0
votes
2answers
27 views
Why every complex of injectives is homotopically injective (provided that, the injective dimension is finite)?
Let $\scr A$ be an abelian category with exact products and a cogenerator (e.g. $\scr A$ is a category of modules). Let ${\mathbf K}(\scr A)$ be the homotopy category of cochain c …
2
votes
0answers
100 views
Monoidal category
Let $(M,\otimes)$ be a small symmetric monoidal category.
Is it possible to choose in each isomorphy class $[A]$ a representative $A_0$ and for each $A\in M$ an isomorphism $\phi_A …
11
votes
1answer
249 views
Lawvere’s fixed point theorem and the Recursion Theorem
Building off of Qiaochu's comment on my answer to a previous mathoverflow question, I would like to know: can the Recursion Theorem, $$\forall e\exists k[\Phi_e\text{ is total }\im …
1
vote
0answers
113 views
How do I find abelian subcategories of periodic triangulated categories?
If $T$ is a triangulated category, then the formalism of $t$-structures gives a way to find abelian subcategories inside. You're supposed to find two strictly full subcategories, …
2
votes
1answer
111 views
Composition in the category quotient
I would like to understand the accounts of P. Gabriel (link text), pag 365, when he shows that the composition of this category is well defined.
Definition: Given a Serre subcateg …
2
votes
1answer
93 views
Directed colimits of maps in a combinatorial model category
I have the following situation. $M$ is a combinatorial model category, or if you like a locally presentable $(\infty,1)$-category. I have a set of maps $S$ and I let $C$ be the cla …
1
vote
1answer
177 views
Drect limit of sequences
Let $\mathcal{C}$ is a grothendiect category and consider all of what follows in $\mathcal{C}$.
Let $${\varepsilon_i: 0\to A_i \to B_i \to C_i\to 0\ ,\ \phi_i^j}$$ be a direct sy …
0
votes
1answer
126 views
Groupoids vs. action groupoids
Let $A\rightrightarrows X$ be a groupoid, where $X$ is the set of objects and $A$ is the set of arrows.
My favorite example of a groupoid is an action groupoid. If a group $G$ acts …
3
votes
1answer
144 views
Reference request: sheaves on closed sets
I am faced with a context in which the most natural notion of a sheaf $\mathcal F$ is as a functor on the category of compact subsets of a (locally compact Hausdorff) space $X$. S …
6
votes
1answer
247 views
Grothendieck fibrations and classifying spaces
Suppose that $$F:\mathcal{D} \to \mathcal{C}$$ is a Grothendieck fibration of small categories, whose fibers are groupoids. Is there anything sensible which can be said about the i …
7
votes
4answers
483 views
On the large cardinals foundations of categories
(This question was posted on math.SE over two weeks ago, but received no answer. I am therefore posting it here as well.)
It is well-known that there are difficulties in developin …
1
vote
1answer
147 views
What structure has been found for functions with this relationship.
Given $f$ and $g$
$\forall x y. f(x) = f(y) \Longrightarrow f(g(x)) = f(g(y))$
Or equivalently
$ker\ f \subseteq ker\ (f \circ g)$.
Note: if $f$ is injective then this holds fo …
2
votes
1answer
94 views
Equivalence and weak equivalence of groupoids
Let $A\rightrightarrows X$ be a groupoid, where $X$ is the set of objects and $A$ is the set of arrows.
My favorite example of a groupoid is an action groupoid. If a group $G$ act …
7
votes
1answer
127 views
Reference for “lax monoidal functors” = “monoids under Day convolution”
Suppose $A$ and $C$ two symmetric monoidal categories. Let's say that $A$ is small and $C$ is locally presentable, and let's assume also that the tensor product on $C$ preserves co …
7
votes
0answers
198 views
Can a composition with itself of a universal self-map be non-universal?
I have formulated (and published) the notion of a universal map (and of a universal morphism), and the problems below, in the early 1960-ies.
DEFINITION A continuous map &n …

