Questions tagged [ct.category-theory]
Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.
6,395
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Lemma 5.4.5.11 of HTT
In Lemma 5.4.5.11 of HTT, the proof given relies on Lemma 5.4.5.10. However it seems that Lurie applies Lemma 5.4.5.10, which requires the given simplicial set to be contractable, to an arbitrary $\...
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About the dual of the cube lemma in homotopy theory
Consider the Reedy category $2\rightarrow 1 \leftarrow 0$. Consider a map of diagrams of topological spaces $D\to E$ over this Reedy category:
The maps which are fibrations are depicted with the ...
- 3,759
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1
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296
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Weil cohomologies with given field of definition and coefficient field
Fix a perfect field $k$. Fix a field $K$ of characteristic $0$.
A Weil cohomology induces a functor from the category of smooth projective geometrically connected $k$-schemes to the category of $\...
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3
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What is the geometric realization of the the nerve of a fundamental groupoid of a space?
It can be easily seen that there exists a functor $F:Top \rightarrow Grpd$ from the category of topological spaces to the category of groupoids defined as follows:
Obj: $X \mapsto \pi_{\leq 1}(X)$, ...
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Can a locally presentable category have a proper class of accessible localizations?
Question: What is an example of a locally presentable category $\mathcal C$ such that there exists a proper class of accessible localizations $(\mathcal C \to \mathcal D_i)_{i < ORD}$?
In other ...
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Reduced products of complete Boolean algebras
I expect that complete Boolean algebras are not closed under reduced products modulo $\kappa$-complete filters, for any regular cardinal $\kappa$. Is it true? And, is a reference for this?
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When are projective modules closed under highly-filtered colimits?
Let $R$ be a ring. Let $Mod(R)$ be the category of left $R$-modules, and let $Proj(R) \subseteq Mod(R)$ be the full subcategory of projective $R$-modules. Let's say that $R$-projectives are closed ...
- 61.6k
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Nonbraided rigid monoidal category where left and right duals coincide
In a braided rigid monoidal category $(\mathcal{M},\otimes)$ left and right duals coincide. What is an example of a rigid monoidal category where left and right duals coincide but there exist no ...
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Is the 2-сategory of groupoids locally presentable?
I am wondering if the 2-сategory of groupoids is locally presentable. Locally presentable means the category is accessible and co-complete.
It has been pointed out that the category of groupoids is ...
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Does the Dwyer-Kan model structure make dgCat a model $2$-category?
Let dgCat be the category of small dg-categories. The well-known Dwyer-Kan model structure makes dgCat a model category.
Now we consider dgCat as a 2-category, which objects small dg-categories, $1$-...
- 8,707
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The skew monoidal structure induced by a functor
$\require{AMScd}$Let $\cal K$ be a 2-category, and $j : A\to B$ one of its 1-cells. Assume that the induced map
$$
j^* : {\cal K}(B,B)\to {\cal K}(A,B)
$$ precomposing with $j$ has a left adjoint $j_!$...
- 13.2k
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Cogenerator of Categories of Topological Spaces Satisfying Some Separation Axiom
This question begins with a sort of mysterious comment at the bottom of this Wikipedia page on injective cogenerators. There, it is said, without citation or proof, that as a result of the Tietze ...
- 10.1k
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Freely adding finite limits preserves some colimits?
Let $\mathcal{K}$ be a category and $\mathcal{K}_{\text{fin}}$ its free completion with finite limits.
Does the embedding $\mathcal{K} \hookrightarrow \mathcal{K}_{\text{fin}}$ preserve some ...
- 8,186
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2
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Limit of a sequence of locally presentable categories
Let $\dotsc \to \mathcal{C}_2 \xrightarrow{F_1} \mathcal{C}_1 \xrightarrow{F_0}\mathcal{C}_0$ be a sequence of cocontinuous functors between locally presentable categories. Consider the limit $\...
- 5,402
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What is a universal tree?
I came across some slides talking about the Hrushovski construction. One of the examples was the construction of a "universal tree".
I was curious because the collection of finite trees does not ...
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Factorization system "tilted" by $(L,R)$
Suppose you have a pair of orthogonal factorization systems, $(E_0, M_0), (E_1, M_1)$ in a category $\cal C$ such that $M_0\subseteq M_1$; this entails that there is a ternary factorization
$$
X\...
- 13.2k
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Do $RHom(C,D)$ and $DG(C,D)$ have equivalent homotopy categories?
Toen in The homotopy theory of dg-categories and derived Morita theory Section 6 introduced the internal Hom's between dg-categories. Actually for two dg-categories $C$ and $D$, Toen defined
$$
RHom(C,...
- 8,707
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Establishing Duality in Tannakian Categories
I sometimes need to prove a category is Tannakian. Part of the definition of a Tannakian category is that it is rigid.
However, I find the definition of rigid categories somewhat difficult. I don't ...
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355
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Push-outs of fully faithful (enriched) functors
I'd like to know a reference where the following property, that I believe to be true, is checked: given a diagram of categories and functors $B\leftarrow A\rightarrow C$ (I'm actually interested in ...
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400
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Are multicolimits suitable colimits?
Today I encountered the notion of multicolimit.
Lacking a standard reference for this notion, let me give a self-contained definition of this gadget.
If $S\colon \cal K\to E$ is a diagram, we ...
- 13.2k
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2
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606
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On internal functions and arrows in a Topos
I am not an expert on elementary topos (meaning by this to work with the internal language in a Grothendiek topos) and I rather be told than struggle with the following:
Consider an elementary topos ...
- 599
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When is a cube of cofibrations are "lattice"?
Let $C$ be a category with cofibrations in the sense of (Waldhausen, Algebraic K-Theory of Spaces) and denote by $F_n(C)$ the category with cofibrations consisting of sequences of $n$ cofibrations $...
- 61.5k
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Why no morphisms from the contradictory proposition to the inconsistent context?
Consider Higher order predicate logic over dependent type theory (DPL) as defined in Chapter 11 of B. Jacobs's book "Categorical Logic and Type Theory" (though I think this question applies to first-...
- 1,419
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2
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535
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Universal cocompletion without leaving our universe
Let $\mathcal{U}$ be a universe and $\mathcal{U}^+$ a universe with $\mathcal{U} \in \mathcal{U}^+$. Denote by $\text{Cat}(\mathcal{U})$ the $\mathcal{U}^+$-category of all $\mathcal{U}$-categories, ...
- 61.5k
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2
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Colimits in a bigger universe
Fix a universe $\mathcal{U}$. Call a category $\mathcal{U}$-complete if every diagram indexed by a $\mathcal{U}$-small category has a limit, and a functor $\mathcal{U}$-continuous if it preserves $\...
- 61.5k
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347
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Applications of lax 2-limits which are not pseudo 2-limits
One application of pseudo 2-limits (bilimits) in algebraic geometry is already found in the definition itself of stacks with values in a 2-category admitting bilimits (i.e. a discrete 1-cell-...
- 91
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Does there exist any massive proper $C^*$-subalgebra?
Definition 1: Suppose $B$ is a $C^* $-algebra. $A$ is massive $C^* $-subalgebra of $B$ iff
1. $A$ is a subalgebra of $B$;
2. for each irreducible representation $\pi$ of $B$ representation $\pi|_A$ is ...
- 1,264
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3
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626
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Fibered category with an adjoint inclusion
Suppose $X:D \to C$ is a fibered category (I do not assume the fibers to be groupoids). Suppose that $X$ is actually left adjoint to a fully faithful embedding $C \hookrightarrow D$. Is there a ...
- 15.3k
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Fibrations of Simplicial sets
Hello,
Maybe it is too vague a question, but I would like to ask if anybody could say some explanatory words about the importance (for infinity category study) of studying all the kinds of fibrations ...
- 5,492
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votes
4
answers
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Kan extensions and the yoneda embedding.
[Edit: For a category $C$ let $C^\wedge$ denote the category of presheaves on $C$.]
Let $f:C\to D$ be a functor. Precomposition with $f^{op}$ induces a functor
$f^\wedge:D^\wedge \to C^\wedge$. This ...
- 3,161
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Induced Grothendieck topology on a presheaf or sheaf category of a site?
While reading Demazure-Gabriel's construction of $\mathcal{S}ch$ as a full subcategory of $\mathcal{P}sh(CRing^{op})$, I've been trying to translate their exposition into the language of covering ...
6
votes
1
answer
350
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Are all coproducts of 1 in a topos distinct ?
Inspired by the two solutions to Harry's question
Can a topos ever be an abelian category?
I was wondering whether all coproducts of 1 in a topos are distinct up to isomorphism? That is $1 + 1 + \...
- 12k
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3
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Quotient of a category by a free group action
Let Cat denote the 1-category of small categories. The functor Mor : Cat -> Set which assigns to a category its set of morphisms (aka Hom([• -> •], -)) does not commute with most colimits. ...
- 24.9k
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Consequences of imposing conditions on the restricted Yoneda embedding of a functor
$\newcommand{\op}{\mathsf{op}}\newcommand{\yo}{よ}$Given a functor $F\colon\mathcal{C}\to\mathcal{D}$, the composition
$$\mathcal{D}\overset{\yo}{\hookrightarrow}\mathsf{PSh}(\mathcal{D})\xrightarrow{F^...
- 10.7k
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Hopf monads in categorical probability theory
1. Context. According to [1], probability monads are arguably the most important concept in categorical probability theory. In [2] Fritz and Perrone argue that "in order for a monad to really ...
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$\infty$-topos as an internal $\infty$-category in itself
I'm interested (both autonomously and directly related to my work) in the natural internalization of $\infty$-topos sheaves in it (as usual, assuming Grothendieck universes). Is there any literature ...
- 6,484
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310
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Do the various notions of morphism spaces of simplicial sets agree on the underived level?
$\DeclareMathOperator\Ho{Ho}\DeclareMathOperator\Hom{Hom}$There are (at least) seven kinds of morphism spaces for a simplicial set $X$:
The left-pinched morphism space $\Hom^L_X(x,y)$,
The right-...
- 10.7k
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votes
1
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391
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Tensor triangulated categories associated to schemes and their families
For any essentially small, rigid and idempotent-complete tensor triangulated (TT for short) category $\mathcal{T}$ Balmer (The spectrum of prime ideals in tensor triangulated categories) constructs a ...
- 1,267
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Groupoids as models of symmetric simplicial sets
In the Elephant, Peter Johnston remarks that internal categories may be regarded as simplicial objects that “preserve all limits that happen to exist in $\Delta^{op}$“ (I guess you might call this a ...
- 1,253
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Can conservativity depend on the universe?
Question 1: Let $F: C \to D$ be a conservative, $\kappa$-cocontinuous functor between small, $\kappa$-cocomplete categories. Is the induced functor $Ind_\kappa(F): Ind_\kappa(C) \to Ind_\kappa(D)$ ...
- 61.6k
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Is there such a thing as a weighted Kan extension?
The title pretty much sums it up.
More in detail. Let $C$, $D$ and $E$ be categories, let $F:C\to D$ and $G:C\to E$ be functors, and let $P:C^{op}\to \mathrm{Set}$ be a presheaf. The colimit of $F$ ...
- 2,129
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318
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Is it possible to use Feynman diagrams to represent a dot product $a \cdot b$?
Feynman diagrams are topological entities, but they describe linear
operators
It has been observed that Feynman diagrams are in particular string diagrams (morphisms in monoidal categories)) in ...
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answer
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Why does every chain complex have a map into its cone?
In Weibel's An introduction to homological algebra he defines a cone as an explicit chain complex associated to the given one -i.e. for a chain $C=(C_i, d)$ he defines $Cone(C)=\left(C_{i-1} \oplus ...
- 475
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Basic example of a formal affine scheme, functorial point of view
$\let\opn=\operatorname$For my BA thesis I have to describe formal groups from the functorial point of view. I am hence reading Strickland - Formal Schemes and Formal Groups, which is apparently the ...
- 455
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Motivation for using etale topology in representability of functors problems
I am reading a paper that proves the representability for certain functors whose domain is the category of superschemes. The paper claims that to prove representability of functors (or possibly just ...
- 255
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Map from the Multiset Monad to the Giry Monad: From Data to Probabilities
The Mulitiset monad, aka the free commutative monoid monad or "Bag" monad, takes a set to the set of all Multisets for that set. A Multiset is like a set, but can have duplicates. It is used in ...
- 1,319
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votes
2
answers
287
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Combinatorial proof that some model categories are monoidal/enriched?
I'm looking for examples of proofs that some Quillen model categories are monoidal, or enriched over an other model category, which are based on explicit computation of the "pushout product" of the ...
- 40.2k
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votes
2
answers
285
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Construction of combinatorial model categories with all objects fibrant
By abstract construction of a combinatorial model category, I mean starting from a locally presentable category satisfying some assumptions, e.g. equipped with a cylinder or a cocylinder satisfying ...
- 3,759
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votes
1
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How to understand the Deligne' tensor product of finite abelian category
In the sec 1.11. "Delignes' tensor product of locally finite abelian categories" of the book "Tensor Categories" of EGNO, the deligne's tensor
product $C \boxtimes D$ of two k-linear locally finite ...
- 481
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303
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Diagrams in an Elementary Topos
Let $T$ be a sheaf topos and $I$ a small category. Then the functor category $[I,T]$ is also a sheaf topos. Now let $E$ be an elementary topos (cartesian closed category with finite limits + subobject ...
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