Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

A $\Delta$-set is a contravariant functor from the category $\Delta'$ of order-preserving injections to the category of sets (this is essentially what Allen Hatcher calls a $\Delta$-complex).

A main reason for working with simplicial sets instead of $\Delta$-sets should be that they allow quotients (see e.g. Allen Hatcher's nice appendix "CW complexes with simplicial structure" to his Algebraic Topology book: "A major disadvantage of $\Delta$-complexes is that they do not allow quotient constructions"), How does this go well with the fact that the category of functors $\Delta'op\to Sets$ has colimits?

(This question was already asked in a comment on Allen Hatcher's answer to this question on the definition of simplicial complexes. I apologize for asking it twice but there has been no answer given and I am afraid that the reason is - if it's not the silliness of my question - that the comment appears only after pressing the "more comments" button. However, I apologize.)

share|improve this question

1 Answer 1

up vote 5 down vote accepted

The basic issue is that not every function that we would like to describe between $\Delta$-complexes can be realized by a natural transformation between functors. The lack of degeneracy maps means that no map $X \to Y$ of $\Delta$-complexes that sends any simplex down to a degenerate simplex can be realized by a natural transformation of functors. For example, if $X$ is a $\Delta$-complex interval realizing $[0,1]$ and $Y$ is a $\Delta$-complex realizing $[0,1]^2$, then there is no natural transformation of functors realizing the projection maps $p_i:[0,1]^2 \to [0,1]$.

As a consequence, the category of $\Delta$-complexes does not have enough immediately-available maps between objects to construct the kinds of colimit diagrams one would like to realize.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.