Significance of subdirect irreducibility outside universal algebra, particularly in category theory? Subdirect irreducibility is a central notion in universal algebra. Trying to formulate it as abstractly as possible, your object is subdirectly irreducible if, in the opposite category, it does not have any nontrivial cover by regular subobjects. That is, in a sufficiently (co?)regular category $\mathbf C$, an object $X$ is subdirectly irreducible if, in the opposite category, for any family $X_i\rightarrowtail X$ of regular subobjects with the induced map $\coprod_iX_i\to X$ (regular?) epi, one of the $X_i$ must be $X$ itself.
Thus an "universal algebra-like" category is such that in its opposite, every object is covered by subobjects which are subdirectly irreducible in the above sense.
While this looks (very!) familiar, to my perplexion I cannot really come up with a specific interesting examples of categories like this which would not come from universal algebra. I have vague feeling this must have to do with local compactness and such, but I am blocked by the fact that subdirect irreducibility of finite unions of subdirectly irreducibles is still another condition which I don't know where to place.
Do you have any interesting examples of this kind, from whatever field?
 A: I cannot really come up with a specific interesting examples of categories like this which would not come from universal algebra
One has to be more precise about this requirement before the question can be answered. I would say that a small category is a $2$-sorted partial algebra $\langle \textrm{Ob}, \textrm{Mor}; \circ, \textrm{Id}, \textrm{dom}, \textrm{cod}\rangle$, where $\circ\colon \textrm{Mor}\times \textrm{Mor}\to \textrm{Mor}$ is a partial operation while $\textrm{Id}\colon \textrm{Ob}\to \textrm{Mor}$, $\textrm{dom}\colon \textrm{Mor}\to \textrm{Ob}$, and $\textrm{cod}\colon \textrm{Mor}\to \textrm{Ob}$ are total operations. The study of small categories then may be viewed as a part of universal algebra (if your universal algebra allows multisorted partial algebras).
I will describe a class of examples and let others tell me whether this class comes from universal algebra.
Let $L$ be a bounded lattice considered as a category.
Us usual, the statement that $L$ is being considered as a category
means that the objects of the category are the elements of $L$
and whenever $a\leq b$ in $L$ we demand that
$\textrm{Hom}(a,b)$ have a unique element,
otherwise $\textrm{Hom}(a,b)$ is empty.
There are no nontrivial pairs of parallel arrows in
categories like this. Thus, the only (co)equalizers
are the identity maps. While every
morphism is both a monomorphism and an epimorphism, the
only regular monomorphisms or regular epimorphisms are the identity maps.
The only regular subobject of $a\in L$ is $\textrm{id}_a\colon a\to a$.
The opposite category is a category of the same type.

That is, in a sufficiently (co?)regular category $\mathbf{C}$,
an object $X$ is subdirectly irreducible if, in the opposite category, for any family $X_i\rightarrowtail X$ of regular subobjects with the
induced map $\bigsqcup X_i\to X$ (regular?) epi, one of the
$X_i$ must be $X$ itself.
If you test the example against this definition
you get that each regular
subobject $X_i\rightarrowtail X$ is an instance of
$\textrm{id}_X\colon X\to X$. The
induced map $\bigsqcup X_i\to X$ will also be
an identity map, hence it will be regular epi. We will have
one of the $X_i$ equal to $X$ itself, since all $X_i$
must be $X$ itself.
The conclusion is that, under the given definition,
all elements of $L$ are subdirectly irreducible.
If you drop the demand that $X_i\rightarrowtail X$ be a regular monomorphism and only require that it be a monomorphism, an element of $L$ will be subdirectly irreducible if and only if it is meet-irreducible in $L$. In that case, if you start with an algebraic lattice $L$, where every element is a complete meet of meet-irreducibles, you will get that every object of $L$ is a subdirect product of subdirectly irreducible objects of $L$.
