How to show only Post titles on category pages

This little piece of code in your post.php displays the title:

 <h2 class="title">" title="<?php printf( esc_attr__( 'Permalink to %s', 'themater' ), the_title_attribute( 'echo=0' ) ); ?>" rel="bookmark"><?php the_title(); ?></h2>

To be more precise, this bit does:

<?php the_title(); ?>

So if you only want the post titles visible in only archive.php. Just replace the line:

get_template_part('post', 'archive');

with:

the_title();

Or, even better. Copy post.php to post-archive.php & remove everything from post-archive.php you don’t want.

post-archive.php will look like something like this:

<div <?php post_class('post clearfix'); ?> id="post-<?php the_ID(); ?>">
    <h2 class="title">" title="<?php printf( esc_attr__( 'Permalink to %s', 'themater' ), the_title_attribute( 'echo=0' ) ); ?>" rel="bookmark"><?php the_title(); ?></h2>

</div><!-- Post ID <?php the_ID(); ?> -->