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Yes - these terms are equivalent. First, it is easy to see that any left-factoring morphism is a retraction: Suppose $l: A \to B$ and pick $Z:= B$ in your definition above and $f:= \mathsf{id}_B$. Si …

Is there an example of a category, and a monomorphism $m:X\to Y$ between two objects such that $m$ is extremal, but not regular? (A monomorphism $m:X\to Y$ is said to be extremal if whenever $m=g\circ …

In the question Fixed points and universal maps for posets, we find the following definition: if $P, Q$ are partially ordered sets, an order-preserving map $u:P\to Q$ is said to be universal if for ev …

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\} …

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 …

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 …

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 …

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

Is every monomorphism in $\mathbf{Frm}$, the category of frames, regular?

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 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 $G_i = (V_i, E_i)$ 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 $\text{Hom}(G_1, G_2)$ …

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 $(X,\tau)$ be a topological space. We say that $A\subseteq X$ is a topological retract if there is a continuous map $r:X\to A$ onto a subspace $A \subseteq X$ such that for all $a\in A$ we have …

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 …