Questions tagged [locally-presentable-categories]

For questions about locally presentable categories.

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Why injectivity class doesn't imply weakly reflective

In the snippet below (taken from Rosicky, Adamek: On injectivity in locally presentable categories) in the context of locally presentable categories it seems that injectivity class doesn't imply ...
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43 views

Grothendieck categories are locally presentable

The way I've read it, many fundamental facts about Grothendieck categories, such as completeness, are special cases of those about locally presentable categories (see, for example, this answer). ...
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2answers
90 views

Why do small objects have to preverse colimits from directed sets, rather than colimits from semilattices?

Let $\kappa$ be a regular cardinal. Let $\mathcal C$ be a category. Let $A : \mathcal C$ be an object of $\mathcal C$. As best I can tell, we say that $A$ is a small object of $\mathcal C$ if, for all ...
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1answer
40 views

$\omega$-pure subgraph in $\mathbf{Grph}$

Here on the page 7: I do not follow how $B$ looks like. It is written there: $B$ is a strong subgraph of $A$ over all nodes distinct from $x_k$ for $k\geq 1$". So $B$ is only $x_0$ as we have ...
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1answer
50 views

Essential smallness of finitely presentable subcategory

Consider the presheaf category $\left[C, \mathbf{Set}\right]$ where $C$ is small. I have read that this is a locally finitely presentable category. This makes sense to me except one detail: Why is ...
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0answers
56 views

Which limit sketches produce Grothendieck toposes?

A limit sketch $\mathcal S=(\mathcal A,L)$ consists of a small category $\mathcal{A}$, together with a set $L$ of cones in $\mathcal A$. A model (in the category of sets) of a limit sketch is a ...
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34 views

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 $i:Y→X$ ...
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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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1answer
50 views

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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2answers
43 views

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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1answer
70 views

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 ...
2
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1answer
64 views

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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1answer
49 views

Categories/Varieties and Monads

What is the difference of $\text{CAT}^{\mathbb T}$ from $\text{VAR}$ in this paper sketched below?
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1answer
52 views

WIth pushout each orthogonality class is an injectivity class

I cannot see how follows the equality $\cal M^{\bot}=M^*$-Inj below: (both inclusions are wanted) where $X \in {\cal H}^\bot \iff \forall f\in{\cal H}: \mathrm{Hom}(f,X)$ bijective and $X \in {\...
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1answer
79 views

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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91 views

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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1answer
56 views

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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1answer
146 views

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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1answer
243 views

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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63 views

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 ...
2
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1answer
75 views

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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1answer
92 views

Subobjects in Locally Presentable categories.

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 ...
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1answer
111 views

$\lambda$-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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1answer
65 views

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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48 views

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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2answers
148 views

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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1answer
42 views

When does this converse of Vopěnka's principle hold?

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,...
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177 views

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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1answer
156 views

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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1answer
215 views

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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1answer
101 views

$S$-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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1answer
180 views

Finitely presentable objects

After introducing the notion of finitely presentable object as an object $A$ such that ${\sf Hom}(A, -)$ preserves directed colimits, an "explicit" form of it is given: $A$ is finitely presentable ...
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1answer
220 views

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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1answer
104 views

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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0answers
84 views

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
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1answer
459 views

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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3answers
412 views

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 ...