All Questions
Tagged with limits-colimits functors
51
questions
3
votes
1
answer
57
views
Does a functor which reflects limits also reflect cones?
Following Borceux's Categorical Algebra Definition 2.9.6:
Let $F: \mathcal{C}\to\mathcal{B}$ be a functor. $F$ reflects limits when, for every functor $G: \mathcal{D}\to\mathcal{A}$ with $\mathcal{D}$...
- 121
3
votes
1
answer
44
views
A sort of Day convolution without enrichment
Some time ago I was trying to define a monoidal structure on a functor category $[\mathcal{C},\mathcal{D}]$ between two monoidal categories $\mathcal{C}$ and $\mathcal{D}$, such that the monoid ...
- 26.3k
3
votes
1
answer
69
views
Orbit functor is not co-representable
Let ${1}\neq H\le G$ be groups. Denote by $G\textit{-}\mathsf{Set}$ the category of sets with a $G$ action, with $G$-equivariant maps as morphisms. Let $(-)/H: G\textit{-}\mathsf{Set}\to \mathsf{Set}$ ...
- 596
2
votes
2
answers
83
views
(Co)Products are bifunctors, but are general (co)limits also functors?
In a category with all products or coproducts, the (co)product operation can be understood as a bifunctor. More generally let $\mathcal{C}$ be a category with all limits of shape $D$, where for ...
1
vote
1
answer
63
views
$F : I → C$ is a functor from a filtered category with any colimit. $ι : J → I$ is the embedding with $J$ cofinal. Then colim $F \cong$ colim($F ◦ ι$)
Let $I$ be a filtered category. We say a full subcategory $J$ is cofinal if for
every object $A \in I$, there is an object $B \in J$ so that $\text{Hom}(A, B) \neq \emptyset$. Let $F : I \to \mathcal ...
- 313
0
votes
0
answers
54
views
Every object of category $\mathcal{C}$ is compact in $Ind(\mathcal{C})$
Let $\mathrm{Ind}(\mathcal{C})$ be the Ind-completion. We can define it in two different (but similar) ways: as filtered colimits of representable presheaves and as the category of diagrams over ...
- 332
1
vote
1
answer
43
views
Condition that limit of a diagram $F$ exists if $\alpha: Mor(-,L) \to \lim Mor(X,F(i))$
Suppose $F : I \to \mathcal C$ is a diagram for a small category $I$ and a category $\mathcal C$. Suppose also that there is an object $L$ of $\mathcal C$ such that for all $X \in Obj(\mathcal C)$ ...
- 313
4
votes
1
answer
100
views
Subcategory of functor category is complete
Let $D$ be a complete and cocomplete symmetric monoidal closed category (a Bénabou cosmos). Let $P$ be the permutation category. Consider the substitution product $\circ$ on the functor category $[P^{...
- 1,769
3
votes
2
answers
161
views
Size-issues in the definition of categorical limits
I know (at least) two definitions of categorical limits. The setting is always as follows. Let $F:J\rightarrow C$ be a functor between not necessarily small nor locally small categories.
For the first ...
- 1,769
3
votes
1
answer
63
views
Confusion on Exercise 3.6.ii in Riehl: showing the product is associative via a unique nat. isomorphism
This is an exercise (I believe) to practice methods using the functoriality of limits. The exercise is as follows:
For any pair of objects $X,Y,Z$ in a category $\mathsf{C}$ with binary products, ...
- 2,504
1
vote
1
answer
490
views
Limits and colimits in the category of presheaves
We know that the category of presheaves (i.e. $Fct(C^{Op}, Set)$ ) is an elementary topos and in particular it is finitely complete and finitely cocomplete and so it has all finite limits and colimits....
- 11
1
vote
1
answer
183
views
Equivalence of categories and colimits
Consider functors $F,G:J\to Cat$ which admit colimits and $\eta:F\to G$ a natural transformation such that $\forall i\in J$ $\eta_i:F(i)\to G(i)$ is an equivalence of categories. Is it true that then ...
- 463
0
votes
1
answer
56
views
Limit of sequential diagram
Consider $X_0\leftarrow X_1\leftarrow \dots$ be a sequential diagram in a category $C$ and suppose it has a limit. I want to show that if $(n_k)$ is a sequence of increasing natural integers, then the ...
- 463
1
vote
1
answer
45
views
Proving a formula involving Hom-set go colimit and constant functor
Let $I$ and $C$ be categories, assume $I$ is small and denote $\Delta$ the functor $C \to C^I$ that sends $Y\in C$ to the constant functor $I\to C$, i.e. $\Delta(Y)(i) = Y$ and $(i \to j) \mapsto \...
- 1,048
1
vote
1
answer
192
views
Adjunctions b/w constant diagram functor and limit/colimit functors for fixed index category
Let $\mathcal{C}$ be a locally small category and let $\mathcal{J}$ be a small category. Assume that $\mathcal{C}$ has all $\mathcal{J}$-shaped limits and colimits. Describe the unit and counit for ...
