31

I think that Penelope Maddy's article What Do We Want a Foundation to Do? is a good starting point if you want to read some literature. I don't agree with all of Maddy's conclusions but the terminology that she introduces in this article is exceedingly helpful, as well as the very simple but often overlooked point that the concept of a "foundation of ...

26

Category theory and set theory are complementary to one another, not in competition. I think this 'debate' is a bit of academic controversialising rather than an actual difference. If you've done a bit of category theory, you will realize how important the category of sets is (for Yoneda's lemma, representability, existence of generators, etc).
Even if ...

19

$\require{AMScd}$
A 1974 paper of R. Lagrange, Amalgamation and epimorphisms in $\mathfrak{m}$-complete Boolean algebras (Algebra Universalis 4 (1974), 277–279, DOI link), settled this affirmatively. In the cited paper, Lagrange shows that for any infinite cardinal $\mathfrak{m}$, the category of $\mathfrak{m}$-complete Boolean algebras with $\mathfrak{m}$-...

15

Theorem: Given $A$ a boolean ring/boolean algebra then there is an equivalence of categories between the category of $A$-modules and the category of sheaves of $\mathbb{F}_2$-vector spaces on Spec $A$. The equivalence sends every sheaf $\mathcal{M}$ of $\mathbb{F}_2$-vector space to its space of section, $\Gamma(\mathcal{M})$ which is a module over $\Gamma(\...

14

The answer to the question posed in the title of your post is yes, the tensor product of chain complexes is a Day convolution product. The important thing to note is that, to define a Day convolution monoidal structure on the $\mathcal{V}$-enriched functor category $[\mathcal{C},\mathcal{V}]$ (where $\mathcal{V}$ is a complete and cocomplete symmetric ...

14

Yes, it is. The reason is:
every object of your presheaf category is a colimit of representables;
so, every object is a filtered colimit of objects which are finite colimits of representables;
so, applying the definition of a compact object, you get a split monomorphism from your compact object $X$ to a finite colimit $T$ of representables. To conclude, ...

12

The best reference I can think of for this is MathOverflow.
Contrary to some of the comments made above, foundational issues are today often a concern in mathematics and computer science. Contrasting foundational schemes is an activity not just limited to researchers in metamathematics or in mathematical logic. It occurs in computer science repeatedly as ...

10

I will show that this is not the case for $F=\mathbb{F}_9$. The proof generalize to any $\mathbb{F}_{p^k}$, with $k >1$.
I'm starting from the observation that:
$\mathbb{F}_9 \simeq \mathbb{Z}[i]/(3) \simeq \mathbb{Z}[\sqrt{2}]/(3)$. where I'm using the isomorphism that identifies $i$ and $\sqrt{2}$ to identify $\mathbb{Z}[i]/(3)$ and $\mathbb{Z}[\sqrt{2}...

10

The informal analogue, is simply the notion of topos of sheaves.
If I work in a "ground" topos (whose object I call set), then a "forcing extention" would be just a Grothendieck topos, that is a topos of sheaves on a small site.
If you want to adopt an external point of view and start from an elementary topos $\mathcal{E}$ , then a forcing extension of $\...

8

The general problem of giving a categorical construction of the free category with finite coproducts and products (or "free sum–product category") seems to still be open, though there are several works on special cases of the problem.
Cockett–Santocanale's On the word problem for ΣΠ-categories, and the properties of two-way communication gives a good ...

7

I just realized that the OP links to a YouTube video and to some slides, but the two don't match—they're two different talks by Buzzard.
For completeness, let me therefore mention some results by James Arthur, which are mentioned in the linked slides but not the linked YouTube video.
On page 13 of Abelian Surfaces over totally real fields are ...

7

I believe the answer is yes. Assume, by way of contradiction, that some finitely presented group cannot be so expressed. Then we can choose such a group $G$ where for any generating set of the form $x_1,y_1, x_2,y_2,\ldots, x_n,y_n$ the number of non-permissible relations needed to define $G$ (together with some finite number of permissible relations) is ...

7

Here's a pretty direct way to do this. Choose any presentation by generators $x_1,\ldots,x_n$ and relations $r_1,\ldots,r_m$; say $r_i = z_{i,1} \cdots z_{i,k}$ for $z_{i,1},\ldots,z_{i,k} \in \{x_1^{\pm 1}, \ldots, x_n^{\pm 1}\}$. Firstly, we may assume all $r_i$ have length $k = 3$: the new variables
$$x_{i,j} = z_{i,1} \cdots z_{i,j}$$
for $0 \leq j \leq ...

6

You need more assumptions for this to be true.
Consider the ring
$$A = \begin{bmatrix} \mathbb k & \mathbb k \\ 0 & \mathbb k \end{bmatrix},$$ where $\mathbb k$ is some field, and let $\mathcal A= \operatorname{mod} A$. There is a non-split exact sequence
$$0 \to \begin{bmatrix} \mathbb k \\ 0 \end{bmatrix} \to \begin{bmatrix} \mathbb k \\ \mathbb ...

6

The rings satisfying your condition (for right modules) are the right pure semisimple rings. There are many equivalent conditions. You can find a lot of information in Section 4.5 of the book
Prest, Mike, Purity, spectra and localisation., Encyclopedia of Mathematics and its Applications 121. Cambridge: Cambridge University Press (ISBN 978-0-521-87308-6/hbk)...

6

I think that Aurelien Djament's answer is essentially correct, but I want to nitpick a bit.
If $\mathcal A$ is any locally finitely presentable category and $\mathcal C \subseteq \mathcal A$ is any strong generator of finitely-presentable objects, then every finitely-presentable object $X \in \mathcal A$ lies in the closure of $\mathcal C$ under finite ...

5

If you use the category $C$ to represent chain complexes and you mean day convolution using a functor $C \otimes C\to C$ it is not possible. This boils down to whether you can obtain the totalization functor from bi-complexes to chain complexes, as a left adjoint to restriction for some functor $m: C \otimes C \to C$.
You cannot do this because the left ...

5

Assume $\mathcal D$ is a presentable stable $\infty$-category with a $\mathrm t$-structure (which is accessible and compatible with filtered colimits), and let $\mathcal A$ be its heart, $\mathcal{D(A)}$ its derived $\infty$-category.
Note that under those hypotheses, $\mathcal A$ is Grothendieck abelian (Higher Algebra, 1.3.5.23.).
You get a natural ...

5

Are there any places one has to be careful to not allow large categories?
No. For the purposes of forming the 2-category of algebraic/topological/differentiable stacks, or more generally, some kind of presentable stacks over a large category there are no size issues. Naively, the 2-category of stacks on $S$ is carved out from the presheaf category $[S^{op},\...

4

Linear categories are not an independent subject of study, they usually appear in combination with other types of categories, for example abelian and/or monoidal. For all this stuff see:
P.Etingof, S.Gelaki, D.Nikshych, V.Ostrik, Tensor Categories (2015)
Note, that the concept of a linear category is a special case of that of enriched category. It is ...

4

I had reason to think about this a few years ago. When $\mathcal D$ arises as the derived category of an abelian category (with a possibly exotic $t$-structure), a construction of a realization functor $D^b(A) \to \mathcal D$ can be found already in Beilinson-Bernstein-Deligne-Gabber. (For them $\mathcal D$ is the derived category of constructible sheaves, ...

4

For your first question:
They are essentially all the same thing: some globular, some simplicial (taking the nerve goes from the former to the latter). The only subtlety is perhaps in the requirement on the maps $d_{2,0}, d_{2,2}$ to be surjective submersions in the del Hoyo–Stefani paper. This is not unusual for the simplicial approach to n-groupoids in ...

4

You can always reduce an arbitrary "zigzag" as in the first definition, to one of length two, as in the second definition, by applying the following trick:
Whenever you encounter morphisms $E_{j-1}\to E_j\to E_{j+1}$ or $E_{j-1}\leftarrow E_j\leftarrow E_{j+1}$ that go in the same direction, take their composite.
Whenever you encounter morphisms of the form ...

4

One way to write the universal property of this object $E(a)$ is as follows:
a map $x \to E(a)$ is the same as an isomorphism class of epimorphism $x \times a \twoheadrightarrow k$ in $Set/x$, that is a diagram
$$ x \times a \twoheadrightarrow k \to x$$
whose composite is the first projection.
So it does "feel like" a subobject classifier, but it is not ...

3

Given $C$ a small category (eventually, a small simplicial category) I denote by $UC$ the projective model structure on the category of simplicial presheaves on $C$ as in the paper. Using the kind of argument you have in mind we obtain the following theorem:
Theorem: If $M$ is a simplicial model category, then there is an equivalence of categories between:
...

3

The inclusion of groupoids into simplicial sets is fully faithful. Its left adjoint, $\Pi_1$ is given by left Kan extension of the functor $\Delta\to \mathcal{Gpd}$ sending the n-simplex to the contractible groupoid with objects $\{0,...,n\}$.
The entirety of the data of the homotopy type of the space $X$ is contained in its singular simplicial set, which ...

3

Let $\mathcal C$ be the category of finite dimensional left modules over a finite dimensional ring $R$. Let $G: \mathcal C \to \mathrm{Vec}$ be an exact and faithful functor to finite dimensional vector spaces. We use $V^*$ to denote the dual vector space. For motivation, notice that if we had a representing object $M$, we would have $$G(R^*) = Hom_R(M,R^*...

3

You should read Section 4.5 of Olsson's book Algebraic Spaces and Stacks.
The notion of a site is a piece of category theory with no intrinsic geometry, so it doesn't really make sense to ask for a geometric description of torsors for a general site.
However, in the concrete geometric contexts where site theory is typically applied, you can generalize ...

3

It's just a Kan fibration with all fibres principal homogeneous $G$-spaces. Take an $\infty$-category $C$ and a functor $C\to BG,$ where BG is the classifying groupoid of an $\infty$-group (a grouplike $E_1$-space). Pulling back the overcategory projection $EG=BG_{/\ast}\to BG,$ where $\ast$ is the unique object of $BG$, gets you the Kan fibration you ...

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