All Questions
Tagged with reference-request ct.category-theory
630 questions
4
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Bicategories in which the composition functors $\circ_{A, B, C}$ admit right adjoints
In Bénabou's 1967 Introduction to Bicategories, he mentions that, in a forthcoming part II, he would study bicategories $\mathcal K$ in which each composition functor $$\circ_{A, B, C} \colon \mathcal ...
- 10.6k
4
votes
1
answer
214
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Reference request: Algebras over monoid objects in a monoidal category [duplicate]
Looking for a reference for the following easy-to-prove fact:
Say $T$ and $S$ are monads on $\text{Set}$ admitting a monoid homomorphism $\phi : S \to T$ (i.e., a morphism in $\text{Mon}([\text{Set},\...
2
votes
1
answer
149
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Baer sums of extensions
Apologies in advance if this question is too basic, I looked briefly through Weibel and the stacks project and couldn't find any relevant reference.
Let $\mathcal{A}$ denote an abelian category, and ...
- 2,907
6
votes
1
answer
131
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Condition for a functor to induce a cartesian closed functor between categories of presheaves
We denote the category of presheaves on a small category ${\cal C}$ (set-valued functor-category) by $$\widehat{\cal C}:=[{\cal C}^{op},{\bf Set}].$$
Such a category is cartesian closed, i.e. it ...
- 567
3
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0
answers
133
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Grothendieck spectral sequence (cohomology version) for posets with functor coefficient
In this paper, Quillen mentioned a spectral sequence as follows.
Let $f:X\to Y$ is a poset map and $\mathcal{F}:X\to Ab$, where $Ab$ is the category of abelian groups, a functor which is contravariant ...
- 139
4
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1
answer
92
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Does an indexed functor $C \rightarrow \mathbb{B}$ extend to $\operatorname{Psh}(C) \rightarrow \mathbb{B}$?
This question concerns indexed categories and functors, as well as internal categories and functors.
$\newcommand{\Psh}{{\operatorname{Psh}}}$
$\newcommand{\Id}{{\operatorname{Id}}}$
$\newcommand{\...
- 285
6
votes
2
answers
320
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Set theoretical foundations for derived categories
A modern approach to derived functors, that has been shown to be useful in a number of different circunstances is that of a derived category (see the book by Yakutieli, for example, here).
However, it ...
- 3,318
5
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0
answers
179
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Left Adjoint From the Category of Topological Groups to the Category of Condensed Groups
In Scholze's Lecture Notes on Condensed Sets, the author states that (Remark 1.8) the functor that takes a topological group $G$ to its condensation $\underline{G}$ has a left-adjoint, but we do not ...
- 241
2
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0
answers
26
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Reference for the biequivalence between the bicategory of distributors and the bicategory of two-sided discrete fibrations
It is well known that a distributor/profunctor $A \not\rightarrow B$, i.e. a functor $B^{\text{op}} \times A \to \mathrm{Set}$, is equivalent to a two-sided discrete fibration from $A$ to $B$. ...
- 10.6k
11
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3
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671
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Merging single-sorted and multi-sorted theories
The general theory of single-sorted (say, algebraic) theories is very similar to the general theory of multi-sorted (algebraic) theories. Each variable gets a sort, but apart from that nothing really ...
- 63.1k
7
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0
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160
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What happens if we add an initial object to a Lawvere theory?
Motivation
There is a theory of smooth spaces (for example, diffeological spaces) as certain sheaves on the site $\mathrm{Cart}$, which is the "category of cartesian spaces" whose objects ...
- 355
3
votes
1
answer
327
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Holomorphic homotopy conjecture
Let $X$ be a smooth projective variety over the complex numbers, and let $\text{Coh}(X)$ be the category of coherent sheaves on $X$. Consider the dg-category $\text{Perf}(X)$ of perfect complexes on $...
- 31
7
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2
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284
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Bounded geometric morphisms, origin and motivation for the terminology
Bounded geometric morphisms serve as a generalization of Grothendieck topoi; $T$ being a Grothendieck topos iff global section $T \rightarrow Set$ is bounded. I managed to only track this down to the ...
- 1,347
3
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0
answers
97
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Notion of a finite generator in an abelian category
Let us say that an abelian category admits a generator $g$, if for every object $x$ in the category there is an epimorphism $g^{\oplus I} \to x$ for some index set $I$. I am interested in weakening ...
- 1,474
3
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0
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47
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Lax morphism classifiers via lax-idempotentification
Let $T$ be a 2-monad on a nice 2-category $\mathcal K$, so that the inclusion $T\text{-}\mathbf{Alg}_s \to T\text{-}\mathbf{Alg}_l$ of the 2-category of (strict) $T$-algebras and strict $T$-algebra ...
- 10.6k
3
votes
1
answer
162
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Ind-completion commutes with category product
$\def\A{\mathcal{A}}
\def\C{\mathcal{C}}
\def\D{\mathcal{D}}
\def\ind{\operatorname{Ind}}
\def\op{\mathrm{op}}
\def\Hom{\operatorname{Hom}}$I am trying to understand the following result from ...
13
votes
3
answers
670
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How algebraic can the dual of a topological category be?
(I'm going to try to use definitions from Abstract and Concrete Categories: The Joy of Cats by Adámek, Herrlich, and Strecker, since both of the adjectives in the title of my question seem to have at ...
- 12.4k
7
votes
2
answers
292
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Quotient topoi as quotient objects
In Lawvere's Open problems in topos theory; quotient topoi are treated as connected geometric morphisms of Grothendieck topoi.
Is there a good reference for where these come from? Is there any sense ...
- 1,347
5
votes
1
answer
367
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Reference request: locally erasable delta-functor is universal
It is well-documented that an erasable delta-functor $(F^n,\delta)$ is universal. However, in 'small' abelian categories (in the technical sense or otherwise), there aren't always enough objects to ...
- 28.7k
3
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0
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270
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Categorical General Relativity
What are some good references for GR from a categorical point of view?
This is essentially just a big-list reference request.
I'm aware that the subject exists and can do some basic sleuthing to find ...
3
votes
0
answers
92
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Reference for the monoidal category structure $X \otimes Y = X + Y + X \times Y$ on a distributive category
Given a distributive category $\mathscr C$ (more generally a rig category), we can define a (semicocartesian) monoidal category structure on $\mathscr C$ with tensor product given by $X \otimes Y := X ...
- 10.6k
8
votes
1
answer
326
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Standard reference for double category theory
Is there a ‘standard reference’ for double category theory?
Ideally something along the lines of CWM for $1$-category theory or Johnson and Yau’s book for $2$-category theory; some reference that ...
- 10.1k
6
votes
1
answer
237
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Resource on how the definitions of subobjects for various categories can vary
I am looking for a reference on the different possible definitions of subobjects. According to a particular friend of mine, subobjects should be at least monomorphisms (up to slice isomorphism) and at ...
- 591
4
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0
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284
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Institutional approach to linear algebra
In Diaconescu's book Institution Independent Model Theory, it is mentioned on p. 37 that linear algebra can be viewed as an institution. Specifically, we have the following
Definition. An institution ...
- 10.1k
3
votes
0
answers
117
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2-cells in the double category of 2-functors
Mike Shulman has in the answer to my previous question argued that for 2-categories $C$ and $K$ there is a double category whose objects are 2-functors between them and morphisms are lax and colax ...
- 1,347
9
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0
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85
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Reference for the tricategory of elements associated to a trifunctor
The theory of bicategorical fibrations has been relatively well studied, e.g. by Baković and by Buckley. In particular, given a trifunctor $F : \mathcal K \to \mathbf{Bicat}$ from a bicategory $\...
- 10.6k
7
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140
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Is there a synthetic approach to (symmetric) monoidal infinity-categories?
Recent work of Riehl and Verity (e.g. the book "Elements of $\infty$-category theory") has established a "synthetic" / model-independent approach to the study of $\infty$-...
- 101
1
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1
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235
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Notion of $\kappa$-sifted categories?
Let $\kappa$ be a regular cardinal. It seems reasonable to introduce the following definition:
Definition. A simplicial set $K$ is $\kappa$-sifted if, for every set $E$ with $\lvert E\rvert<\kappa$...
- 2,806
3
votes
1
answer
194
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Double category of monads and pseudo monad-morphisms
We can construct bicategories of monads in a bicategory $B$, $Mnd_l(B)$/$Mnd_c(B)$ with lax and colax monad-morphisms respectively.
I am failing to find a good notion of pseudo monad-morphisms.
Is ...
- 1,347
14
votes
1
answer
295
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When were Allegories first introduced?
I’m doing some bibliographic work for my PhD and I’m struggling to find the earliest resources on Allegories.
They were surely made famous by the 90s book “Categories, Allegories” by Freyd and Scedrov....
- 249
2
votes
0
answers
123
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If-and-only-if Linton monadicity theorem over presheaves
This is almost exactly the same question as The (co)monadicity theorem relative to a presheaf topos, but the accepted answer there does not cover the full extension of Linton's monadicity that is a ...
- 1,347
3
votes
0
answers
152
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My category is rigid: what this implies for representation theory?
I am studying a subcategory $\mathcal{C}$ of modules for an associative noncommutative algebra $A$ (which is in fact also a Hopf algebra).
It is clear from our definition of $\mathcal{C}$ that it is ...
- 3,318
6
votes
1
answer
273
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Reference: the category of derived affine schemes is extensive
The category (that is, $(\infty, 1)$-category) of derived affine schemes is the opposite category of the localization of simplicial commutative rings in weak equivalences.
See extensive category. Does ...
- 6,962
4
votes
1
answer
140
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Reference for Reedy weak factorization systems, not Reedy model structure?
What paper/book is appropriate as the standard reference for Reedy weak factorization systems, not Reedy model structure? Specifically, I would like a reference for the following Propositions 1 and 2:
...
- 366
7
votes
0
answers
141
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Quillen's theorem A/B for simplicially enriched categories
I'm looking for any reference which states some version of Quillen's theorem A/B for simplicially enriched categories (so to be clear, simplicial objects of $\mathrm{Cat}$ whose simplicial set of ...
- 171
6
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0
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78
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A distributor between categories induces a distributor between their categories of presheaves
Let $P$ be a distributor/profunctor from a small category $A$ to a small category $B$, i.e. a functor $P : B^\circ \times A \to \mathrm{Set}$.
We may then define a distributor from $[A^\circ, \mathrm{...
- 10.6k
6
votes
0
answers
86
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Reciprocity for algebra objects in two algebraic categories
I think this question Compact Hausdorff and C^*-algebra "objects" in a category. shows that there is no reciprocity between categories of algebra-objects of two algebraic categories.
So, ...
- 519
2
votes
1
answer
214
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Is the category of simplicial $R$-modules closed monoidal?
I am trying to understand if the simplicial mapping space for simplicial $R$-Modules (or at least simplicial vector spaces) is adjoint to the (level-wise) tensor product. For simplicity, let me state ...
- 91
0
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0
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56
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Names for product-like algebras involving a "duo of directed pseudoforests"
I am looking for the names (and/or for any information regarding) two algebras, one "free" and one "restricted" by an equivalence class.
In both cases, there is an (infix) binary ...
16
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2
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2k
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Is Freyd's thesis available online anywhere?
Peter Freyd is a great category theorist. His PhD dissertation, Functor Theory, dates from Princeton in 1960. It's cited as [14] in Mitchell's book Theory of categories. In fact, Google scholar says ...
- 30.3k
1
vote
1
answer
219
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Reference request regarding faithfully exact functors between abelian categories
I am looking for a reference for the following result (or any subresult) in any book or notes:
Lemma. Let $F:\mathcal{A}\to\mathcal{B}$ be an exact functor between abelian categories. The following ...
8
votes
1
answer
253
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Compact objects in slice categories of finitely presentable categories
Given a locally finitely presentable category $\mathscr C$ and an object $X \in \mathscr C$, it is not so hard to show that a morphism $(X \to Y)$ is compact in $\mathscr C_{X/}$ if it can be obtained ...
- 28.7k
6
votes
0
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143
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Looking for Lawvere's "Closed categories and biclosed bicategories" lecture notes
The nLab page on closed bicategories reads
Closed bicategories were introduced by Lawvere in unpublished lecture notes Closed categories and biclosed bicategories (1971).
This work has also been ...
- 11.8k
1
vote
0
answers
87
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Beck's original formulation of the precise tripleability theorem. Reference when considering reflexive pairs?
Thanks to MO's user Varkor, we have access to Beck's original untitled manuscript where Beck first stated his precise tripleability theorem. Up to terminological isomorphism, the PTT as stated in p. 3 ...
13
votes
2
answers
1k
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Categories on which one can determine all model structures?
Famously, there are exactly nine model structures on the category of sets, which are detailed here. In this case, one can exhaustively determine all six weak factorization systems and then see which ...
- 63.9k
5
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0
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249
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Classical first-order model theory via hyperdoctrines
I have been reading this discussion by John Baez and Michael Weiss and I find this approach to model theory using boolean hyper-doctrines very interesting. One of their goal was to arrive at a proof ...
- 3,446
5
votes
1
answer
226
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Categorical description of umbral calculus?
The theme of my current research project is related to umbral calculus in the context of more general algebraic structures, like Hopf algebras (and, in particular, shuffle algebras), so I am trying to ...
- 491
6
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0
answers
184
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Drinfeld center of non-rigid closed monoidal categories
Context.
The Drinfeld center of a rigid monoidal category $\mathcal{C}$ is again rigid. This is not hard to see: Given an object $X\in \mathcal{C}$ together with a half-braiding $\phi_X:X\otimes-\...
- 764
11
votes
1
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482
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What is the commutative coproduct and where can I learn more about it?
This is a reformulation of a Mathematics Stack Exchange question that got no answer there, and, at the same time, a follow-up to another question on MSE.
The original problem was to prove $U(\mathfrak{...
- 491
2
votes
0
answers
92
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Are covering families of localizations stable under pushouts?
For a commutative ring $A$, we call a finite family of localizations $A \to A_{S_i}$ (where $S_i$ are some subsets of $A$) a covering if the canonical morphism $A \to \prod A_{S_i}$ is an effective ...
- 6,962