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For any simple, undirected graphs $G, H$, let $G\times H$ denote their category-theoretical product. What is an example of an infinite connected graph $G$ with $G \cong G \times G$? (Note that the tot …

Let $G_i = (V_i, E_i)$ be simple, undirected graphs for $i=1,2$. A graph homomorphism is a map $f:V_1\to V_2$ such that $\{f(v), f(w)\}\in E_2$ whenever $\{v,w\}\in E_1$. By $\text{Hom}(G_1, G_2)$ …

Let $\text{Meas}$ be the category of measurable spaces with measurable functions as morphisms. Does $\text{Meas}$ have exponential objects?

Let the category $\mathbf{Gr}$ consist of simple, undirected graphs, together with graph homomorphisms. We say that a graph $P$ is projective if for all graphs $A, B$ with a surjective graph homomorph …

Let $\mathrm{Graph}$ be the category of simple, undirected graphs with graph homomorphisms. For any graphs $G, H$ we denote by $\text{Hom}(G, H)$ the set of graph homomorphisms $f:G\to H$. (Note that …

If $H_i = (V_i, E_i)$ for $i=1,2$ are hypergraphs then a map $f:V_1\to V_2$ is said to be a hypergraph homomorphism if $f(e_1)\in E_2$ for all $e_1\in E_1$. Hypergraphs together with hypergraph homomo …

Let $X$ be a set and let $\Phi(X)$ denote the collection of filters on $X$. For $x\in X$ we denote by $P_x$ the filter $P_x=\{A\subseteq X:x\in A\}$. A convergence space is a pair $(X,\to)$, where $X$ …

Convergence spaces are a generalization of topological spaces; we denote the category of convergence spaces with continuous maps with ${\bf Conv}$. Is ${\bf Conv}$ cartesian-closed?

Given a poset $(P,\leq)$ the interval topology $\tau_i(P)$ on $P$ is generated by $$\{P\setminus{\downarrow x} : x\in P\} \cup \{P\setminus{\uparrow x} : x\in P\},$$ where $\downarrow x = \{y\in P: y\ …

Yes - the product is the categorical product of graphs, which is also bipartite (it's easy to see that if $G, H$ are bipartite, then $G\times H$ cannot contain odd circles), and the coproduct is the d …

A space $(X,\tau)$ is called a $D$-space if whenever one is given a neighborhood $N(x)$ of $x$ for each $x\in X$, then there is a closed discrete subset $D\subseteq X$ such that $\{N(x): x\in D\}$ cov …

First, let me state that I don't believe this question is suitable for MO - but I'll give an answer anyway. The categorical product (in any category) of a family of objects $(G_i)_{i\in I}$ is charac …

By a graph I mean a pair $G = (V, E)$ where $V$ is a set and $E \subseteq [V]^2 := \{\{a,b\}: a\neq b \in V\}$. A graph homomorphism between graphs $G, H$ is a map $f:V(G)\to V(H)$ such that $\{v, w\} …

Let $\textbf{Grph}$ be the category whose objects are graphs $G = (V,E)$ such that $V$ is a set and $E \subseteq \mathcal{P}_2(V) := \{\{a,b\} \subseteq V: a\neq b\}$. We sometimes write $E(G)$ for $E …

Let $\textbf{Grph}$ be the category whose objects are graphs $G = (V,E)$ such that $V$ is a set and $E \subseteq \mathcal{P}_2(V) := \{\{a,b\} \subseteq V: a\neq b\}$. We sometimes write $E(G)$ for $E …