Answer by R. van Dobben de Bruyn for Categories that admit all products but...
Now that the question was changed to existence of finite (co)products, here is a well-known example: the category $\mathbf{Gp}_{\text{fin}}$ of finite groups has finite limits, but not finite...
View ArticleAnswer by R. van Dobben de Bruyn for Categories that admit all products but...
Here is an example appearing 'in nature': the category $\mathbf{Sch}$ of schemes has arbitrary (small) coproducts (disjoint unions), but existence of (small) products is a subtle question (on the other...
View ArticleAnswer by Tim Campion for Categories that admit all products but not all...
Jeremy Rickard gives a nice example of a (locally small) category with all small colimits but not all small limits in an answer to Cocomplete but not complete abelian category. His category is even...
View ArticleAnswer by Todd Trimble for Categories that admit all products but not all...
If $X$ is a large set and $\mathsf{Set}$ is the category of small sets, then $\mathsf{Set}/X$ is small-cocomplete, but it fails to have a terminal object. Now dualize this.
View ArticleAnswer by Max New for Categories that admit all products but not all coproducts
As the question is stated, there are actually no examples, at least assuming classical logic.Classically, every small complete category is thin, i.e., a preorder with arbitrary meets....
View ArticleAnswer by Alex Kruckman for Categories that admit all products but not all...
For the version of the question about finite products and coproducts:Any meet-semilattice that does not have joins. For example, a linear order with no least element (which has no empty join).Take any...
View ArticleCategories that admit all products but not all coproducts
What are examples for categories that admit all products but not all coproducts.
View ArticleAnswer by Dario Stein for Categories that admit all products but not all...
Opposites of Kleisli categories.If C is a category with coproducts, and T a monad on C, then it is easy to see that the Kleisli category Kl(T) will inherit the coproducts from C. On the other hand,...
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