My past research interests included differential algebraic equations, nonlinear analysis and relations between symmetries and structural properties.

I recently investigated hierarchical structures, starting from group cohomology, continuing with semi-group theory and ending with lattices and universal algebra.

1d
answered Which constructions on a category are still interesting for a groupoid?
1d
answered Is a homomorphism expected to be a (structure-preserving) map?
2d
comment Why is there apparently no general notion of structure-homomorphism?
So the first "of course" should be translated as "because there are no relational symbols for algebraic structures". I got confused, because this weak homomorphism notion is the appropriate notion of morphism for universal Horn structures, and in that case there can be relational symbols.
2d
comment Why is there apparently no general notion of structure-homomorphism?
Wow, so many "of course" and "elementary" (not just in your answer). Only missing "the right definition of morphism in model theory is that of an elementary function", which you wrote in a comment to another question. Do you mean something like "element wise" when you write "elementary" here, or does this refer to the non-logical symbols of the first order language, i.e. analogous to how it is used in "elementarily equivalent structures".
2d
comment Is a homomorphism expected to be a (structure-preserving) map?
@tomasz My basic problem is that no single definition of homomorphism for first-order structures seems to fit for all "applications", contrary to the definition of isomorphism: Structure (mathematical logic). So I wondered whether I'm at least allowed to assume that a homomorphism will always be a mapping.
Apr
16
comment Which constructions on a category are still interesting for a groupoid?
@magma The 5th German edition (2007) of "Einführung in die mathematische Logik" by Ebbinghaus et. al. Don't confuse this with the second edition (1996) of the English translation of that book. The amazon reviews of the 5th edition all have 5 stars, the 4th edition (1996) is rather at 3 stars, and the English edition is even worse than the 4th German edition. The most significant difference between the 4th and 5th edition are complete solutions for all excercises.
Apr
16
accepted Which constructions on a category are still interesting for a groupoid?
Apr
16
revised Is a homomorphism expected to be a (structure-preserving) map?
edited tags
Apr
16
comment Is a homomorphism expected to be a (structure-preserving) map?
@Christoph Already the information that this is a soft-question would be part of an answer. But I agree that I should add a terminology tag (if there exists one).
Apr
16
comment Is a homomorphism expected to be a (structure-preserving) map?
@Christoph Well, let's say the "appropriate" use of the word homomorphism, instead of "conventional". It certainly implies "structure-preserving" (even so this is not a well defined notion), I just wonder whether it also implies "map".
Apr
16
asked Which constructions on a category are still interesting for a groupoid?
Apr
16
revised Is a homomorphism expected to be a (structure-preserving) map?
make it more clear, what I want to know
Apr
16
asked Is a homomorphism expected to be a (structure-preserving) map?
Apr
15
comment Why use languages in Complexity theory
@DavidRicherby The point I wanted to bring across is that nailing down the appropriate morphisms is more challenging than nailing down the appropriate objects (=languages). (Especially since there is normally more than one appropriate notion of morphisms.) Without morphisms, you can't talk about isomorphic problems (or algorithms). However, languages give you a way to still talk about equivalence of problems. Perhaps I didn't explain this properly, but (for me) this is a good reason for "using languages in complexity theory".
Apr
15
answered Why use languages in Complexity theory
Apr
14
comment Has there been any more progress on P vs. PSPACE compared to P vs. NP?
As a note of caution, we have the following theorem: "There exists $A \subseteq \{0, 1\}^∗$, such that $\mathsf{PH}^A \neq \mathsf{PSPACE}^A$. More generally, for each $k$ there exists an oracle, relative to which the polynomial hierarchy has exactly $k$ levels."
Apr
13
comment Has there been any more progress on P vs. PSPACE compared to P vs. NP?
I know this is a problematic answer in certain aspects, but it should explain why we don't really expect $\mathsf{P}\neq\mathsf{PSPACE}$ to be much easier to prove than $\mathsf{P}\neq\mathsf{NP}$.
Apr
13
answered Has there been any more progress on P vs. PSPACE compared to P vs. NP?
Apr
12
comment How do we know that P != LINSPACE without knowing if one is a subset of the other?
@ScottAaronson ping (see also the two other answers)
Apr
11
revised Is the categorical product for projective spaces essentially the tensor product?
removed suggestion that categorical product can't be embedded into smaller dimension, because it is false
1 2 3 4 5