15
votes
Accepted
Is there a large colimit-sketch for topological spaces?
The answer is no. I think this example illustrates why pure colimit sketches are rarely studied; one generally allows limits into the sketch before generalizing to allow colimits into the sketch. That ...
- 63.9k
9
votes
Is there a large colimit-sketch for topological spaces?
Isbell shows in Function spaces and adjoints (1975) that every cocontinuous functor $\mathbf{Top} \to \mathbf{Set}$ is a coproduct of copies of the forgetful functor. (I have found an alternative ...
8
votes
Accepted
Example of a non-cocomplete model category of a realized limit sketch
Here is a nice trick to construct an example. But maybe there are more naturally occuring examples. I feel like there should be a better way to explain the construction, but I don't know how for now.
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- 42.4k
7
votes
Example: Accessible category without colimits
The category Hil of Hilbert spaces, considered as a full subcategory of Ban is $\aleph_1$-accessible but not locally presentable, in fact it is self dual.
The category Lin of linear orders and ...
- 9,111
3
votes
Accepted
How to turn a limit sketch into an essentially algebraic theory?
We'll add a new sort $E$ and force it to be isomorphic (by a specified isomorphism) to the equalizer of $f$ and $g$. Then we can replace $h$ and $k$ by total functions $E \to C$, and so $p$ will have ...
- 25.2k
3
votes
Example: Accessible category without colimits
Here's a general way to get an accessible category that likely will not have many colimits: let $C$ be a locally presentable category, and then let $C^{mono}$ be the category with the same objects as $...
- 63.9k
2
votes
Define a sketch $s_{\mathbf{Grp}}$ such that $\mathbf{Grp}\backsimeq \mathbf{Mod}(s_{\mathbf{Grp}},\mathbf{Set})$
The case of semigroups is spelled out in all details in Sec 2 of Chap 7 of Category theory for Computer Science by Barr and Wells.
- 9,111
2
votes
Accepted
A notion of limit sketches that makes theories unique up to equivalence
If I understand correctly your question you are looking for some definition of limit-sketches such that if two sketches $\mathcal T$ and $\mathcal T'$ are Morita-equivalent (that is the categories of ...
- 3,349
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