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Motivation. Piggybacking a previous question that I posted, and drawing the colimit diagram for reference and notation:

enter image description here

I wanted to be able to formally say that $\phi_{X},\phi_{Y}$ were injections (in the usual sense) so that, for example, taking a pushout would give me the "natural" pushout (as opposed to some other colimit using different $\phi_{X}$ and $\phi_{Y}$ mappings which are not the injection mappings). In some cases, I would explicitly be in some category (like spaces) and I could say something like $\phi_{X}(y) = y$ in order to force my map to be the injection map; but in other cases where the arrows between objects are not functions (like poset categories) I'm not sure how to say this formally.

Question. When one talks about taking "the usual" pushout, how does one define the injective arrows $\phi_{X}$ and $\phi_{Y}$ for categories where the arrows are not functions (as in the category of posets).

I'm a bit new at this stuff, so if my question is not well-stated or garbled I will attempt to reword it!

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2 Answers 2

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To help you with wording, what you are asking is how the inclusions into the colimit are defined in categories which aren't concrete.

In any category, you have a notion of what a morphism is, and to define a morphism you simply have to specify which of these morphisms your inclusion is. In a concrete category, this means that you can define a morphism by specifying what it does to each object in a set, but that's not the only way!

Consider the category whose objects are real numbers and we have a unique morphism $x\to y$ if $x\leq y$. Then and subset $X\subseteq \mathbb R$ can be thought of as a diagram in our category, and the colimit of this diagram would be a real number $L$ such that for every $x\in X$, we have a map $x\to L$, and if we have any other number $M$ such that each $x\in X$ has a map $x\to M$, then we have a factorization $x\to L\to M$. Since the existence of a map $x\to y$ just means $x\leq y$, this is another way of saying that $L=\sup X$ (if it exists! This category isn't complete or cocomplete meaning limits and colimits need not exist). In this case, the maps into the colimit are the only maps they can be. I.e. the map $\phi_x\colon x\to L$ is the unique map given since $x\leq L$.

As an exercise, try writing down something similar for the case of limits, and then work out what limits and colimits are in an arbitrary poset.

As another side note, technically a poset is a concrete category, meaning it's equivalent to one whose underlying objects are sets. See if you can work out how this is done.

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Thank you, this is exactly what I was trying to ask. This is also a pretty nice example --- though, at the end, you note that the maps are "the only maps they can be"; this is nearer to what I'd like to look at. What if there was not an obvious choice for maps? Is there a "good" way to choose a mapping? "Good" is obviously a loaded word, but I mean in the sense that in, say, the category of spaces we have the standard inclusion mappings into the colimit. Or is it the case that in this non-concrete category it does not matter much the mapping you take since the diagram will force one? –  james Apr 12 '12 at 16:54
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@james: The definition of colimit is usually a cocone; it's the object $L$ together with the family of insertion maps $\phi_X$. Saying the colimit is just $L$ is an abuse of notation. –  Hurkyl Apr 12 '12 at 17:43
    
@Hurkyl Ah, alright. Is there any way to force the insertion mappings to be similar to the usual inclusion mappings we use for spaces and groups? Maybe by requiring that they be monic mappings or something like this...? –  james Apr 12 '12 at 19:05
    
Actually, after re-reading the other answer, I think that answer answers my last question! I can't always make them inclusion-like. Sad. –  james Apr 12 '12 at 19:23
    
@james Exactly. We generally think of the maps into the colimit as inclusions because that's exactly what they are in most categories people care about, but in some contexts this 'inclusion' isn't an underlying set theoretic inclusion. It's an abuse of terminology I guess, but it's fairly standard. –  SL2 Apr 12 '12 at 19:34

In category theory, there is a notion of a monic map, which is typically used in place of the notion of an injective function. If both $\phi$ and $\psi$ are cocones, then $\phi_X$ is monic if and only if $\psi_X$ is monic.

However, I think you didn't mean "injection", but instead "inclusion". In general, things in category theory are only defined up to isomorphism. There typically is no literal analog of the set-theoretic notion of "subset".

However, it's not hard to use monic maps to define a notion of "subobject": a subobject of $X$ is just an isomorphism class of "monic maps to $X$" Specifically, if $f : Y \to X$ and $g : Z \to X$ are monic, then a morphism of "monic maps to $X$" from $f$ to $g$ is given by a morphism $h : Y \to Z$ such that $gh = f$. It's an isomorphism of "monic maps to $X$" iff $h$ is an isomorphism.

This reproduces the notion of inclusion when you couple it with the notion of a generalized element. A generalized element of $X$ is simply a morphism $a : Y \to X$. If $S$ is a subobject of $X$, then we can $a \in S$ iff there is a factorization $a = fb$ (where $f$ is any representative of the isomorphism class $S$).

Also, if $g : X \to Z$ is a morphism, we write $g(a)$ for the generalized element $ga$ of $Z$.


That said, the maps $\phi_X$ aren't always monic, so even in $\mathbf{Set}$, you can't arrange for them to be inclusions (or even injections). (e.g. in a pushout diagram, if all of the arrows are epimorphisms, then the maps $\phi_X$ are epimorphisms too)

Even if they are monic, you still can't always arrange for them to be inclusions. For example, consider the pushout of the diagram

$$ \begin{array}{ccc} \{ x \} &\to& \{ y \} \\ \downarrow \\ \{ x \} \end{array} $$

The pushout is a one-element set, but if $x \neq y$, it cannot contain both $\{x\}$ and $\{ y \}$.

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