All Questions
Tagged with examples ct.category-theory
46 questions
5
votes
1
answer
195
views
What is the most general notion of exactness for functors between triangulated categories?
For triangulated categories $T,T'$ I would like to define "weakly exact" functors as those that respect cones, that is, $F(Cone f)\cong Cone(F(f))$ for any $T$-morphism $f$, and I do not ...
- 16.9k
6
votes
1
answer
207
views
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 ...
- 764
12
votes
2
answers
786
views
Examples of non-polynomial comonads on Set?
Question: What are examples of comonads on $\mathbf{Set}$ that are not polynomial?
Background: polynomial functors and comonads on Set
A functor $F\colon\mathbf{Set}\to\mathbf{Set}$ is called ...
- 8,659
10
votes
0
answers
228
views
Natural cotransformations and "dual" co/limits
$\DeclareMathOperator{\id}{\mathrm{id}}\DeclareMathOperator{\Hom}{\mathrm{Hom}}\DeclareMathOperator{\UnCoNat}{\mathrm{UnCoNat}}\DeclareMathOperator{\UnNat}{\mathrm{UnNat}}\DeclareMathOperator{\CoNat}{\...
- 11.8k
4
votes
1
answer
222
views
$\ast$-autonomous categories with non-invertible dualizing object?
1. Definition
Firstly, recall the following nLab-definition of a $\ast$-autonomous category:
A $\ast$-autonomous category is a symmetric closed monoidal category $(C,\otimes,I,\multimap)$ with a ...
- 764
8
votes
1
answer
284
views
Cartesian monoidal star-autonomous categories
Disclaimer: This is a crosspost (see MathStackexchange). Apologies if cross-posting is frowned upon. However, it seems that on Stackexchange there are not many people familiar with star-autonomous ...
- 764
13
votes
0
answers
213
views
Examples and counterexamples to Lack's coherence observation
In Lack's A 2-categories companion, he states
There are general results asserting that any bicategory is biequivalent to
a 2-category, but in fact naturally occurring bicategories tend to be ...
- 10.6k
7
votes
2
answers
644
views
Existence of nontrivial categories in which every object is atomic
An object $X$ of a cartesian closed category $\mathbf C$ is atomic if $({-})^X \colon \mathbf C \to \mathbf C$ has a right adjoint (hence is also internally tiny). Intuitively, atomic objects are &...
- 10.6k
7
votes
2
answers
237
views
Examples of 2-categories with multiple interesting proarrow equipment structures
Proarrow equipments (also known as framed bicategories) are identity-on-objects locally fully faithful pseudofunctors $({-})_* \colon \mathcal K \to \mathcal M$ for which every 1-cell $f_*$ in the ...
- 10.6k
9
votes
1
answer
388
views
Simple example of nontrivial simplicial localization
Does anyone has a simple example of a 1-category $\mathcal{C}$ and a collection of morphisms W such that the infinity-categorical / simplicial localization $\mathcal{C}\left[W^{-1}\right]$ is not a 1-...
- 834
5
votes
1
answer
256
views
Example of a non-cocomplete model category of a realized limit sketch
Let $(\mathcal{E},\mathcal{S})$ be a realized limit sketch, i.e. a locally small category $\mathcal{E}$ with a class $\mathcal{S}$ of limit cones in it. It is not assumed that $\mathcal{E}$ is small, ...
- 63.1k
27
votes
1
answer
1k
views
"Non-categorical" examples of $(\infty, \infty)$-categories
This title probably seems strange, so let me explain.
Out of the several different ways of modeling $(\infty, n)$-categories, complicial
sets and comical sets allow $n = \infty$,
providing ...
0
votes
1
answer
362
views
Examples of additive categories [closed]
I already this question here but I didn't get any satisfactory answer, so I will try in MO now.
There are a lot of interesting and creative examples of categories, such as for example, the category ...
1
vote
1
answer
932
views
Examples of faithful functors not injective on objects
As is well-known, a faithful functor need not be injective on objects. What are some good examples to illustrate this point?
9
votes
1
answer
328
views
Horizontal categorification: Two questions
According to the nlab, horizontal categorification is a process in which a concept is realized to be equivalent to a certain type of category with a single object, and then this concept is generalized ...
- 63.1k
4
votes
1
answer
331
views
Very canonical constructions
You have two categories $C_1$ and $C_2$. We call a map of the classes $\mathrm{Ob}(C_1)\rightarrow \mathrm{Ob}(C_2)$ a construction. Sometimes you can find a functor $C_1\rightarrow C_2$ inducing this ...
38
votes
10
answers
4k
views
Big list of comonads
The concept of a monad is very well established, and there are very many examples of monads pertaining almost all areas of mathematics.
The dual concept, a comonad, is less popular.
What are examples ...
10
votes
1
answer
534
views
Intuition behind orthogonality in category theory, and origin of name
In category theory, two morphisms $e:A\to B$ and $m:C\to D$ are said to be orthogonal if for any $f:A\to C$ and $g:B\to D$ with $m\circ f=g\circ e$, there exists a unique morphism $d:B\to C$ such that ...
- 2,129
6
votes
2
answers
295
views
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 ...
- 42.4k
9
votes
1
answer
737
views
Example of an abelian category with enough projectives and injectives which are not dual
For trying to understand how general a certain theorem is, I'm looking for an example of an essentially small abelian category which has enough projectives and enough injectives, but whose category of ...
- 2,450
2
votes
0
answers
195
views
Right adjoint completions
Forgive me if this question is not well thought out. I don't know how else to ask it.
The nlab page on completion gives some examples of completions which are left adjoints. These completions are "...
9
votes
2
answers
672
views
An example of two cofibrant dg categories whose tensor product is not cofibrant
I have been reading the paper by Toën "The homotopy theory of dg categories and derived Morita theory" where in chapter 4 it is stated that the tensor product of two cofibrant dg categories $C$ and $D$...
- 91
14
votes
3
answers
1k
views
Are all vector-space valued functors on sets free?
Let $\mathbf{Set}$ be the category of finite sets and functions between them, and let $\mathbf{Vect}$ be the category of finite-dimensional complex vector spaces and linear transformations between ...
- 3,937
2
votes
2
answers
369
views
Example(s) of monoidal symmetric closed category with NNO without infinite coproducts?
The question is in the title, here is my motivation:
$\require{AMScd}$Let $(\mathcal C,\otimes,I)$ be a monoidal symmetric closed category. Then, the tensor product commutes with colimits, and if $\...
- 277
5
votes
1
answer
393
views
Not quite adjoint functors
What are standard and/or natural examples of pairs of functors $F:C\leftrightarrows D:G$ and unnatural bijections $\hom_D(Fx,y)\to\hom_C(x,Gy)$ for all $x$ and $y$? Can one do this so that the ...
- 47.3k
30
votes
11
answers
5k
views
What are your favorite concrete examples of limits or colimits that you would compute during lunch?
(The title was initially "What are your favorite concrete examples that you would compute on the table during lunch to convince a working mathematician that the notions of limits and colimits are not ...
5
votes
1
answer
459
views
Example of a non-closed cocomplete symmetric monoidal category
Background
By a cocomplete symmetric monoidal category $C$ I mean a symmetric monoidal category whose underlying category is cocomplete and such that $- \otimes X : C \to C$ is cocontinuous for all $X ...
- 63.1k
10
votes
3
answers
665
views
Abelian category which is not well-powered
Can you give an example of an abelian category which is not well-powered? If not, maybe you can give any reason why there are such abelian categories?
- 101
24
votes
6
answers
4k
views
Any example of a non-strong monad?
Looking for an example of a monad that is not strong.
The reason being, a strong monad (wrt cartesian product) is an "applicative functor" (in functional programming); an example of a non-strong ...
- 562
5
votes
3
answers
235
views
two essentially different concretizaions
It is sometimes emphasized that a "concrete category" is not a property of a category $C$, but rather a structure, i.e. a faithful functor from $C$ to $Set$. Thus, When people talk about a concrete ...
- 2,027
70
votes
28
answers
7k
views
Examples where it's useful to know that a mathematical object belongs to some family of objects
For an expository piece I'm writing, it would be useful to have good examples of the following phenomenon:
(1) ${\cal X}$ is a parameterized family of somethings. (Varieties, schemes, manifolds, ...
4
votes
1
answer
464
views
Examples of "inner products" of parallel morphisms in a dagger category
There is a very interesting abstract notion of the trace of an endomorphism $f : c \to c$ of an object $c$ in a braided monoidal category (although the symmetric case is easier): see, for example, ...
- 118k
4
votes
1
answer
500
views
Is there a category with a subobject classifier but which is not finitely complete?
This is a reverse of the question “Is there a finitely complete category with terminal object but NO subobject classifier?” From “An informal introduction to topos theory” by Tom Leinster I learned ...
- 530
20
votes
3
answers
3k
views
Is there an additive functor between abelian categories which isn't exact in the middle?
Suppose $F: C\to D$ is an additive functor between abelian categories and that
$$0\to X\xrightarrow f Y\xrightarrow g Z\to 0$$
is and exact sequence in $C$. Does it follow that $F(X)\xrightarrow{F(f)...
1
vote
1
answer
1k
views
Classification Problems [closed]
I was thinking about the famous question in philosophy of mathematics: "When are two proofs the same?" and I was wondering if we could somehow "classify" proofs by establishing some sort of functorial ...
5
votes
1
answer
653
views
Categories with products that preserve quotients
It is well known that in the category of all topological spaces, quotient maps aren't preserved by products (this follows from the simpler fact that $X\times (-):Top\to Top$ doesn't preserve quotients)...
- 35.5k
4
votes
0
answers
518
views
Example of a Grothendieck pretopology satisfying a weak saturation condition
Recall that a singleton Grothendieck pretopology (henceforth 'singleton pretopology') on a category $C$ is a collection of maps $J$ containing the isomorphisms, closed under composition and stable ...
- 35.5k
1
vote
2
answers
736
views
Weakly initial sets - examples and nonexamples
A weakly initial set in a category C is a set of objects I of C such that every object a of C has at least one arrow from an object contained in I.
The question is then, does Fields have a weakly ...
- 35.5k
7
votes
9
answers
3k
views
What category without initial object do you care about?
Recently I have been listening to some constructions that have been designed to accommodate categories without an initial object. The speaker has given some idea of a category or two that he cares ...
5
votes
6
answers
2k
views
Is there a finitely complete category with terminal object but NO subobject classifier?
This came up today while thinking about topoi in seminar, as the title suggests my question is;
Is there a finitely complete category with terminal object but NO subobject classifier?
Hopefully if ...
- 4,842
48
votes
6
answers
7k
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What is Yoneda's Lemma a generalization of?
What is Yoneda's Lemma a generalization of?
I am looking for examples that were known before category theory entered the stage resp. can be known by students before they start with category theory.
...
0
votes
1
answer
216
views
Motivation for Cosuspended Category Axioms
Today I was wondering about the axioms given by Bernhard Keller for Cosuspended Categories.
The axioms of a triangle feel very much like exactness, but not quite. The last axiom about the large ...
- 4,842
18
votes
7
answers
5k
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Simple show cases for the Yoneda lemma
I've been given a very simple motivating and instructive show case for the Yoneda lemma:
Given the category of graphs and a graph object $G$, seen as a quadruple $(V_G,\ E_G,\ S_G:E\rightarrow V,\ ...
20
votes
6
answers
3k
views
Is a functor which has a left adjoint which is also its right adjoint an equivalence ?
I am looking for a counter-example of two functors F : C -> D and G : D->C such that
1) F is left adjoint to G
2) F is right adjoint to G
3) F is not an equivalence (ie F is not a quasi-inverse of ...
- 1,320
8
votes
2
answers
625
views
Which commutative rigs arise from a distributive category?
A rig is an algebraic object with multiplication and addition, such that multiplication distributes over addition and addition is commutative. However, instead of requiring that the set forms an ...
- 12.6k
63
votes
22
answers
19k
views
What's a groupoid? What's a good example of a groupoid? [closed]
Or more specifically, why do people get so excited about them? And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?