Questions tagged [locally-presentable-categories]

For questions about locally presentable categories.

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A paper: Accessible categories, saturation and categoricity

I've been reading a paper on accessible categories, saturation and categoricity by "Jiří Rosický" for quite some time, but I still cannot understand one detail: In the snippet below, in the ...
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A locally small cocomplete quasitopos with strong generator is locally presentable

In Theorem C.2.2.13 of Johnstone's Elephant, it is asserted that any locally small cocomplete quasitopos with a strong generator (a "generating set" in Johnstone's terminology) is locally ...
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Models of Limit Sketches

How do look the categories equivalent to models of finitary and infinitary limit sketches, respectively ? There are at least $2$ interpretations of the word finite here: the cardinality of the set of ...
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What are the categories whose sheaves are representable?

Let $\mathcal{A}$ be a category. The Yoneda embedding $Y : \mathcal{A} \hookrightarrow \mathrm{Hom}(\mathcal{A}^{\mathrm{op}},\mathbf{Set})$ corestricts to an embedding $$y : \mathcal{A} \...
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The metric spaces and nonexpansive maps: no non-empty space is finitelly presentable

I have asked this question elsewhere, but I wrongly formulated it. Let me now correct it. Why is no nonempty metric space in the category of metric spaces and non-expansive maps finitely presentable ? ...
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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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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
95 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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$\omega$-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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53 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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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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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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54 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
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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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108 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 ...
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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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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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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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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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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
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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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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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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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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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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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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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$\lambda$-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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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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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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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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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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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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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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133 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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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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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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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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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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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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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 ...