# Questions tagged [locally-presentable-categories]

For questions about locally presentable categories.

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### Why is it crucial that $\kappa$ is a regular cardinal in the definition of $\kappa$-accessible categories?

In the definition of a $\kappa$-accessible (or presentable) category, the cardinal $\kappa$ is always supposed to be regular. What happens in the irregular case?
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### $\lambda$-presentable objects in a locally $\lambda$-presentable category

In Adamek and Rosickys' Locally Presentable and Accessible Categories, I came across the following statement (I'm paraphrasing), every $\mu$-presentable object in a locally $\lambda$-presentable ...
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### Definition of locally presentable category

The standard book on locally presentable categories defines them as : cocomplete categories with a small set of $\lambda$-small objets generating objects of the category under $\lambda$-filtered ...
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### Epi-Mono factorization in presentable categories

If $\mathcal{C}$ is a locally presentable category, then it seems to be well-known that (Strong Epi, Mono) is a factorization system on $\mathcal{C}$. Where can I find a proof of this fact? Actually I ...
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### Question on the definition of a locally presentable category

According to nlab, a category $C$ is called locally presentable if it is accessible and has all small colimits. Moreover, one can show, that this conditions are equivalent to the condition of $C$ ...
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### Presheaf category is locally finitely presentable

Let $S$ be a small category. Consider its presheaf category $\widehat{S} = [S^{\mathrm{op}},\mathsf{Set}]$. Is there a direct way to see that $\widehat{S}_{\mathrm{fp}}$ is essentially small, and that ...
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### Finitely presentable objects and the Kleisli category

There is a correspendence between Lawvere theories $L$ and finitary monads $\mathbb{T}_L$ (associated to $L$), due to Lawvere: the category $Mod(L)$ of models of $L$ (in $\mathbf{Set}$) is equivalent ...
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### Isomorphisms in a reflective subcategory

Let $S$ be a small family of arrows in a locally presentable category $\mathcal{K}$. It is known that the category $\mathcal{K}[S^{-1}]$ is reflective in $\mathcal{K}$ and correspond to the solution ...
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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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### Category of profinite groups

My question is simple: Is the category of profinite groups an accessible category? Thank you Edit: I will add the (hopefully simpler) question: Is the category of profinite groups complete and ...
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### Compact objects and locally finitely presentable categories (the Category of Groups)

I am trying to understand the concept of locally finitely presentable categories. I have discovered the concept of compact object here. I have discovered that for groups, the finitely presented ...
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### Reflections in locally presentable categories-unclear step in the proof

Here in the paper by Rosicky Adamek, Reflections in locally presentable categories on the page 90 in the proof theorem on the page 89, I do not follow the 3rd line there: ... and hence is reflective ...
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### W-split coequalizers

The following snippet is from Adamek, Rosicky:Algebra and local presentability,how algebraic are. It is unclear to me the end of Example 5.1: Since $e$ is the coequalizer of $\bar{u}_1,\bar{u}_2$ in ...
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### $S$-local objects of presheaves are reflective and characterize local presentability

Let $PSh(\mathcal{C})$ be the category of presheaves on a small category $\mathcal{C}$. Let also $S$ be any fixed set of morphisms in $PSh(\mathcal{C})$. I say that an object $F\in PSh(\mathcal{C})$ ...
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### Locally finitely presented category

In page 1 of "Locally finitely presented additive categories", author says that a locally finitely presented category $\mathcal{A}$ is one for which every object can be expressed as a direct limit of ...
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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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### Is the category of chain complexes over an ring $R$ a locally presentable category?

I wonder if the category of chain complexes over an ring $R$ is a locally presentable category. I am trying proving that this category is combinatorial, I have seen some reference for the cofibrantly ...
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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
Take a proper subobject $m: A \to B$ in a locally presentable category. Since the category is locally presentable, $B = \text{colim} B_i$ where $B_i$ are presentables. Under which hypothesis m ...