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Basically, my question is simple: why does the definition of a locally presentable category require all colimits exist?

The motivation for this is that I was learning about algebraic posets, and had a certain idea of a Grothendieck topology in mind - namely, a covering family of $X$ must contain every compact subobject of $X$. I wanted to know what it would take for representable functors to be sheaves in this topology: after unwrapping definitions, this topology is subcanonical if and only if the poset is algebraic.

I was then studying a somewhat natural generalization of this, where the poset is replaced with a category, and covering sieves need to contain all maps from compact objects. (i.e., a subfunctor $S$ of $h_X$ is a covering sieve on $X$ iff $S(F)=Hom(F,X)$ for each compact $F$). You can then show (I'm less confident about this, but I think it holds!) that this topology is subcanonical if and only if every object is a filtered colimit of compact objects. (This can be thought of as a generalization of the previous paragraph, with filtered colimits replacing suprema).

So I had another statement about a certain kind of category, which I had translated as saying this topology was subcanonical. Indeed, these categories satisfy all the requirements of being locally finitely presentable - except that they don't have all their colimits (though they do have all their filtered colimits).

Now, I'm a little unclear on the definition of locally presentable categories - namely, some sources (such as ncatlab) only require that every object being a (suitably small) colimit of compact objects, while others (see for example here ) require that every object be a filtered colimit. In the latter definition, it would seem to make sense to only require (suitably small) filtered colimits to exist.

So returning to my original question: what is the intuition/motivation behind a locally presentable category, and how does the existence of colimits match that motivation?

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Well, there is also the notion of accessible category, which is basically a locally presentable category without the hypothesis that all colimits exist. –  Zhen Lin Jul 31 at 10:01
    
Yep, that's exactly what I wanted. Thanks! –  Andrew Smith Jul 31 at 10:06
    
You should get your hands on Locally Presentable and Accessible Categories by Adamek and Rosicky, which I'm sure could answer many questions you might have. –  Todd Trimble Jul 31 at 11:17
    
@ZhenLin: What about making this an answer? –  Martin Brandenburg Aug 14 at 12:55
1  
It doesn't answer the question that is actually asked, but rather the question that should have been asked. –  Zhen Lin Aug 14 at 14:22

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