### From Sets to Algebras

For an integral domain $$A$$ we consider the contravariant functor $$\hom(\_, A)$$ from the category of sets to the category of commutative unital $$A$$-algebras. We note that since $$A$$ is an integral domain the idempotents of $$\hom(L, A)$$ for any set $$L$$ are precisely the maps from $$L$$ to $$A$$ with values in $$\{0, 1\}$$.

1. Lemma. $$\hom(\_, A)$$ is pseudomonic.

• Proof. $$\hom(\_, A)$$ is faithful since for any map $$m \colon L \rightarrow K$$ and $$k \in K$$ we have $$m^{-1} (k) = \big(\hom(m, A) (1_k)\big)^{-1} (1)$$ where $$1_k := 1_{\{k\}}$$ and $$1_{K'}$$ is the indicator function for any subset $$K' \subseteq K$$.

Now suppose $$\varphi$$ is an isomorphism from $$\hom(K, A)$$ to $$\hom(L, A)$$ then $$\varphi$$ induces a bijection between the non-zero centrally primitive idempotents of $$\hom(K, A)$$ and $$\hom(L, A)$$. Now the non-zero centrally primitive idempotents of $$\hom(K, A)$$ are just the maps of the form $$1_k$$ for some $$k \in K$$ and similarly for $$\hom(L, A)$$. Let $$m \colon L \rightarrow K$$ be the corresponding inverse bijection, then for any $$c \in \hom(K, A)$$ and $$l \in L$$ we have $\begin{split} \varphi(c) \cdot 1_l & = \varphi(c \cdot 1_{m(l)}) = \varphi(c(m(l)) 1_{m(l)}) \\ & = c(m(l)) \varphi(1_{m(l)}) = c(m(l)) 1_l \\ & = \hom(m, A) (c) \cdot 1_l \end{split}$ and thus $$\varphi = \hom(m, A)$$.

• Corollary. The functor $$\hom(\_, A)$$ induces an anti-equivalence between the category of sets and the replete image of $$\hom(\_, A)$$.

• Corollary. For any category $$\mathcal{C}$$ the functor $$\hom(\_, A)$$ induces an anti-equivalence between the category of set-valued precosheaves on $$\mathcal{C}$$ and the category of presheaves with values in the replete image of $$\hom(\_, A)$$.

• Lemma. $$\hom(\_, A)$$ is full when restricted to the category of finite sets.

• Proof. Let $$\varphi \colon \hom(K, A) \rightarrow \hom(L, A)$$ be a homomorphism with $$K$$ and $$L$$ finite, then $$\varphi(1_k)$$ is an idempotent for each $$k \in K$$ and thus we have subsets $$L_k \subseteq L$$ such that $$\varphi(1_k) = 1_{L_k}$$. Further we have $$\sum_{l \in L} 1_l = 1 = \varphi(1) = \varphi\big(\sum_{k \in K} 1_k) = \sum_{k \in K} \varphi(1_k) = \sum_{k \in K} 1_{L_k}$$ and thus $$L = \bigcup_{k \in K} L_k$$. Now for any $$k, k' \in K$$ with $$k \neq k'$$ we have $$0 = \varphi(0) = \varphi (1_k \cdot 1_{k'}) = 1_{L_k} \cdot 1_{L_k'}$$, hence $$L_k$$ and $$L_{k'}$$ are disjoint. Altogether we obtain that the subsets $$L_k$$ with $$k \in K$$ form a partition of $$L$$ and we may define a map $$m \colon L \rightarrow K$$ such that $$m(l) = k$$ for $$l \in L_k$$ for all $$k \in K$$. With this definition we have $$\varphi = \hom(m, A)$$ since the two maps agree on a basis of $$\hom(K, A)$$.

The following example shows that we cannot assume the unrestricted functor $$\hom(\_, A)$$ to be full, if $$A$$ is a general ring.

• Example. We consider $$\hom ({\mathbb{N}}, {\mathbb{Z}}/ p {\mathbb{Z}})$$. Let $$\mathfrak{a}$$ be the ideal of all $$c \in \hom ({\mathbb{N}}, {\mathbb{Z}}/ p {\mathbb{Z}})$$ with $$c^{-1} (0)$$ cofinite. By Krull’s theorem $$\hom ({\mathbb{N}}, {\mathbb{Z}}/ p {\mathbb{Z}})$$ has a maximal ideal $$\mathfrak{m}$$ with $$\mathfrak{a} \subset \mathfrak{m}$$ and this gives a homomorphism of fields $$i \colon {\mathbb{Z}}/ p {\mathbb{Z}}\rightarrow \hom ({\mathbb{N}}, {\mathbb{Z}}/ p {\mathbb{Z}}) / \mathfrak{m}$$. We further have $$[c]^p - [c] = [c^p - c] = 0$$ for all $$[c] \in \hom ({\mathbb{N}}, {\mathbb{Z}}/ p {\mathbb{Z}}) / \mathfrak{m}$$ and as $$X^p - X$$ is a polynomial of degree $$p$$ it has at most $$p$$ roots in $$\hom ({\mathbb{N}}, {\mathbb{Z}}/ p {\mathbb{Z}}) / \mathfrak{m}$$ and thus $$i$$ is a bijection. Now the canonical homomorphism from $$\hom ({\mathbb{N}}, {\mathbb{Z}}/ p {\mathbb{Z}})$$ to the quotient $$\hom ({\mathbb{N}}, {\mathbb{Z}}/ p {\mathbb{Z}}) / \mathfrak{m}$$ yields a homomorphism $$\varphi \colon \hom ({\mathbb{N}}, {\mathbb{Z}}/ p {\mathbb{Z}}) \rightarrow \hom(\{1\}, {\mathbb{Z}}/ p {\mathbb{Z}}) \cong {\mathbb{Z}}/ p {\mathbb{Z}}$$ which is not in the image of $$\hom(\_, {\mathbb{Z}}/ p {\mathbb{Z}})$$, since for any map $$m \colon \{1\} \rightarrow {\mathbb{N}}$$ the element $$1_{m(1)} \in \mathfrak{a} \subset \mathfrak{m}$$ is mapped to $$1 \in {\mathbb{Z}}/ p {\mathbb{Z}}$$ under $$\hom(m, {\mathbb{Z}}/ p {\mathbb{Z}})$$.

• Remark. From a discussion similar to that of the previous lemma and example we can conclude that for sets $$K$$ and $$L$$ with $$L$$ non-empty, the map from $$\hom(L, K)$$ to $$\hom_{A\text{-algebras}} (\hom(K, A), \hom(L, A))$$ induced by $$\hom(\_, A)$$ is surjective if and only if all ideals $$\mathfrak{p}$$ of $$\hom(K, A)$$, with $$\hom(K, A) / \mathfrak{p} \cong A$$ as $$A$$-algebras, are of the form $${\{c \in \hom(K, A) ~|~ c(k) = 0\}}$$ for some $$k \in K$$.

• Lemma. $$\hom(\_, A)$$ is continuous as a functor from the opposed category of sets to the category of $$A$$-algebras.

• Proof. We argue that $$\hom(\_, A)$$ is continuous as a functor to the category of commutative rings, the lemma then follows by a general result about limits in the under category. We fix a small category $$D$$. For an object $$X$$ of any category $$\mathcal{C}$$ we denote by $$\Delta (X)$$ the constant functor from $$D$$ to $$\mathcal{C}$$ that maps any object of $$D$$ to $$X$$ and any morphism of $$D$$ to the identity. Let $$F$$ be a functor from $$D$$ to the category of sets, then we have the canonical natural transformation $$t \colon F \rightarrow \Delta( {\operatorname{colim}}(F) )$$. Now $$\hom(\Delta({\operatorname{colim}}(F)), A) = \Delta(\hom({\operatorname{colim}}(F), A))$$ and by the universal property of the limit of $$\hom(F(\_), A)$$ we have a homomorphism of rings $$s \colon \lim(\hom(F(\_), A)) \rightarrow \hom({\operatorname{colim}}(F), A)$$ such that $$(\hom(\_, A) \circ t) \circ \Delta(s)$$ is the canonical natural transformation from $$\Delta(\lim(\hom(F(\_), A)))$$ to $$\hom(F(\_), A)$$. Now the forgetful functor from the category of commutative rings to the category of sets is continuous as well as $$\hom(\_, A)$$ as a functor to the category of sets, hence in the category of sets both $$(\hom(\_, A) \circ t) \circ \Delta(s)$$ and $$\hom(\_, A) \circ t$$ itself satisfy the universal property of the limit of $$\hom(F(\_), A)$$, and thus $$s$$ is a bijection.

• Corollary. If $$D$$ is a set-valued cosheaf, then $$\hom(D(\_), A)$$ defines a sheaf with values in the category of $$A$$-algebras.

• Example. For any locally path connected topological space $$X$$ the singular homology $$H_0 (X)$$ is naturally isomorphic to the free abelian group with basis $$\Lambda(X)$$ and by the universal property of the free ablian group the restriction from $$\hom_{{\mathbb{Z}}} (H_0 (X), A)$$ to $$\hom(\Lambda(X), A)$$ is an isomorphism of $$A$$-modules. Further we have a natural isomorphism of $$A$$-modules from $$H^0 (X, A)$$ to $$\hom_{{\mathbb{Z}}} (H_0 (X), A)$$ by the universal coefficient theorem and since for any $$x \in X$$ and $$\alpha, \beta \in H^0 (X, A)$$ we have $\begin{split} \langle \alpha \cup \beta, [x] \rangle & = \langle H^0 (d, A) (\alpha \times \beta), [x] \rangle = \langle \alpha \times \beta, H_0 (d) ([x]) \rangle \\ & = \langle \alpha \times \beta, [(x, x)] \rangle = \langle \alpha \times \beta, [x] \times [x] \rangle \\ & = \langle \alpha, [x] \rangle \langle \beta, [x] \rangle, \end{split}$ where $$d \colon X \rightarrow X \times X, x \mapsto (x, x)$$ is the diagonal map, the composition of these two isomorphisms is an ismorphism of $$A$$-algebras. Since the above identifications are natural in $$X$$, the functors $$\hom(\Lambda(\_), A)$$ and $$H^0 (\_, A)$$ define isomorphic sheaves on any locally path connected topological space.

Given a continuous function $$f \colon X \rightarrow M$$ from a locally path connected topological space $$X$$ to $$M$$, the sheaves $$f_* \hom(\Lambda(\_), A) \cong f_* H^0 (\_, A)$$ and $$\hom(\lambda_f (\_), A)$$ are identical. Bubenik, de Silva, and Scott (2014) define a generalized persistence module on the poset of open sets of $$M$$ to be a functor to another category, thus $$\lambda_f$$ is a generalized persistence module with values in the opposed category of sets and $$f_* H^0 (\_, A)$$ is a persistence module with values in the category of $$A$$-algebras. A functor from a category $$\mathcal{C}$$ to a category $$\mathcal{D}$$ then gives rise to a map from the generalized persistence modules with values in $$\mathcal{C}$$ to persistence modules with values in $$\mathcal{D}$$, so in their language $$f_* H^0 (\_, A)$$ is the image of $$\lambda_f$$ under the map induced by $$\hom(\_, A)$$ and thus their theory can be used to relate these two constructions in the context of topological persistence.

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