Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Lascar groups and the 1st homology groups in model theory   Let $p$ be a strong type of an algebraically closed tuple over
$B=\acl^{\eq}(B)$ in any theory $T$. Depending on a ternary relation $\indo^*$
satisfying some basic axioms (there is at least one such, namely the trivial
independence in $T$), the first homology group $H^*_1(p)$ can be introduced,
similarly to \cite{GKK1}. We show that there is a canonical surjective
homomorphism from the Lascar group over $B$ to $H^*_1(p)$. We also notice that
the map factors naturally via a surjection from the `relativised' Lascar group
of the type (which we define in analogy with the Lascar group of the theory)
onto the homology group, and we give an explicit description of its kernel. Due
to this characterization, it follows that the first homology group of $p$ is
independent from the choice of $\indo^*$, and can be written simply as
$H_1(p)$. As consequences, in any $T$, we show that $|H_1(p)|\geq 2^{\aleph_0}$
unless $H_1(p)$ is trivial, and we give a criterion for the equality of stp and
Lstp of algebraically closed tuples using the notions of the first homology
group and a relativised Lascar group. We also argue how any abelian connected
compact group can appear as the first homology group of the type of a model.

1
The categories of this paper are: cs
20
The categories of this paper are: acc-phys
16
The categories of this paper are: stat
5
The categories of this paper are: gr-qc
9
The categories of this paper are: nlin
34
The categories of this paper are: chem-ph
13
The categories of this paper are: quant-ph
14
The categories of this paper are: math-ph
30
The categories of this paper are: q-alg
18
The categories of this paper are: econ
32
The categories of this paper are: atom-ph
10
The categories of this paper are: q-bio
33
The categories of this paper are: plasm-ph
27
The categories of this paper are: comp-gas
31
The categories of this paper are: ao-sci
29
The categories of this paper are: funct-an