psi : F_2 \to \IZ \wr \IZ$ from the proof that $\Grp_\fg$ has no kernels, which maps $x \mapsto t$ and $y \mapsto e_0$. It is surjective, since $\IZ \wr \IZ$ is generated by $t$ and the elements $t^n e_0 t^{-n} = e_n$, so it is an epimorphism. By a theorem of Baumslag, the restricted wreath product $\IZ \wr \IZ$ is not finitely presented (he shows more generally that a restricted wreath product $A \wr B$ of non-trivial finitely presented groups is finitely presented if and only if $B$ is finite). If $\psi$ were regular, its kernel would be the normal closure of finitely many elements (see below), so that $\IZ \wr \IZ \cong F_2 / \ker(\psi)$ would be finitely presented. + + Alternatively, here is a counting argument: Since there are only countably many finite presentations, there are only countably many finitely presented groups up to isomorphism. Since there are uncountably many finitely generated groups up to isomorphism (see the proof that $\Grp_\fg$ is not essentially countable), there is a finitely generated group $H$ that is not finitely presented. Choose a surjective homomorphism $f : F_n \to H$ from a free group of finite rank. As above, it is an epimorphism, but not a regular one. + references: + - grp_fg_no_kernels + - grp_fg_uncountably_many_objects + label: grp_fg_not_epi_regular + + - property: kernel pairs + proof: >- + We apply Lemma 12 here to the inclusion $\Grp_\fg \hookrightarrow \Grp$. It preserves limits (see the proof that $\Grp_\fg$ has no kernels), in particular kernel pairs. It maps epimorphisms to regular epimorphisms, since epimorphisms in $\Grp_\fg$ are surjective (see below), and surjective homomorphisms are regular epimorphisms in $\Grp$. Since we already know that $\Grp_\fg$ is not epi-regular, it does not have kernel pairs. More precisely, every epimorphism in $\Grp_\fg$ that is not regular, such as $F_2 \to \IZ \wr \IZ$, has no kernel pair. + references: + - grp_fg_no_kernels + + - property: coquotients of cocongruences + proof: >- + We show that if $G$ is a finitely generated group and $L \subseteq G$ is a subgroup that is not finitely generated, then the cokernel pair of $L \hookrightarrow G$ in $\Grp$ is a cocongruence in $\Grp_\fg$ without an equalizer. Such subgroups exist, for example the kernel $L \subseteq F_2$ from the proof that $\Grp_\fg$ has no kernels. + + Let $E \coloneqq G \sqcup_L G$ be the pushout in $\Grp$ with its two inclusions $i_1,i_2 : G \rightrightarrows E$. Then $E$ is finitely generated, since it is generated by $i_1(G) \cup i_2(G)$, and $i_1,i_2$ is a cocongruence in $\Grp$. Since $\Grp_\fg$ is a full subcategory containing $G$ and $E$, it is also a cocongruence in $\Grp_\fg$. + + Since $\Grp$ is mono-regular, $L \hookrightarrow G$ is a regular monomorphism in $\Grp$. By this lemma, it is therefore the equalizer of its cokernel pair $i_1,i_2$ in $\Grp$. Now assume that $i_1,i_2$ have an equalizer $e : K \to G$ in $\Grp_\fg$. Since the inclusion $\Grp_\fg \hookrightarrow \Grp$ preserves limits (see the proof that $\Grp_\fg$ has no kernels), $e$ is also an equalizer of $i_1,i_2$ in $\Grp$. So $K \cong L$, which is not finitely generated, a contradiction. + references: + - grp_fg_no_kernels + + - property: counital + proof: The canonical morphism $F_2 = \IZ \sqcup \IZ \to \IZ \times \IZ$ is not a monomorphism since $F_2$ is not abelian. + + - property: cogenerator + proof: 'This is the same argument as for $\Grp_\c$. Assume that a cogenerator $Q$ exists in $\Grp_\fg$. Since $Q$ is countable, there are only countably many finitely generated subgroups of $Q$. But there are continuum many finitely generated simple groups; this follows from Hyde–Lodha, Corollary 1.5. Hence, there is a finitely generated simple group $H$ which does not embed into $Q$. Since $H$ is simple, any non-trivial homomorphism $H \to Q$ would be injective, so every homomorphism $H \to Q$ must be trivial. But then $\id_H, 1 : H \rightrightarrows H$ are not separated by a homomorphism $H \to Q$.' + references: + - grpc_no_cogenerator + + - property: coregular + proof: Pushouts of injective homomorphisms between finitely generated groups do not need to be injective, see MSE/5088032. + + - property: regular quotient object classifier + proof: >- + We adapt the proof for $\Grp$. Assume that $\Grp_\fg$ has a regular quotient object classifier, i.e. a finitely generated group $P$ such that every regular epimorphism $G \to H$ in $\Grp_\fg$ is the cokernel of a unique homomorphism $\varphi : P \to G$; here we use that pushouts in $\Grp_\fg$ are computed as in $\Grp$. By the classification of regular epimorphisms (see below), this means equivalently that every normal subgroup $N \subseteq G$ which is the normal closure of finitely many elements is $\langle \langle \varphi(P) \rangle \rangle$ for a unique homomorphism $\varphi : P \to G$, where $\langle \langle - \rangle \rangle$ denotes the normal closure. If $c_g : G \to G$ denotes the conjugation by $g \in G$, then the images of $\varphi$ and $c_g \circ \varphi$ have the same normal closure, so the homomorphisms must be equal. In other words, $\varphi$ factors through the center $Z(G)$. But then every normal subgroup of $G$ that is the normal closure of finitely many elements, in particular $G$ itself, would be contained in $Z(G)$, which is wrong for every non-abelian finitely generated group $G$, such as $S_3$. + references: + - Grp_no_regular_quotient_object_classifier + +special_objects: + initial object: + description: trivial group + terminal object: + description: trivial group + coproducts: + description: '[finite case] free products' + products: + description: '[finite case] direct products with pointwise operations' + +special_morphisms: + isomorphisms: + description: bijective homomorphisms + proof: It is a full subcategory of $\Grp$, for which we know that isomorphisms are bijective homomorphisms. + monomorphisms: + description: injective homomorphisms + proof: For the non-trivial direction, the forgetful functor to $\Set$ is representable (by the finitely generated group $\IZ$), hence preserves monomorphisms. + epimorphisms: + description: surjective homomorphisms + proof: 'This is the same argument as for $\Grp_\c$. For the non-trivial direction, if $f : G \to H$ is an epimorphism, we may factor it as $G \to f(G) \to H$, where $f(G)$ is finitely generated as a quotient of $G$. Then $f(G) \to H$ is still an epimorphism, but also an inclusion and hence a monomorphism. Since we already know that the category is mono-regular, $f(G) \to H$ must be an isomorphism.' + regular epimorphisms: + description: surjective homomorphisms whose kernel is the normal closure of finitely many elements + proof: >- + If $f : G \to H$ is a regular epimorphism, it is the coequalizer of some pair $a,b : K \rightrightarrows G$ in $\Grp_\fg$. We already know that coequalizers in $\Grp_\fg$ are computed as in $\Grp$. Therefore, $f$ is surjective, and its kernel $N$ is the normal closure of $\{a(k) b(k)^{-1} : k \in K\}$. Let $k_1,\dotsc,k_n$ be generators of $K$, and let $M \subseteq N$ be the normal closure of $a(k_1) b(k_1)^{-1}, \dotsc, a(k_n) b(k_n)^{-1}$. The two homomorphisms $K \rightrightarrows G \to G/M$ agree on the generators $k_i$, hence they are equal, which means $a(k) b(k)^{-1} \in M$ for all $k \in K$. Thus, $N = M$. This proves that $N$ is the normal closure of finitely many elements. + + Conversely, let $f : G \to H$ be a surjective homomorphism whose kernel is the normal closure of $g_1,\dotsc,g_n \in G$. Let $a : F_n \to G$ be the homomorphism from the free group $F_n$ on $e_1,\dotsc,e_n$ with $a(e_i) = g_i$, and let $1 : F_n \to G$ be the trivial homomorphism. Then $f$ is the coequalizer of $a,1$ in $\Grp$, and hence also in $\Grp_\fg$, since $F_n$ is finitely generated. diff --git a/database/data/category-implications/congruences.yaml b/database/data/category-implications/congruences.yaml index ad5a60b86..965e3311b 100644 --- a/database/data/category-implications/congruences.yaml +++ b/database/data/category-implications/congruences.yaml @@ -116,7 +116,7 @@ proof: >- Suppose we have a coreflexive corelation $$X+X' \xtwoheadrightarrow{p} E \xtwoheadrightarrow{r} X$$ - on $X$, where we let $X'$ be an isomorphic copy of $X$ for clarity below. Let $Y$ be the equalizer of $p\circ i_1, p\circ i_2 : X \rightrightarrows E$. That means that for a generalized element $x \in X(T)$, $x \in Y(T)$ if and only if $p(x) = p(x')$. Then by the assumptions $p$ is a regular epimorphism. By regularity, $p$ is the coequalizer of its kernel pair, which can be expressed as the equalizer $K$ of + on $X$, where we let $X'$ be an isomorphic copy of $X$ for clarity below. Let $Y$ be the equalizer of $p\circ i_1, p\circ i_2 : X \rightrightarrows E$. That means that for a generalized element $x \in X(T)$, $x \in Y(T)$ if and only if $p(x) = p(x')$. Then by the assumptions $p$ is a regular epimorphism. By this lemma, $p$ is the coequalizer of its kernel pair, which can be expressed as the equalizer $K$ of $$p \circ \pi_1,\, p\circ \pi_2 : (X+X') \times (X+X') \rightrightarrows E,$$ where $\pi_1, \pi_2$ are the projections. By distributivity and extensivity, it is sufficient to calculate the equalizer on each "quadrant" of $(X+X') \times (X+X')$, i.e. the four copies of $X \times X$. diff --git a/database/data/functor-implications/limits preservation.yaml b/database/data/functor-implications/limits preservation.yaml index fa2b1222c..374cb021d 100644 --- a/database/data/functor-implications/limits preservation.yaml +++ b/database/data/functor-implications/limits preservation.yaml @@ -149,10 +149,10 @@ - preserves coreflexive equalizers associated_assumptions: domain: - - pushouts + - cokernel pairs conclusions: - preserves regular monomorphisms - proof: 'Let $F : \C \to \D$ be a functor preserving coreflexive equalizers, where $\C$ has pushouts; we only need that $\C$ has cokernel pairs. Let $i : X \to Y$ be a regular monomorphism in $\C$. By Prop. 3.2 at the nLab, $i$ is the equalizer of the two canonical morphisms $u_1,u_2 : Y \rightrightarrows Y \sqcup_X Y$ into the cokernel pair of $i$. By the universal property of the pushout, there is a morphism $\nabla : Y \sqcup_X Y \to Y$ with $\nabla u_1 = \nabla u_2 = \id_Y$. Thus, $u_1,u_2$ is a coreflexive pair. Since $F$ preserves coreflexive equalizers by assumption, $F(i)$ is the equalizer of $F(u_1), F(u_2)$. In particular, $F(i)$ is a regular monomorphism.' + proof: 'Let $F : \C \to \D$ be a functor preserving coreflexive equalizers, where $\C$ has cokernel pairs. Let $i : X \to Y$ be a regular monomorphism in $\C$. By this lemma, $i$ is the equalizer of the cokernel pair $u_1,u_2 : Y \rightrightarrows Y \sqcup_X Y$ of $i$. By the universal property of the pushout, there is a morphism $\nabla : Y \sqcup_X Y \to Y$ with $\nabla u_1 = \nabla u_2 = \id_Y$. Thus, $u_1,u_2$ is a coreflexive pair. Since $F$ preserves coreflexive equalizers by assumption, $F(i)$ is the equalizer of $F(u_1), F(u_2)$. In particular, $F(i)$ is a regular monomorphism.' - id: zero_preserving_condition assumptions: diff --git a/shared/structure.history.json b/shared/structure.history.json index dbe579f93..e9a7f1c32 100644 --- a/shared/structure.history.json +++ b/shared/structure.history.json @@ -214,5 +214,6 @@ "Bin_symm", "Bin_symm_refl", "Set_bij", - "Set_family_bij" + "Set_family_bij", + "Grp_fg" ]