Let $\mathbf{Poset}$ denote the category of partially ordered sets and order-preserving maps. Does $\mathbf{Poset}$ have quotients?



Quotients of posets
(There may be some users who think this would have been better asked at Mathematics StackExchange, but I'll go ahead and answer because there are several ways of looking at it.) 


The answer is "of course". See [The Joy of Cats, p. 119](http://katmat.math.uni-bremen.de/acc/acc.pdf#119). The coequalizer of two maps $f, g: X \rightrightarrows Y$ in $\mathbf{Poset}$ is computed in two steps: first take the coequalizer of $f, g$ as if in the category $\mathbf{Preord}$ of preordered sets (sets with reflexive transitive relations). This is done by taking the coequalizer in $\mathbf{Set}$, say $q: Y \to Q$, and then endowing $Q$ with the smallest reflexive transitive relation that makes $q$ an order-preserving map. Second, take the "posetal reflection of $Q$", i.e., the quotient $R$ where $x, y \in Q$ are identified if $x \leq y$ and $y \leq x$ in $Q$; the order relation on $R$ inherited from $Q$ makes $R$ the coequalizer of $f, g$ in $\mathbf{Pos}$. 


In other language: $\mathbf{Preord}$ is topological over $\mathbf{Set}$ (see The Joy of Cats for details), and therefore is cocomplete (and much more). The full inclusion $\mathbf{Pos} \hookrightarrow \mathbf{Preord}$ is a reflective subcategory, and therefore $\mathbf{Pos}$ is also cocomplete, although not topological over $\mathbf{Set}$. 


In still other language: it may be shown that $\mathbf{Preord}$ and $\mathbf{Pos}$ are locally finitely presentable, i.e., they are categories of structures defined entirely in terms of suitable finite limit diagrams in $\mathbf{Set}$, and on general abstract grounds such categories are cocomplete. See Adámek & Rosický, *Locally presentable and accessible categories*, Cambridge University Press 1994. 


