# Questions tagged [locally-presentable-categories]

For questions about locally presentable categories.

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### Reflections in locally presentable categories

In this paper, -4th line in the first paragraph on the (first) page 89, then each full subcategory of $\cal H$ closed under limits ... should or should not the word reflective be present: then each ...
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### Dual version of Adjoint Functor theorems

I am trying to dualize three versions of the adjoint functor theorem. If $C$ and $D$ are locally small, $C$ is total (meaning the yoneda functor has left adjoint) then $F:C\rightarrow D$ has a right ...
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### Commutation of filtered colimits with pullbacks

It is known that in a locally $\lambda$-presentable category $\lambda$-filtered colimits commute with $\lambda$-small limits, and hence with pullbacks. I guess that this fails for $\mu$-filtered ...
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### Exercise 1.d.1 in Locally Presentable and Accessible Categories

Find a category $K$ which is cocomplete and in which every object is a directed colimit of finitely presentable objects, although $K$ is not locally presentable. My attempt was the category Ord, the ...
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### Why is every object in a locally presentable category small

The definition I am working with is the following, a category $\mathcal{C}$ with all small colmits is called locally presentable if it has a set of small objects $S\subset Obj(\mathcal{C})$ every ...
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### Is the 2-Category of Groupoids Locally Presentable?

I am wondering if the 2-Category of groupoids is Locally Presentable? Locally presentable means the category is accessible and co-complete. Edit: It has been pointed out that the category of ...
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What is the difference of $\text{CAT}^{\mathbb T}$ from $\text{VAR}$ in this paper sketched below?
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### Nearly locally presentable categories

Here1, in the remark $2.3 (1)$ how from the fact that ${\cal K}(A,-)$ does not preserve coproducts it follows that ${\cal K}(A,-)$ sends special $\lambda$-directed colimits to $\lambda$-directed ...
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### Comma categories of locally finitely presentable categories

Let $\mathbf{C}$ be a locally finitely presentable category, and let $A$ be an object of $\mathbf{C}$. The slice category $\mathbf{C}/A$ is locally finitely presentable. Is this also true for the co-...
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### Cogenerating sets in l.f.p. categories?

Locally finitely presentable categories have generating sets by definition. I wonder if there are any examples (or if there is a known classification) of l.f.p categories which have a cogenerating set....
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### Is $MonCat$ locally presentable?

Is the category of monoidal categories and strict monoidal functors locally presentable? Recall this means that there is a small set of small objects $S$ such that any object in $MonCat$ can be ...
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### Grpd as a locally presentable category

Is the category of groupoids Grpd a locally presentable category? If the answer is yes, can someone sketch a proof or point a reference out? Thanks
### directed colimit in $\mathsf{Set}$
In my previous question here, how can I prove that $FC$ doesn't idenity too much from the fact that $F$ is bounded, i.e. by this fact: for every $X$ and $x∈FX$ there is a finite $Y$ and $i:Y→X$ ...
The $n$Lab page on coreflective subcategories cites a theorem of Adamek and Rosický showing that every colimit-closed full subcategory of a locally presentable category is coreflective. My question is,...