<p>Yes, you can make a fully convolutional classifier, one example is <a href="https://arxiv.org/abs/1602.07360" rel="nofollow noreferrer">SqueezeNet</a>.</p>

<p>The basic working principle is that at the end of the network you insert a convolutional layer with C output channels, where C is the number of classes. Then you proceed to apply global average pooling, which will produce a 1D vector of C elements (independent of input feature map width/height), and you can apply the softmax function to that vector to produce output class probabilities.</p>
Are fully connected layers really required in Deep neural networks?
<p>I mean to ask that can I have a neural network classifier with a large number of layers without fully connected layers?</p>